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\topmatter 
\title \fourteenpoint\bf HYPQ
\endtitle 
\author C.~Krattenthaler
\endauthor 
\affil 
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.\\
e-mail: KRATT\@AP.UNIVIE.AC.AT\\
WWW: \tt http://www.mat.univie.ac.at/People/kratt
\endaffil 
%\thanks{}\endthanks
\endtopmatter
\leftheadtext{The MATHEMATICA package \tenpoint\bf HYPQ}
\rightheadtext{The MATHEMATICA package \tenpoint\bf HYPQ}
\document


This is a MATHEMATICA package for handling basic hypergeometric
series. It provides quite a few tools for
\roster
\item "(A)" manipulating $q$-factorial expressions
\item "(B)" transforming $q$-binomial sums into basic hypergeometric
notation
\item "(C)" summing basic hypergeometric series
\item "(D)" transforming basic hypergeometric series
\item "(E)" applying contiguous relations
\item "(F)" doing formal limits of basic hypergeometric
expressions
\item "(G)" transforming basic hypergeometric MATHEMATICA expressions
into \TeX-code.
\endroster
The tools for items (A), (B), (F), (G) are contained in the file \hbox{\tt
hyp.q}, the basic package. This file must be loaded at the very
beginning of your MATHEMATICA session. (Ignore error messages
occuring when loading \hbox{\tt hyp.q}.) The file \hbox{\tt hyp.q}
 defines the basic
objects, the rules and functions for items (A), (B), (F), and (G), and
predefines all the remaining ones. The tools for (C) are the contents
of the file \hbox{\tt summatio.q}, those for (D) are the contents of the
files \hbox{\tt transfor.q} and \hbox{\tt transfor.qli}, 
and those for (E) are the contents of the
file \hbox{\tt contig.q}. You also have access to summation and
transformation formulas in form of equations. This is the contents of
the files \hbox{\tt summatio.qgl} and \hbox{\tt transfor.qgl}, respectively.
The file \hbox{\tt output} defines some nice MATHEMATICA output features
for \hbox{\tt SUM}, \hbox{\tt Product}, \hbox{\tt Integrate},
\hbox{\tt Abs}, \hbox{\tt Floor}, \hbox{\tt Ceiling}, \hbox{\tt Pi},
and \hbox{\tt Infinity}. 

However, the philosophy of this package is:
\bigskip
\centerline{\twelvepoint\it Do it by yourself!}
\bigskip
The idea is that you should be able to control each
step in a series of manipulations by yourself.
So, for instance, this package does not make any attempt to sum or
transform a series automatically. So, it is you who has to tell the
package which command, summation, or transformation has to be applied
next. Therefore a basic knowledge of basic hypergeometric series is
required (cf\. \cite{\GaRaAA, pp.~1--6}). This handbook provides you
with a list of the rules, functions, summations, transformations that
are available. The main source for identities that are included in
this package has been the 
book \cite{\GaRaAA}, which contains a fairly comprehensive collection
of known summation and transformation formulas for basic
hypergeometric series. In particular, the (almost) complete Appendix 
of \cite{\GaRaAA} is included in this package. 

Finally you should be warned that there is no guarantee that a
formula that has been obtained using this package is actually valid.
Many formulas or operations are only valid under certain restrictions
for the parameters. This package only helps you
to do calculations fast. It is up to you to check that the manipulations
you are doing are actually being allowed. 

For a brief summary of the main features of this package the user is
referred to \cite{\KratAT} which is the contents of the \AmSTeX\ file
\hbox{\tt hyp$_{\text {--}}$hypq.tex}.

\head Basic hypergeometric notation\endhead
All the notation and terminology is adopted from \cite{\GaRaAA, pp.~1--6}.
Given a (fixed) complex number $q$ with $|q|<1$,
the {\it basic hypergeometric series\/} is defined by
$${}_r\phi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q,
z\right]=\sum _{n=0} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}}
{\poq{q}{n}\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom
n2}\right)^{s-r+1}z^n\ ,$$
where the rising $q$-factorial $(a;q)_n$ is given by 
$(a;q)_n:=(1-a)(1-aq)\cdots(1-aq^{n-1})$, $n\ge1$, $(a)_0:=1$. Also the
infinite $q$-factorial $(a;q)_\infty:=\prod _{i=0} ^{\infty}(1-aq^{i})$
is used.
A basic hypergeometric series $_{r+1}\phi_r$ is called {\it very
well-poised\/} if $a_ib_i=qa_0$ for $i=1,2,\dots,r$,
and among the parameters $a_i$ occur both $q\sqrt{a_0}$ and
$-q\sqrt{a_0}$. We
use the standard abbreviation for very well-poised basic hypergeometric
series,
$$_{r+1}W_r(a_0;a_3,a_4,\dots,a_r;z):={}_{r+1}\phi_r\!\[\matrix
a_0,q\sqrt{a_0},-q\sqrt{a_0},a_3,a_4,\dots,a_r\\
\sqrt{a_0},-\sqrt{a_0},qa_0/a_2,qa_0/a_3,\dots,qa_0/a_r\endmatrix; z\].$$
A convenient notation for {\it basic hypergeometric series\/} with
several bases is
$$\multline 
{}_r\Phi_s\!\left[\matrix a^{(1)}_1,\dots,a^{(1)}_{r_1};
\dots ;a^{(p)}_1,\dots,a^{(p)}_{r_p}\\
 b^{(1)}_1,\dots,b^{(1)}_{s_1};\dots;
 b^{(p)}_1,\dots,b^{(p)}_{s_p}\endmatrix; q_1,\dots,q_p;
z\right]\\
=\sum _{n=0} ^{\infty}\frac {z^n} {(q_1;q_1)_n}(-1)^nq_1^{\binom n2}
\prod _{i=1} ^{p}
\frac {(a^{(i)}_1;q_i)_n\cdots(a^{(i)}_{r_i};q_i)_n}
{(b^{(i)}_1;q_i)_n\cdots(b^{(i)}_{s_i};q_i)_n}
\left((-1)^nq^{\binom
n2}\right)^{s_i-r_i}\ .
\endmultline$$
The {\it bilateral basic hypergeometric series\/} is defined by
$${}_r\psi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q,
z\right]=\sum _{n=-\infty} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}}
{\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom
n2}\right)^{s-r+1}z^n\ ,$$
We also use the compact
Gasper-Rahman notation
$$ {(a_1,a_2,\dots,a_r;q)_n}
:=(a_1;q)_n\,(a_2;q)_n\,\cdots\,(a_r;q)_n.$$


\head The file \tt hyp.q\endhead
The objects which are defined in the file hyp.q are
\medskip
{\leftskip20pt \rightskip20pt \noindent
\hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller}, \hbox{\tt
AbsUndetermined},
\hbox{\tt Add}, \hbox{\tt AmSLaTeX}, \hbox{\tt AmSTeX}, 
\hbox{\tt baszerl1}, \hbox{\tt baszerl2}, \hbox{\tt
baszus1},\linebreak 
\hbox{\tt baszus2}, \hbox{\tt Binomialpq}, \hbox{\tt
Binomialq}, \hbox{\tt Div}, \hbox{\tt Drucke},
\hbox{\tt Ers}, \hbox{\tt erw1}, \hbox{\tt erw2}, \hbox{\tt Expandq},
\hbox{\tt Factorialpq}, \hbox{\tt Factorialq}, 
\hbox{\tt Gleichung}, \hbox{\tt GlTausche}, 
\hbox{\tt Hoch}, \hbox{\tt
hypqAttributes}, \hbox{\tt inv1}, \hbox{\tt inv2}, \hbox{\tt LaTeX}, 
\hbox{\tt Limes}, \hbox{\tt lina1},
\hbox{\tt lina2}, \hbox{\tt linz},
\hbox{\tt LS}, \hbox{\tt Mal}, \hbox{\tt ManipulationsListe},
\hbox{\tt MinusOne}, \hbox{\tt Multinomialpq}, \hbox{\tt Multinomialq},
\hbox{\tt neg1}, \hbox{\tt neg2}, \hbox{\tt ph}, \hbox{\tt Ph}, 
\hbox{\tt phCancel}, \hbox{\tt PhEinf}, 
\hbox{\tt phEinf}, \hbox{\tt phFormat}, \hbox{\tt Phinv}, \hbox{\tt phinv},
\hbox{\tt PhOrdne}, \hbox{\tt phOrdne}, \hbox{\tt PhPerm}, \hbox{\tt phPerm}, 
\hbox{\tt Phph}, \hbox{\tt phPh}, \hbox{\tt phps}, \hbox{\tt PhSUM}, 
\hbox{\tt phSUM},
\hbox{\tt phTausche}, \hbox{\tt PosListe}, \hbox{\tt pq}, \hbox{\tt
PQ}, \hbox{\tt pqaufl}, \hbox{\tt pqinf},
\hbox{\tt pqinfzerl}, \hbox{\tt pqinfzus}, 
\hbox{\tt PQSort}, \hbox{\tt pqzerl}, \hbox{\tt pqzus}, \hbox{\tt ps}, 
\hbox{\tt psEinf}, \hbox{\tt psinv}, \hbox{\tt psOrdne}, 
\hbox{\tt psPerm}, \hbox{\tt psph}, 
\hbox{\tt psShift}, \hbox{\tt psSUM}, \hbox{\tt RS}, 
\hbox{\tt SchreibeZahl}, 
\hbox{\tt SimplifyPQ}, \hbox{\tt Sub}, \hbox{\tt Subst},
\hbox{\tt SUM}, \hbox{\tt SUMErw1}, \hbox{\tt SUMErw2}, \hbox{\tt SUMExpand}, 
\hbox{\tt SUMInfinity}, \hbox{\tt SUMph}, \hbox{\tt SUMPh}, 
\hbox{\tt SUMps}, \hbox{\tt SUMRegeln},
\hbox{\tt SUMSammle}, \hbox{\tt SUMShift}, \hbox{\tt SUMTausche}, 
\hbox{\tt SUMUmkehr}, \hbox{\tt SUMZerl}, \hbox{\tt TeX}, \hbox{\tt
TeXMat}, \hbox{\tt TeXphW},
\hbox{\tt trans}, \hbox{\tt W}, \hbox{\tt zerl}, \hbox{\tt zus1},
\hbox{\tt zus2}, \hbox{\tt zus3}.  \par}
\medskip\noindent
These objects can be divided into 9 groups: There are the basic
objects,
\medskip\noindent
{\leftskip20pt\rightskip20pt\noindent
\hbox{\tt Binomialpq}, \hbox{\tt Binomialq}, \hbox{\tt Factorialpq},
\hbox{\tt Factorialq}, \hbox{\tt Multinomialpq}, \hbox{\tt Multinomialq},
\hbox{\tt ph}, \hbox{\tt Ph}, 
\hbox{\tt pq}, \hbox{\tt pqinf}, \hbox{\tt ps}, 
\hbox{\tt SUM}, \hbox{\tt W},\par}
\medskip\noindent
the rules for manipulating $q$-factorial expressions
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt baszerl1}, \hbox{\tt baszerl2}, \hbox{\tt baszus1},
\hbox{\tt baszus2}, 
\hbox{\tt erw1}, \hbox{\tt erw2}, \hbox{\tt Expandq}, 
\hbox{\tt inv1}, \hbox{\tt inv2},
\hbox{\tt lina1}, \hbox{\tt lina2}, \hbox{\tt linz}, \hbox{\tt MinusOne},
\hbox{\tt neg1}, \hbox{\tt neg2}, \hbox{\tt pqaufl}, 
\hbox{\tt pqinfzerl}, \hbox{\tt pqinfzus}, 
\hbox{\tt pqzerl}, \hbox{\tt pqzus}, 
\hbox{\tt trans}, \hbox{\tt zerl}, \hbox{\tt zus1}, \hbox{\tt zus2},
\hbox{\tt zus3},  \par}
\medskip\noindent
the rules for manipulating sums and basic hypergeometric series,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt phCancel}, \hbox{\tt phEinf}, \hbox{\tt PhEinf}, 
\hbox{\tt phFormat}, \hbox{\tt Phinv}, \hbox{\tt phinv},
\hbox{\tt PhOrdne}, \hbox{\tt phOrdne}, \hbox{\tt PhPerm}, 
\hbox{\tt phPerm}, \hbox{\tt Phph}, \hbox{\tt phPh}, 
\hbox{\tt phps}, \hbox{\tt PhSUM}, 
\hbox{\tt phSUM}, \hbox{\tt phTausche}, \hbox{\tt ps}, 
\hbox{\tt psEinf}, \hbox{\tt psinv}, \hbox{\tt psOrdne}, 
\hbox{\tt psPerm}, \hbox{\tt psph}, 
\hbox{\tt psShift}, \hbox{\tt psSUM},
\hbox{\tt SUMErw1}, \hbox{\tt SUMErw2}, \hbox{\tt SUMExpand}, 
\hbox{\tt SUMInfinity}, 
\hbox{\tt SUMph}, \hbox{\tt SUMPh}, \hbox{\tt SUMps}, \hbox{\tt SUMRegeln},
\hbox{\tt SUMSammle}, \hbox{\tt SUMShift}, \hbox{\tt SUMTausche}, 
\hbox{\tt SUMUmkehr}, \hbox{\tt SUMZerl}, \par}
\medskip\noindent
two functions for controlled use of rules,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Ers}, \hbox{\tt Posliste},  \par}
\medskip\noindent
one function for substitution of an expression instead of another
expression,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Subst}, \par}
\medskip\noindent
some objects for doing limits of basic hypergeometric expressions,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller}, \hbox{\tt
AbsUndetermined}, \hbox{\tt Limes}, \par}
\medskip\noindent
one object for simplifying arguments in basic hypergeometric expressions,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt SimplifyPQ},  \par}
\medskip\noindent
some objects for converting expressions into \TeX-code,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt AmSLaTeX}, \hbox{\tt AmSTeX}, \hbox{\tt LaTeX}, \hbox{\tt TeX}, 
\hbox{\tt TeXMat}, \hbox{\tt TeXphW}, \par}
\medskip\noindent
two objects for on-line help,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt hypqAttributes}, \hbox{\tt ManipulationsListe}, \par}
\medskip\noindent
and the function 
\medskip\noindent
{\leftskip20pt\rightskip20pt
\tt Drucke \par} 
\medskip\noindent
which enables you to directly send an expression to the printer.
Finally there are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Add}, \hbox{\tt Div}, \hbox{\tt Gleichung}, \hbox{\tt
GlTausche}, \hbox{\tt Hoch},
\hbox{\tt LS}, \hbox{\tt Mal}, \hbox{\tt PQSort}, \hbox{\tt RS}, \hbox{\tt Sub}\par}
\medskip\noindent
for manipulating equations and writing expressions in a ``normalized"
form (\hbox{\tt PQSort}) in order to be able to quickly check if two
expressions agree. These objects are particularly important when
using objects from \hbox{\tt summatio.qgl} and\linebreak \hbox{\tt
transfor.qgl}.

\bigskip
Most of the tools for manipulating expressions that are provided by
this package are rules. This has the advantage that very often you do
not have to specify to which part of an expression you want to apply
a rule, since there is just one subexpression to which the
rule applies. However, if there are more subexpressions to which a
rule applies, you will sometimes want to apply the rule only to
some of the subexpressions. To handle this conveniently, there are
the functions \hbox{\tt Ers} and \hbox{\tt PosListe}. 


\head The file \tt summatio.q\endhead
This file contains most of the summation formulas of the book \cite{\GaRaAA},
including the (almost) complete Appendix~II of \cite{\GaRaAA}, in form of
rules. 
You do not have to load this file by hand since it is loaded
automatically once an object of this file is called.
The objects that are defined by \hbox{\tt summatio.q} are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt S0110}, \hbox{\tt S1001}, \hbox{\tt S1101}, \hbox{\tt S1102}, 
\hbox{\tt S1110},
\hbox{\tt S2101}, \hbox{\tt S2102},
\hbox{\tt S2103}, \hbox{\tt S2104}, \hbox{\tt S2105}, \hbox{\tt S2106}, 
\hbox{\tt S2107}, \hbox{\tt S2161}, 
\hbox{\tt S2201}, \hbox{\tt S2202}, \hbox{\tt S2210},
\hbox{\tt S3201}, \hbox{\tt S3202}, \hbox{\tt S3203}, \hbox{\tt S3204}, 
\hbox{\tt S3261}, \hbox{\tt S3310}, \hbox{\tt S4301}, \hbox{\tt S4302},
\hbox{\tt S4303}, \hbox{\tt S4304}, \hbox{\tt S4305}, \hbox{\tt S4306}, 
\hbox{\tt S4307}, \hbox{\tt S4308}, \hbox{\tt S4361}, 
\hbox{\tt S4410}, \hbox{\tt S5401}, 
\hbox{\tt S5402}, \hbox{\tt S5501}, 
\hbox{\tt S6501}, \hbox{\tt S6502}, \hbox{\tt S6610}, 
\hbox{\tt S8701}, \hbox{\tt S8702}, 
\hbox{\tt S8703}, \hbox{\tt S8704}, \hbox{\tt S8761}, \hbox{\tt S10901}, 
\hbox{\tt SListe}, \hbox{\tt SumListe}.  \par}
\medskip\noindent
The numbering of each rule is 
{\tt S$\langle$d1$\rangle$$\langle$d2$\rangle
$$\langle$n1$\rangle$}
following the following system: The number
{\tt $\langle$d1$\rangle$} 
is the number of the upper parameters, the number
{\tt $\langle$d2$\rangle$} is the number of the lower parameters of the basic hypergeometric
series to which the rule applies. The number $\langle$n1$\rangle$ allows to
distinguish the rules applying to basic hypergeometric series with
equal numbers of upper and lower parameters. 
$\langle$n1$\rangle$ is within the
range 01--30 if the summation is a one-term summation, it is within
the range 61--90 if the summation is a two- or more-term summation.

For terminating series there is a check if one of the parameters is
of the form $q^{-n}$ where $n$ is a nonnegative integer. Depending on
your input you might be asked if some expression is a nonnegative
integer (see the examples for \hbox{\tt S3201}). 
Be sure to give an
affirmative answer only for {\it one} of several expressions, otherwise the
package will try to find the minimum of all of these, which might cause
problems. This remark also applies to other rules which put this
question, e.g\. \hbox{\tt SUMUmkehr}, \hbox{\tt phSUM}, or in case
that automatic evaluating is active (cf\. \hbox{\tt PQ}). The rule
\hbox{\tt SListe} enables you to quickly check if one of the
summation rules \hbox{\tt S0110}--\hbox{\tt S8761} can be directly applied.


\head The file \tt transfor.q\endhead
This file contains most of the transformation formulas of the book
\cite{\GaRaAA}, including the (almost) complete Appendix~III of 
\cite{\GaRaAA}, in form of rules. 
You do not have to load this file by hand since it is loaded
automatically once an object of this file is called.
The objects that are defined by \hbox{\tt transfor.q} are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt T2101}, \hbox{\tt T2102}, \hbox{\tt T2103}, \hbox{\tt T2104},
\hbox{\tt T2105}, \hbox{\tt T2106}, \hbox{\tt T2107}, \hbox{\tt T2108},
\hbox{\tt T2109}, \hbox{\tt T2110}, \hbox{\tt T2111}, \hbox{\tt T2112},
\hbox{\tt T2161},
\hbox{\tt T2162}, \hbox{\tt T2163}, 
\hbox{\tt T2201}, \hbox{\tt T2202}, \hbox{\tt T3101}, 
\hbox{\tt T3201}, \hbox{\tt T3202},
\hbox{\tt T3203}, \hbox{\tt T3204}, \hbox{\tt T3205},
\hbox{\tt T3206}, \hbox{\tt T3207}, \hbox{\tt T3208}, \hbox{\tt T3209},
\hbox{\tt T3210}, \hbox{\tt T3211}, \hbox{\tt T3212}, \hbox{\tt T3213},
\hbox{\tt T3214}, \hbox{\tt T3215}, \hbox{\tt T3216}, \hbox{\tt T3217}, 
\hbox{\tt T3261}, \hbox{\tt T3262}, \hbox{\tt T3263},
\hbox{\tt T3264}, \hbox{\tt T3265},
\hbox{\tt T3266}, \hbox{\tt T3267}, \hbox{\tt T3268},
\hbox{\tt T3269}, 
\hbox{\tt T4301}, \hbox{\tt T4302}, \hbox{\tt T4303}, \hbox{\tt T4304},
\hbox{\tt T4305}, \hbox{\tt T4306}, \hbox{\tt T4307}, 
\hbox{\tt T4308}, \hbox{\tt T4309}, \hbox{\tt T4310}, 
\hbox{\tt T4311}, \hbox{\tt T4312}, \hbox{\tt T4313}, 
\hbox{\tt T4361}, \hbox{\tt T4362}, 
\hbox{\tt T5401}, \hbox{\tt T5402}, \hbox{\tt T5403}, \hbox{\tt T5404},
\hbox{\tt T5405}, \hbox{\tt T5461}, \hbox{\tt T5462}, \hbox{\tt T5463}, 
\hbox{\tt T5464}, \hbox{\tt T5465}, \hbox{\tt T5466}, \hbox{\tt T5467}, 
\hbox{\tt T5468}, \hbox{\tt T5469}, 
\hbox{\tt T6501}, 
\hbox{\tt T7601}, \hbox{\tt T7701}, 
\hbox{\tt T8701}, \hbox{\tt T8702},
\hbox{\tt T8703}, \hbox{\tt T8704}, \hbox{\tt T8705}, 
\hbox{\tt T8706}, \hbox{\tt T8707}, \hbox{\tt T8708}, 
\hbox{\tt T8709}, \hbox{\tt T8710}, \hbox{\tt T8711}, 
\hbox{\tt T8761}, \hbox{\tt T8762}, \hbox{\tt T8763}, \hbox{\tt T8764}, 
\hbox{\tt T8810},
\hbox{\tt T10901}, \hbox{\tt T10902}, \hbox{\tt T10903}, \hbox{\tt T10904}, 
\hbox{\tt T10905}, \hbox{\tt T10906}, \hbox{\tt T10907}, 
\hbox{\tt T10961}, \hbox{\tt T10962}, \hbox{\tt T10963}, 
\hbox{\tt T101010},
\hbox{\tt T121101}, \hbox{\tt T121102}, \hbox{\tt T121103}, 
\hbox{\tt T121104}, \hbox{\tt T121105}, \hbox{\tt T121106}, 
\hbox{\tt T121107}, 
\hbox{\tt T121161}, 
\hbox{\tt TListe}, \hbox{\tt TransListe}.  \par}
\medskip\noindent
The comments for the file \hbox{\tt summatio.q} 
regarding the numbering of the rules and optional
questions for input also apply here.
The rule
\hbox{\tt TListe} enables you to quickly check if one of the
transformation rules \hbox{\tt T2101}--\hbox{\tt T10961} can be directly applied.

\head The file \tt transfor.qli\endhead
Each of the objects of this file corresponds to a transformation rule
of the file \hbox{\tt transfor.q}. Each object gives a list of all
the outcomes under application of a particular transformation 
after before having permuted the upper and lower parameters of
the involved basic hypergeometric series. All the objects in this
file are rules. These rules help to prove conjectured transformation
formulas quickly.
You do not have to load this file by hand since it is loaded
automatically once an object of this file is called.
The objects that are defined by \hbox{\tt transfor.qli} are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Tli2101}, \hbox{\tt Tli2102}, \hbox{\tt Tli2103}, \hbox{\tt Tli2104},
\hbox{\tt Tli2105}, \hbox{\tt Tli2106}, \hbox{\tt Tli2107}, \hbox{\tt Tli2108},
\hbox{\tt Tli2109}, \hbox{\tt Tli2110}, \hbox{\tt Tli2111}, \hbox{\tt Tli2112},
\hbox{\tt Tli2161},
\hbox{\tt Tli2162}, \hbox{\tt Tli2163}, 
\hbox{\tt Tli2201}, \hbox{\tt Tli2202}, \hbox{\tt Tli3101}, 
\hbox{\tt Tli3201}, \hbox{\tt Tli3202},
\hbox{\tt Tli3203}, \hbox{\tt Tli3204}, \hbox{\tt Tli3205},
\hbox{\tt Tli3206}, \hbox{\tt Tli3207}, \hbox{\tt Tli3208}, \hbox{\tt Tli3209},
\hbox{\tt Tli3210}, \hbox{\tt Tli3211}, \hbox{\tt Tli3212}, \hbox{\tt Tli3213},
\hbox{\tt Tli3214}, \hbox{\tt Tli3215}, \hbox{\tt Tli3216}, 
\hbox{\tt Tli3261}, \hbox{\tt Tli3262}, \hbox{\tt Tli3263}, 
\hbox{\tt Tli3264}, \hbox{\tt Tli3265},
\hbox{\tt Tli3266}, \hbox{\tt Tli3267}, \hbox{\tt Tli3268},
\hbox{\tt Tli3269}, 
\hbox{\tt Tli4301}, \hbox{\tt Tli4302}, \hbox{\tt Tli4303}, \hbox{\tt Tli4304},
\hbox{\tt Tli4305}, \hbox{\tt Tli4306}, \hbox{\tt Tli4307}, 
\hbox{\tt Tli4308}, \hbox{\tt Tli4309}, \hbox{\tt Tli4310}, 
\hbox{\tt Tli4311}, \hbox{\tt Tli4312}, \hbox{\tt Tli4313}, 
\hbox{\tt Tli4361}, 
\hbox{\tt Tli4362}, 
\hbox{\tt Tli5401}, \hbox{\tt Tli5402}, \hbox{\tt Tli5403}, \hbox{\tt Tli5404},
\hbox{\tt Tli5405}, \hbox{\tt Tli5461}, \hbox{\tt Tli5462}, \hbox{\tt Tli5463}, 
\hbox{\tt Tli5464}, \hbox{\tt Tli5465}, \hbox{\tt Tli5466}, \hbox{\tt Tli5467}, 
\hbox{\tt Tli5468}, \hbox{\tt Tli5469}, 
\hbox{\tt Tli6501}, 
\hbox{\tt Tli7601}, 
\hbox{\tt Tli8701}, \hbox{\tt Tli8702},
\hbox{\tt Tli8703}, \hbox{\tt Tli8704}, \hbox{\tt Tli8705}, 
\hbox{\tt Tli8706}, \hbox{\tt Tli8707}, \hbox{\tt Tli8708}, 
\hbox{\tt Tli8709}, \hbox{\tt Tli8710}, \hbox{\tt Tli8711}, 
\hbox{\tt Tli8761}, \hbox{\tt Tli8762}, \hbox{\tt Tli8763}, 
\hbox{\tt Tli8764}, \hbox{\tt Tli8810},
\hbox{\tt Tli10901}, \hbox{\tt Tli10902}, \hbox{\tt Tli10903}, 
\hbox{\tt Tli10904}, 
\hbox{\tt Tli10905}, \hbox{\tt Tli10906}, \hbox{\tt Tli10907}, 
\hbox{\tt Tli10961}, \hbox{\tt Tli10962}, \hbox{\tt Tli10963}, 
\hbox{\tt Tli101010},
\hbox{\tt Tli121101}, \hbox{\tt Tli121102}, \hbox{\tt Tli121103}, 
\hbox{\tt Tli121104},
\hbox{\tt Tli121105}, \hbox{\tt Tli121106}, \hbox{\tt Tli121107}, 
\hbox{\tt Tli121161}.  \par}



\head The files \tt summatio.qgl \rm and \tt transfor.qgl\endhead
These files contain the same summations, respectively transformations,
as \hbox{\tt summatio.q}, respectively \hbox{\tt transfor.q}, but in form of
equations. You do not have to load these files by hand since they are loaded
automatically once an object of this file is called.
The respective objects are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Sgl0110}, \hbox{\tt Sgl1001}, \hbox{\tt Sgl1101}, 
\hbox{\tt Sgl1102}, 
\hbox{\tt Sgl1110}, \hbox{\tt Sgl2101}, \hbox{\tt Sgl2102},
\hbox{\tt Sgl2103}, \hbox{\tt Sgl2104}, \hbox{\tt Sgl2105}, \hbox{\tt Sgl2106}, 
\hbox{\tt Sgl2107}, \hbox{\tt Sgl2161}, 
\hbox{\tt Sgl2201}, \hbox{\tt Sgl2202}, \hbox{\tt Sgl2210}, 
\hbox{\tt Sgl3201}, \hbox{\tt Sgl3202}, \hbox{\tt Sgl3203}, \hbox{\tt Sgl3204}, 
\hbox{\tt Sgl3261}, \hbox{\tt Sgl3310}, \hbox{\tt Sgl4301}, \hbox{\tt Sgl4302},
\hbox{\tt Sgl4303}, \hbox{\tt Sgl4304}, \hbox{\tt Sgl4305}, \hbox{\tt Sgl4306}, 
\hbox{\tt Sgl4307}, \hbox{\tt Sgl4308}, \hbox{\tt Sgl4361}, 
\hbox{\tt Sgl4410}, \hbox{\tt Sgl5401}, 
\hbox{\tt Sgl5402}, \hbox{\tt Sgl5501}, 
\hbox{\tt Sgl6501}, \hbox{\tt Sgl6502}, \hbox{\tt Sgl6610}, 
\hbox{\tt Sgl8701}, \hbox{\tt Sgl8702}, 
\hbox{\tt Sgl8703}, \hbox{\tt Sgl8704}, \hbox{\tt Sgl8761}, 
\hbox{\tt Sgl10901},\linebreak \hbox{\tt SumListe\$gl},  \par}
\medskip\noindent
and
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Tgl2101}, \hbox{\tt Tgl2102}, \hbox{\tt Tgl2103}, \hbox{\tt Tgl2104},
\hbox{\tt Tgl2105}, \hbox{\tt Tgl2106}, \hbox{\tt Tgl2107}, \hbox{\tt Tgl2108},
\hbox{\tt Tgl2109}, \hbox{\tt Tgl2110}, \hbox{\tt Tgl2111}, \hbox{\tt Tgl2112},
\hbox{\tt Tgl2161},
\hbox{\tt Tgl2162}, \hbox{\tt Tgl2163}, 
\hbox{\tt Tgl2201}, \hbox{\tt Tgl2202}, \hbox{\tt Tgl3101}, 
\hbox{\tt Tgl3201}, \hbox{\tt Tgl3202},
\hbox{\tt Tgl3203}, \hbox{\tt Tgl3204}, \hbox{\tt Tgl3205},
\hbox{\tt Tgl3206}, \hbox{\tt Tgl3207}, \hbox{\tt Tgl3208}, \hbox{\tt Tgl3209},
\hbox{\tt Tgl3210}, \hbox{\tt Tgl3211}, \hbox{\tt Tgl3212}, \hbox{\tt Tgl3213},
\hbox{\tt Tgl3214}, \hbox{\tt Tgl3215}, \hbox{\tt Tgl3216}, 
\hbox{\tt Tgl3261}, \hbox{\tt Tgl3262}, \hbox{\tt Tgl3263}, 
\hbox{\tt Tgl3264}, \hbox{\tt Tgl3265},
\hbox{\tt Tgl3266}, \hbox{\tt Tgl3267}, \hbox{\tt Tgl3268},
\hbox{\tt Tgl3269}, 
\hbox{\tt Tgl4301}, \hbox{\tt Tgl4302}, \hbox{\tt Tgl4303}, \hbox{\tt Tgl4304},
\hbox{\tt Tgl4305}, \hbox{\tt Tgl4306}, \hbox{\tt Tgl4307}, 
\hbox{\tt Tgl4308}, \hbox{\tt Tgl4309}, \hbox{\tt Tgl4310}, 
\hbox{\tt Tgl4311}, \hbox{\tt Tgl4312}, \hbox{\tt Tgl4313}, 
\hbox{\tt Tgl4361}, 
\hbox{\tt Tgl4362}, 
\hbox{\tt Tgl5401}, \hbox{\tt Tgl5402}, \hbox{\tt Tgl5403}, \hbox{\tt Tgl5404},
\hbox{\tt Tgl5405}, \hbox{\tt Tgl5461}, \hbox{\tt Tgl5462}, \hbox{\tt Tgl5463}, 
\hbox{\tt Tgl5464}, \hbox{\tt Tgl5465}, \hbox{\tt Tgl5466}, \hbox{\tt Tgl5467}, 
\hbox{\tt Tgl5468}, \hbox{\tt Tgl5469}, 
\hbox{\tt Tgl6501}, 
\hbox{\tt Tgl7601}, 
\hbox{\tt Tgl8701}, \hbox{\tt Tgl8702},
\hbox{\tt Tgl8703}, \hbox{\tt Tgl8704}, \hbox{\tt Tgl8705}, 
\hbox{\tt Tgl8706}, \hbox{\tt Tgl8707}, \hbox{\tt Tgl8708}, 
\hbox{\tt Tgl8709}, \hbox{\tt Tgl8710}, \hbox{\tt Tgl8711}, 
\hbox{\tt Tgl8761}, \hbox{\tt Tgl8762}, \hbox{\tt Tgl8763}, 
\hbox{\tt Tgl8764}, \hbox{\tt Tgl8810}, 
\hbox{\tt Tgl10901}, \hbox{\tt Tgl10902}, \hbox{\tt Tgl10903}, \hbox{\tt Tgl10904}, 
\hbox{\tt Tgl10905}, \hbox{\tt Tgl10906}, \hbox{\tt Tgl10907}, 
\hbox{\tt Tgl10961}, \hbox{\tt Tgl10962}, \hbox{\tt Tgl10963}, 
\hbox{\tt Tgl101010}, 
\hbox{\tt Tgl121101}, \hbox{\tt Tgl121102}, \hbox{\tt Tgl121103}, 
\hbox{\tt Tgl121104},
\hbox{\tt Tgl121105}, \hbox{\tt Tgl121106}, \hbox{\tt Tgl121107}, 
\hbox{\tt Tgl121161}, 
\hbox{\tt TransListe\$gl}.  \par}
\medskip\noindent
When calling one of these objects you will be put two questions. If
the variables of the called summation or transformation are
undefined, the first question is
\medskip
\MATH
Do you want to set values for the equation? [y|n]: 
\endMATH
\medskip\noindent
Enter \hbox{\tt y} if you want to set values, even only for some of
them, if you do not need to set values enter \hbox{\tt n}. If some
of the variables of the called summation or transformation are
already defined, you will be asked as first question
\medskip
\MATH
Some variables have a value. Should the variables 
$\{$[V,a,r,i,a,b,l,e,s]$\}$ be cleared? Do you want to set       
values for the equation (v)? [y|n|yv|nv]: 
\endMATH
\medskip\noindent
Now you have four options depending on if you want to set values or
not and if you want to clear the already defined variables or not.
For example, if you want to set values but do not want to clear the
defined variables, enter \hbox{\tt nv}. 

The second question concerns the base \hbox{\tt q}. Either it is
\medskip
\MATH
Do you want to set a value for q in the equation? [y|n]: 
\endMATH
\medskip\noindent
or
\medskip\noindent
\MATH
q has a value. Should q be cleared?        
Do you want to set a value for q in the equation (v)? [y|n|yv|nv]: 
\endMATH
\medskip\noindent
The explanations for the first question also apply here.
(Cf\. the examples in \hbox{\tt Sgl2101}). 

In addition there are the functions and variables
\medskip\noindent
{\leftskip20pt\rightskip20pt\tt
Add, Div, Gleichung, GlTausche, Hoch, LS, Mal, RS, Sub,\par}
\medskip\noindent
for manipulating equations. In fact, once you have called 
one of the objects {\tt Sgl$*$} or {\tt Tgl$*$}, the right-hand side of the
displayed equation will have been assigned to {\tt RS}, the left-hand
side to {\tt LS}, and thus the equation itself to \hbox{\tt
Gleichung} (cf\. the example in \hbox{\tt Gleichung}). The functions
{\tt Add, Div, GlTausche, Hoch, Mal, Sub}, and also {\tt Ers},
allow you to manipulate the equation.


\head The file \tt contig.q\endhead
This file contains a vast number of contiguous relations in form of
rules. You do not have to load this file by hand since it is loaded
automatically once an object of this file is called. The objects that
are defined by {\tt contig.q} are 
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt C01},
\hbox{\tt C02},
\hbox{\tt C11},
\hbox{\tt C12},
\hbox{\tt C13},
\hbox{\tt C14},
\hbox{\tt C15},
\hbox{\tt C16},
\hbox{\tt C17},
\hbox{\tt C18},
\hbox{\tt C19},
\hbox{\tt C20},
\hbox{\tt C21},
\hbox{\tt C22},
\hbox{\tt C23},
\hbox{\tt C24},
\hbox{\tt C25},
\hbox{\tt C26},
\hbox{\tt C27},
\hbox{\tt C28},
\hbox{\tt C29},
\hbox{\tt C30},
\hbox{\tt C31},
\hbox{\tt C32},
\hbox{\tt C33},
\hbox{\tt C34},
\hbox{\tt C35},
\hbox{\tt C36},
\hbox{\tt C37},
\hbox{\tt C38},
\hbox{\tt C39},
\hbox{\tt C40},
\hbox{\tt C41},
\hbox{\tt C42},
\hbox{\tt C43},
\hbox{\tt C44},
\hbox{\tt C45},
\hbox{\tt C46},
\hbox{\tt C47},
\hbox{\tt C48},
\hbox{\tt C49},
\hbox{\tt C50},
\hbox{\tt C51},
\hbox{\tt C52},
\hbox{\tt C53},
\hbox{\tt C54},
\hbox{\tt C55},
\hbox{\tt C56},
\hbox{\tt C57},
\hbox{\tt C58},
\hbox{\tt C59},
\hbox{\tt C60},
\hbox{\tt C61},
\hbox{\tt C62},
\hbox{\tt C63},
\hbox{\tt C64},
\hbox{\tt C65},
\hbox{\tt C66},
\hbox{\tt C67},
\hbox{\tt C68},
\hbox{\tt C69},
\hbox{\tt C70},
\hbox{\tt C71},
\hbox{\tt C72},
\hbox{\tt C73},
\hbox{\tt C74},
\hbox{\tt C75},
\hbox{\tt C76},
\hbox{\tt C77},
\hbox{\tt C78},
\hbox{\tt C79},
\hbox{\tt C80},
\hbox{\tt C81},
\hbox{\tt C82},
\hbox{\tt C83},
\hbox{\tt C84},
\hbox{\tt C85},
\hbox{\tt C86},
\hbox{\tt C87},
\hbox{\tt C88},
\hbox{\tt C89},
\hbox{\tt C90},
\hbox{\tt C91},
\hbox{\tt C92},
\hbox{\tt C93},
\hbox{\tt C94},
\hbox{\tt C95},
\hbox{\tt C96},
\hbox{\tt C97},
\hbox{\tt C98},
\hbox{\tt C99},
\hbox{\tt C100},
\hbox{\tt C101},
\hbox{\tt C102},
\hbox{\tt C103},
\hbox{\tt C104},
\hbox{\tt C105},
\hbox{\tt C106},
\hbox{\tt C107},
\hbox{\tt C108},
\hbox{\tt C109},
\hbox{\tt C110},
\hbox{\tt C111},
\hbox{\tt C112},
\hbox{\tt C113},
\hbox{\tt C114},
\hbox{\tt C115},
\hbox{\tt C116},
\hbox{\tt C117},
\hbox{\tt C118},
\hbox{\tt C119},
\hbox{\tt C120},
\hbox{\tt C121},
\hbox{\tt ContigListe}.  \par}
\medskip\noindent


\head Simultaneous use of HYP and HYPQ \endhead

It is possible to load both packages, HYP and HYPQ. In this case, the
objects of the package that is loaded last will override the
respective
objects of the other package which have identical names. However, you
can use the overrided objects by calling them by their {\it full}
names. To determine the full name of an object the following rule
applies: 
\medskip\noindent
{\leftskip20pt\rightskip20pt
If the object \hbox{\tt Object} is defined in the file \hbox{\tt
File.ext}, then the full name of \hbox{\tt Object} is \hbox{\tt
File`ext`Object}.
\par}
\medskip\noindent
For instance, if you load \hbox{\tt hyp.q} first and then \hbox{\tt
hyp.m} and want to use 
\hbox{\tt Limes} with a basic
hypergeometric expression, then you have to type \hbox{\tt
Hyp`q`Limes} instead of \hbox{\tt Limes}. (Calling \hbox{\tt Limes}
would invoke the {\it ordinary} hypergeometric \hbox{\tt Limes}.) 
For information on contexts in MATHEMATICA confer \cite{\WolfAA}.

\head On-line help \endhead

For each object of this package on-line help is supported in the
usual way. For instance, quick information about \hbox{\tt
Limes} (not having the manual at hand) is available in the
following way.
\MATH

In[1]:= ?Limes
Description: Function for doing formal limits of basic hypergeometric   
  expressions. If required for taking the limit, you will be asked whether   
  or not the absolute value of some variable or expression is smaller than   
  1. Your decision is stored for the rest of your MATHEMATICA session. If   
  you want to change your decision later, use "AbsGreater", "AbsSmaller",   
  or "AbsUndetermined", respectively.   
  By default the absolute value of q is defined to be smaller than 1. Also    
  this can be changed by "AbsGreater", "AbsSmaller", or "AbsUndetermined",   
  respectively.   
Warning: This function uses primitive algebraic techniques to do the    
  limit. There is no check if taking the limit is actually allowed. So it   
  is left to you to check the validity of a result of "Limes".   
Usage: Limes[Expr, x-\MATHgroesser x0].   
See also: AbsGreater, AbsSmaller, AbsUndetermined, MinusOne.

\endMATH

\head The screen output \endhead

The screen output of the examples in this manual imitates the output
under usage of code tables 437, 860, 863, or 865 (cf\. the \hbox{\tt
read.me}). The output under other code tables is a little bit less
attractive, but similar. For instance, the examples for \hbox{\tt ph}
then would read as follows.
\MATH

In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
\goodbreakpoint%
            [            ]
            | a, b       |
Out[1]=  ph |     ; q, z |
        2  1|  c         |
            [            ]
\goodbreakpoint%
In[2]:= ph[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,0%
\MATHrbrace ,q,z]
\goodbreakpoint%
            [               ]
            | a, b, c       |
Out[2]=  ph |        ; q, z |
        3  3| d, e, 0       |
            [               ]
\endMATH
\newpage


\head A brief dictionary \endhead
Most of the names of the objects are obviously German influenced. 
To help those who are not so familiar with German, brief German--English
and English--German vocabularies are provided.

\vbox{
\head \tenpoint \bf A German--English vocabulary \endhead
$$\smatrix \format\sa\l\s\l\s\l\se\\
\hlinefor7\\
&\text {\eightpoint German}&&\text {\eightpoint English}&&
\text {\eightpoint {\sl Mathematica} objects in HYPQ  }&\\
&&&&&\text {\eightpoint containing the word}&\\
\hlinefor7\\
&\text {abspalten}&&       \text {split}&&   \text {\tt lina1, lina2}&\\
\hlinefor7\\
&\text {aufl\"osen}&&       \text {dissolve}&&   \text {\tt pqaufl}&\\
\hlinefor7\\
&\text {drucken}&& \text {print}&&   \text {\tt Drucke}&\\
\hlinefor7\\
&\text {einf\"ugen}&&      \text {insert}&&  \text {\tt phEinf}&\\
\hlinefor7\\
&\text {ersetzen}&&\text {replace}&& \text {\tt Ers}&\\
\hlinefor7\\
&\text {erweitern}&&       \text {extend}&&  \text {\tt erw1, erw2, SUMErw1, SUMErw2}&\\
\hlinefor7\\
&\text {Gleichung}&&       \text {equation}&&\text {\tt Gleichung}&\\
\hlinefor7\\
&\text {"2 hoch 3"}&& \text {"2 to the 3"}&& \text {\tt Hoch}&\\
\hlinefor7\\
&\text {"2 mal 3"}&& \text {"2 times 3"}&& \text {\tt Mal}&\\
\hlinefor7\\
&\text {ordnen}&&  \text {order}&&   \text {\tt phOrdne}&\\
\hlinefor7\\
&\text {Regel}&&   \text {rule}&&    \text {\tt SUMRegeln}&\\
\hlinefor7\\
&\text {sammeln}&& \text {collect}&& \text {\tt SUMSammle}&\\
\hlinefor7\\
&\text {schreiben}&&       \text {write}&&   \text {\tt SchreibeZahl}&\\
\hlinefor7\\
&\text {tauschen}&&\text {exchange, interchange}&& \text {\tt  SUMTausche, GlTausche}&\\
\hlinefor7\\
&\text {umkehren}&&\text {reverse}&& \text {\tt SUMUmkehr}&\\
\hlinefor7\\
&\text {Zahl}&&    \text {number}&&  \text {\tt SchreibeZahl}&\\
\hlinefor7\\
&\text {zerlegen}&&\text {split}&&   \text {\tt zerl1, pqzerl, pqinfzerl, SUMZerl}&\\
\hlinefor7\\
&\text {zusammenfassen}&&\text {put together}&&\text {\tt zus1, 
pqzus, pqinfzus}&\\
\hlinefor7\\
\endsmatrix
$$
}

\newpage

\vbox{
\head \tenpoint \bf An English--German vocabulary \endhead
$$\smatrix \format\sa\l\s\l\s\l\se\\
\hlinefor7\\
&\text {\eightpoint English}&&\text {\eightpoint German}&&
\text {\eightpoint {\sl Mathematica} objects in HYPQ }&\\
&&&&&\text {\eightpoint containing the word}&\\
\hlinefor7\\
&\text {collect}&& \text {sammeln}&& \text {\tt SUMSammle}&\\
\hlinefor7\\
&\text {dissolve}&&   \text {aufl\"osen}&& \text {\tt pqaufl}&\\
\hlinefor7\\
&\text {equation}&&\text {Gleichung}&&       \text {\tt Gleichung}&\\
\hlinefor7\\
&\text {exchange}&&\text {tauschen}&&\text {\tt SUMTausche, GlTausche}&\\
\hlinefor7\\
&\text {extend}&&  \text {erweitern}&&       \text {\tt erw1, erw2, SUMErw1, SUMErw2}&\\
\hlinefor7\\
&\text {insert}&&  \text {einf\"ugen}&&      \text {\tt phEinf}&\\
\hlinefor7\\
&\text {interchange}&&     \text {tauschen}&&\text {\tt SUMTausche, GlTausche}&\\
\hlinefor7\\
&\text {number}&&  \text {Zahl}&&    \text {\tt SchreibeZahl}&\\
\hlinefor7\\
&\text {order}&&   \text {ordnen}&&  \text {\tt phOrdne}&\\
\hlinefor7\\
&\text {print}&&   \text {drucken}&& \text {\tt Drucke}&\\
\hlinefor7\\
&\text {replace}&& \text {ersetzen}&&\text {\tt Ers}&\\
\hlinefor7\\
&\text {reverse}&& \text {umkehren}&&\text {\tt SUMUmkehr     }&\\
\hlinefor7\\
&\text {rule}&&    \text {Regel}&&   \text {\tt SUMRegeln}&\\
\hlinefor7\\
&\text {split}&&   \text {abspalten}&&       \text {\tt lina1, lina2}&\\
\hlinefor7\\
&\text {split}&&   \text {zerlegen}&&\text {\tt zerl1, pqzerl, pqinfzerl, SUMZerl}&\\
\hlinefor7\\
&\text {"2 times 3"}&& \text {"2 mal 3"}&& \text {\tt Mal}&\\
\hlinefor7\\
&\text {"2 to the 3"}&& \text {"2 hoch 3"}&& \text {\tt Hoch}&\\
\hlinefor7\\
&\text {put together}&&\text {zusammenfassen}&&\text {\tt zus1, 
pqzus, pqinfzus}&\\
\hlinefor7\\
&\text {write}&&   \text {schreiben}&&       \text {\tt SchreibeZahl}&\\
\hlinefor7\\
\endsmatrix
$$
}



\newpage

\head
 Alphabetic List of the objects with desriptions \endhead
\vskip1cm
\parindent0pt

%\tracingmacros=2 \tracingcommands=2
\Name AbsGreater
\Description Function for declaring the absolute value of a variable or 
  expression to be greater than 1. This declaration is used by \hbox{\tt Limes}.
  By default the absolute value of {\tt q} is defined to be smaller than 1.
\Usage AbsGreater[Expr].
\Example
\MATH
In[1]:= Limes[pq[a,n,p],n-\MATHgroesser Infinity]
Is %
\MATHvStrich p%
\MATHvStrich  smaller than 1?
[y|n|u]: y
\goodbreakpoint%
Out[1]= (a;p)
             \MATHinfty 
\goodbreakpoint%
In[2]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity]
Is n even, odd, or neither of both?
[e|o|n]: n
Is %
\MATHvStrich a%
\MATHvStrich  smaller than 1?
[y|n|u]: u
Is %
\MATHvStrich a%
\MATHvStrich  smaller than 1?
[y|n|u]: u
\goodbreakpoint%
The expression
\goodbreakpoint%
                  2
    n  n  -n/2 - n /2  p
(-1)  a  p            (-;p)
                       a   \MATHinfty 
\goodbreakpoint%
was obtained.
\goodbreakpoint%
Therefore the limit n -\MATHgroesser  \MATHinfty  could not be determined.
Here is your expression:
\goodbreakpoint%
         a
Out[2]= (--; p)
          n    n
         p
\goodbreakpoint%
In[3]:= AbsGreater[p]
\goodbreakpoint%
In[4]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]: e
\goodbreakpoint%
         a 1
Out[4]= (-;-)
         p p \MATHinfty 
\endMATH
\Seealso AbsSmaller, AbsUndetermined, Limes.
\Name AbsSmaller
\Description Function for declaring the absolute value of a variable or 
  expression to be smaller than 1. This declaration is used by \hbox{\tt Limes}.
  By default the absolute value of {\tt q} is defined to be smaller than 1.
\Usage AbsSmaller[Expr].
\Example
\MATH
In[1]:= Limes[pq[a,n,1/p],n-\MATHgroesser Infinity]
Is %
\MATHvStrich p%
\MATHvStrich  smaller than 1?
[y|n|u]: n
\goodbreakpoint%
           1
Out[1]= (a;-)
           p \MATHinfty 
\goodbreakpoint%
In[2]:= Limes[pq[a*p\MATHhoch n,n,1/p],n-\MATHgroesser Infinity]
Is n even, odd, or neither of both?
[e|o|n]: n
Is %
\MATHvStrich a%
\MATHvStrich  smaller than 1?
[y|n|u]: u
Is %
\MATHvStrich a%
\MATHvStrich  smaller than 1?
[y|n|u]: u
\goodbreakpoint%
The expression
\goodbreakpoint%
                 2
    n  n  n/2 + n /2   1  1
(-1)  a  p           (---;-)
                      a p p \MATHinfty 
\goodbreakpoint%
was obtained.
\goodbreakpoint%
Therefore the limit n -\MATHgroesser  \MATHinfty  could not be determined.
Here is your expression:
\goodbreakpoint%
            n  1
Out[2]= (a p ; -)
               p n
\goodbreakpoint%
In[3]:= AbsSmaller[p]
\goodbreakpoint%
In[4]:= Limes[pq[a*p\MATHhoch n,n,1/p],n-\MATHgroesser Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]: e
\goodbreakpoint%
Out[4]= (a p;p)
               \MATHinfty 
\endMATH
\Seealso AbsGreater, AbsUndetermined, Limes.

\Name AbsUndetermined
\Description Function for declaring the absolute value of a variable or 
  expression to be neither smaller nor greater than 1. 
  This declaration is used by \hbox{\tt Limes}.
  By default the absolute value of {\tt q} is defined to be smaller than 1.
\Usage AbsUndetermined[Expr].
\Example
\MATH
In[1]:= AbsSmaller[p]
\goodbreakpoint%
In[2]:= Limes[pq[a,n,p],n-\MATHgroesser Infinity]
\goodbreakpoint%
Out[2]= (a;p)
             \MATHinfty 
\goodbreakpoint%
In[3]:= AbsGreater[p]
\goodbreakpoint%
In[4]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]: e
\goodbreakpoint%
         a 1
Out[4]= (-;-)
         p p \MATHinfty 
\goodbreakpoint%
In[5]:= AbsUndetermined[p]
\goodbreakpoint%
In[6]:= Limes[pq[a,n,p],n-\MATHgroesser Infinity]
Is %
\MATHvStrich p%
\MATHvStrich  smaller than 1?
[y|n|u]: u
\goodbreakpoint%
The expression
\goodbreakpoint%
Indeterminate
\goodbreakpoint%
was obtained.
\goodbreakpoint%
Therefore the limit n -\MATHgroesser  \MATHinfty  could not be determined.
Here is your expression:
\goodbreakpoint%
Out[6]= (a; p)
              n
\goodbreakpoint%
In[7]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity]
Is %
\MATHvStrich p%
\MATHvStrich  smaller than 1?
[y|n|u]: u
\goodbreakpoint%
The expression
\goodbreakpoint%
Indeterminate
\goodbreakpoint%
was obtained.
\goodbreakpoint%
Therefore the limit n -\MATHgroesser  \MATHinfty  could not be determined.
Here is your expression:
\goodbreakpoint%
         a
Out[7]= (--; p)
          n    n
         p
\endMATH
\Seealso AbsGreater, AbsSmaller, Limes.

\Name Add
\Description Function that adds \hbox{\tt Expr} to \hbox{\tt Gleichung}.
\Usage Add[Expr].
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= Add[pq[c/a,n]]
\goodbreakpoint%
                                                    n  c
           \MATHluEck               \MATHruEck                         a  (-; q)
           \MATHvStrich      -n       \MATHvStrich     c          c            a    n
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  + (-; q)  == (-; q)  + %
----------
        2 1\MATHvStrich               \MATHvStrich     a    n     a    n    (c; q)
           \MATHloEck    c          \MATHroEck                                n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
                                                    n  c
           \MATHluEck               \MATHruEck                         a  (-; q)
           \MATHvStrich      -n       \MATHvStrich     c          c            a    n
Out[3]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  + (-; q)  == (-; q)  + %
----------
        2 1\MATHvStrich               \MATHvStrich     a    n     a    n    (c; q)
           \MATHloEck    c          \MATHroEck                                n
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Div, Sub, 
Hoch, GlTausche, Ers.

\Name AmSLaTeX
\Description Switch that changes the output of TeXForm to be usable with
   \AmS-\LaTeX. By default the output of TeXForm is usable with 
   \AmSTeX.
\Usage AmSLaTeX.
\Example 
\MATH
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= TeXForm[ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[2]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  \MATHbackslash phi \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash  \MATHbackslash let \MATHbackslash over /
   c\MATHbackslash endmatrix ;q, %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ]
\goodbreakpoint%
In[3]:= AmSLaTeX
\goodbreakpoint%
In[4]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[5]:= TeXForm[ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[5]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  \MATHbackslash phi \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash ! \MATHbackslash left [ \MATHbackslash begin%
\MATHlbrace matrix%
\MATHrbrace  \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash  \MATHbackslash let \MATHbackslash over /
   c\MATHbackslash end%
\MATHlbrace matrix%
\MATHrbrace  ;q, %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ]
\endMATH
\Seealso AmSTeX, LaTeX, TeX, TeXMat, TeXphW.

\Name AmSTeX
\Description Switch that changes the output of TeXForm to be usable with
   \AmSTeX. By default the output of TeXForm is usable with \AmSTeX.
\Usage AmSTeX.
\Example 
\MATH
In[1]:= TeX
\goodbreakpoint%
In[2]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[3]:= TeXForm[ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[3]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  \MATHbackslash phi \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash cr \MATHbackslash let \MATHbackslash over / c%
\MATHrbrace 
   ;q, %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ]
\goodbreakpoint%
In[4]:= AmSTeX
\goodbreakpoint%
In[5]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[6]:= TeXForm[ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[6]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  \MATHbackslash phi \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash  \MATHbackslash let \MATHbackslash over /
   c\MATHbackslash endmatrix ;q, %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ]
\endMATH 
\Seealso AmSLaTeX, LaTeX, TeX, TeXMat, TeXphW.

\Name baszerl1
\Description  
\vtab $(a;q)_n \to \prod _{k=0} ^{m-1} (aq^k;q^m)_{n/m}$,\\
             $(a;q)_\infty \to \prod _{k=0} ^{m-1} (aq^k;q^m)_\infty$.
\endvtab
\vskip6pt
\hskip10pt  The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.baszerl1.
\Example 
\MATH 
In[1]:= pq[a,n]
\goodbreakpoint%
Out[1]= (a; q)
              n
\goodbreakpoint%
In[2]:= \%/.baszerl1
split into ? terms: 2
\goodbreakpoint%
             2            2
Out[2]= (a; q )    (a q; q )
               n/2          n/2
\goodbreakpoint%
In[3]:= pq[b,4*m]
\goodbreakpoint%
Out[3]= (b; q)
              4 m
\goodbreakpoint%
In[4]:= \%/.baszerl1
split into ? terms: 4
\goodbreakpoint%
             4          4       2   4       3   4
Out[4]= (b; q )  (b q; q )  (b q ; q )  (b q ; q )
               m          m           m           m
\goodbreakpoint%
In[5]:= pqinf[c,q\MATHhoch (1/2)]
\goodbreakpoint%
Out[5]= (c;Sqrt[q])
                   \MATHinfty 
\goodbreakpoint%
In[6]:= \%/.baszerl1
split into ? terms: 4
\goodbreakpoint%
            2               2         2       3/2  2
Out[6]= (c;q )  (c Sqrt[q];q )  (c q;q )  (c q   ;q )
              \MATHinfty                \MATHinfty          \MATHinfty             \MATHinfty 
\endMATH 
\Seealso baszerl2, baszus1, baszus2, Ers, PosListe, ManipulationsListe.

\Name baszerl2
\Description 
  \vtab $(a;q)_n \to \prod _{k=0} ^{m-1} (e^{2\pi i k/m}a^{1/m};q^{1/m})_n$,\\
  $(a;q)_\infty \to \prod _{k=0}^{m-1}(e^{2\pi i k/m}a^{1/m};q^{1/m})_\infty$.
\endvtab
\vskip6pt
\hskip10pt  The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.baszerl2.
\Example 
\MATH 
In[1]:= pq[a,n]
\goodbreakpoint%
Out[1]= (a; q)
              n
\goodbreakpoint%
In[2]:= \%/.baszerl2
split into ? terms: 2
\goodbreakpoint%
Out[2]= (-Sqrt[a]; Sqrt[q])  (Sqrt[a]; Sqrt[q])
                           n                   n
\goodbreakpoint%
In[3]:= pq[a\MATHhoch 2,n,q\MATHhoch 4]
\goodbreakpoint%
          2   4
Out[3]= (a ; q )
                n
\goodbreakpoint%
In[4]:= \%/.baszerl2
split into ? terms: 4
\goodbreakpoint%
Out[4]= (-Sqrt[a]; q)  (-I Sqrt[a]; q)  (I Sqrt[a]; q)  (Sqrt[a]; q)
                     n                n               n             n
\goodbreakpoint%
In[5]:= pqinf[c,q\MATHhoch 3]
\goodbreakpoint%
            3
Out[5]= (c;q )
              \MATHinfty 
\goodbreakpoint%
In[6]:= \%/.baszerl2
split into ? terms: 3
\goodbreakpoint%
          1/3       (2 I)/3 \MATHpi   1/3       (4 I)/3 \MATHpi   1/3
Out[6]= (c   ;q)  (E          c   ;q)  (E          c   ;q)
                \MATHinfty                     \MATHinfty                     \MATHinfty 
\endMATH 
\Seealso Ers, PosListe, ManipulationsListe.

\Name baszus1
\Description \vtab $(a;q)_n \to (a;q^{1/m})_{mn}/
                                 \prod _{k=1} ^{m-1} (aq^{k/m};q)_n$,\\
             $(a;q)_\infty \to (a;q^{1/m})_\infty/
                                 \prod _{k=1} ^{m-1} (aq^{k/m};q)_\infty$.
\endvtab
\vskip6pt
\hskip10pt  The parameter \hbox{\tt m} has to be entered on request.
  This operation is basically the inverse of \hbox{\tt baszerl1}.
\Usage Expr/.baszus1.
\Example 
\MATH 
In[1]:= pq[a,n,q\MATHhoch 2]*pq[a*q,n,q\MATHhoch 2]
\goodbreakpoint%
             2          2
Out[1]= (a; q )  (a q; q )
               n          n
\goodbreakpoint%
In[2]:= Ers[\%,baszus1,\MATHlbrace 1\MATHrbrace ]
put together ? terms: 2
put together ? terms: 2
\goodbreakpoint%
Out[2]= (a; q)
              2 n
\goodbreakpoint%
In[3]:= pqinf[a/q,q\MATHhoch 2]*pqinf[a,q\MATHhoch 2]*pq[b,m]*pq[b*q\MATHhoch (1/3),m]*pq[b*q\MATHhoch (2/3),m]
\goodbreakpoint%
            2    a  2               1/3          2/3
Out[3]= (a;q )  (-;q )  (b; q)  (b q   ; q)  (b q   ; q)
              \MATHinfty   q    \MATHinfty        m            m            m
\goodbreakpoint%
In[4]:= Ers[\%,baszus1,\MATHlbrace 3\MATHrbrace ]
put together ? terms: 3
put together ? terms: 3
\goodbreakpoint%
            2    a  2        1/3
Out[4]= (a;q )  (-;q )  (b; q   )
              \MATHinfty   q    \MATHinfty           3 m
\goodbreakpoint%
In[5]:= Ers[\%,baszus1,\MATHlbrace 2\MATHrbrace ]
put together ? terms: 2
put together ? terms: 2
\goodbreakpoint%
         a          1/3
Out[5]= (-;q)  (b; q   )
         q   \MATHinfty           3 m
\endMATH 
\Seealso Ers, PosListe, ManipulationsListe.

\Name baszus2
\Description 
  \vtab $(a;q)_n \to (a^m;q^m)_n/\prod _{k=1} ^{m-1}(e^{2\pi i k/m}a;q)_n$,\\
  $(a;q)_\infty \to (a^m;q^m)_\infty/
                       \prod _{k=1} ^{m-1}(e^{2\pi i k/m}a;q)_\infty$.
\endvtab
\vskip6pt
\hskip10pt  The parameter \hbox{\tt m} has to be entered on request.
  This operation is basically the inverse of \hbox{\tt baszerl2}.
\Usage Expr/.baszus2.
\Example 
\MATH 
In[1]:= pq[a,m,q\MATHhoch 2]*pq[-a,m,q\MATHhoch 2]
\goodbreakpoint%
              2        2
Out[1]= (-a; q )  (a; q )
                m        m
\goodbreakpoint%
In[2]:= Ers[\%,baszus2,\MATHlbrace 1\MATHrbrace ]
put together ? terms: 2
put together ? terms: 2
\goodbreakpoint%
          2   4
Out[2]= (a ; q )
                m
\goodbreakpoint%
In[3]:= pqinf[-A,q\MATHhoch (1/2)]*pqinf[A,q\MATHhoch (1/2)]*pq[-a,n]*pq[-I*a,n]*pq[I*a,n]*
        pq[a,n]
\goodbreakpoint%
Out[3]= (-A;Sqrt[q])  (A;Sqrt[q])  (-a; q)  (-I a; q)  (I a; q)  (a; q)
                    \MATHinfty             \MATHinfty         n          n         n       n
\goodbreakpoint%
In[4]:= Ers[\%,baszus2,\MATHlbrace 3\MATHrbrace ]
put together ? terms: 4
put together ? terms: 4
\goodbreakpoint%
                                     4   4
Out[4]= (-A;Sqrt[q])  (A;Sqrt[q])  (a ; q )
                    \MATHinfty             \MATHinfty          n
\goodbreakpoint%
In[5]:= Ers[\%,baszus2,\MATHlbrace 1\MATHrbrace ]
put together ? terms: 2
put together ? terms: 2
\goodbreakpoint%
          2       4   4
Out[5]= (A ;q)  (a ; q )
              \MATHinfty          n
\endMATH 
Whenever you use a rule, that asks you for an input, within \hbox{\tt
Ers}, you will get the reqest for input twice. The value which is
entered last is used.
\Seealso Ers, PosListe, ManipulationsListe.

\Name Binomialpq
\Description \hbox{\tt Binomialpq[n,k,q]} is the $q$-binomial coefficient, written in terms
  of $q$-factorial symbols \hbox{\tt pq}. The parameter \hbox{\tt q} is optional. It will be set 
  equal \hbox{\tt q} if it is omitted. 
\Usage Binomialpq[n,k,q]

   \leavevmode\hphantom{Usa}\rm or: \tt Binomialpq[n,k].
\Example 
\MATH 
In[1]:= Binomialpq[n,k]
\goodbreakpoint%
          1 - k + n
        (q         ; q)
                       k
Out[1]= ----------------
            (q; q)
                  k
\goodbreakpoint%
In[2]:= Binomialpq[6,3]
\goodbreakpoint%
          4
        (q ; q)
               3
Out[2]= --------
        (q; q)
              3
\endMATH 
\Seealso Binomialq, Multinomialpq, Multinomialq, Factorialq, Factorialpq.

\Name Binomialq
\Description \hbox{\tt Binomialq[n,k,q]} is the $q$-binomial coefficient, expanded into
  a $q$-series, if possible. The parameter \hbox{\tt q} is optional. It will be set equal 
  \hbox{\tt q} if it is omitted. 
\Usage Binomialq[n,k,q]

   \leavevmode\hphantom{Usa}\rm or: \tt Binomialq[n,k].
\Example 
\MATH 
In[1]:= Binomialq[n+k,n-k]
\goodbreakpoint%
        \MATHluEck         \MATHruEck  
        \MATHvStrich  k + n  \MATHvStrich  
Out[1]= \MATHvStrich         \MATHvStrich  
        \MATHvStrich  -k + n \MATHvStrich  
        \MATHloEck         \MATHroEck  
                  q
\goodbreakpoint%
In[2]:= Binomialq[6,3]
\goodbreakpoint%
                   2      3      4      5      6      7    8    9
Out[2]= 1 + q + 2 q  + 3 q  + 3 q  + 3 q  + 3 q  + 2 q  + q  + q
\endMATH 
\Seealso Binomialpq, Multinomialpq, Multinomialq, Factorialq, Factorialpq.

\Name C01
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  1 + {{{{\left( -1 \right) }^{1 - r + s}} z 
       }\over 
     {\left( 1 - q \right)   }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / q, \qAi\\
        \let \over / {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}}
        z} \right ]$$
\Usage Expr/.C01.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C02
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai, q\\ \let \over/
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  -{{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
         }\over 
{z  }} 
{{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
+    {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
       }\over {z }
     }
{{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over /\Aiq, q\\ \let \over /
        \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]
$$
\Usage Expr/.C02.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C11
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
    \Bi\endmatrix ;q, {\displaystyle q z} \right ] + 
{{\left( -1 \right) }^{1 - r + s}} z 
   {{       \prodl_{i = 1}^{r} (1 - \ai )  
       }\over 
     {\prodl_{i = 1}^{s}(1 - \bi)}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over /
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$
\Usage Expr/.C11.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C12
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
    \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - 
{{{{\left( -1 \right) }^{1 - r + s}} z }\over q}
   {{       \prodl_{i = 1}^{r} (1 - \ai )  
       }\over 
     { \prodl_{i = 1}^{s} (1 - \bi ) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over /
        \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ]$$
\Usage Expr/.C12.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C13
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
{{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z}
  {{       \prodl_{i = 1}^{s} (1 - {{\bi}\over q} )  
       }\over 
     { \prodl_{i = 1}^{r} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]
\\-   {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z}
 {{       \prodl_{i = 1}^{s} (1 - {{\bi}\over q} )  
       }\over 
     { \prodl_{i = 1}^{r} (1 - {{\ai}\over q} ) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C13.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C14
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
{{{{\left( -1 \right) }^{1 - r + s}} a z }\over q}
  {{       \prodl_{i = 1}^{r-1} (1 - \ai )  
       }\over 
     { \prodl_{i = 1}^{s} (1 - \bi ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \qAi\\ \let \over
        / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]+ 
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let
    \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]
$$
\Usage Expr/.C14[m1].\newline
\rm {\tt m1} is the position of the special upper parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C15
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over /
    \Bi\endmatrix ;q, {\displaystyle z} \right ] - 
{{\left( -1 \right) }^{1 - r + s}} a z 
   {{       \prodl_{i = 1}^{r - 1} (1 - \ai )  
       }
      \over {\prodl_{i = 1}^{s}(1 - \bi)}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \qAi\\ \let
        \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$
\Usage Expr/.C15[m1].\newline
\rm {\tt m1} is the position of the special upper parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C16
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
{{{{\left( -1 \right) }^{1 - r + s}} {q^{2 - r + s}} }\over {az}}
  {{       \prodl_{i = 1}^{s} (1 - {{\bi}\over q} )  
       }\over
       {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over
        / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]
\\-   {{{{\left( -1 \right) }^{1 - r + s}} {q^{2 - r + s}}}\over{az}}
 {{       \prodl_{i = 1}^{s} (1 - {{\bi}\over q} )  
       }\over 
     {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\
        \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z}
        \right ]
\endmultline$$
\Usage Expr/.C16[m1].\newline
\rm {\tt m1} is the position of the special upper parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C17
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
{{\left( -1 \right) }^{1 - r + s}} z 
  {{       \prodl_{i = 1}^{r - 1} (1 - \ai )  
       }\over 
     {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \qAi\\ \let \over
        / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]+ 
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let
    \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ]
$$
\Usage Expr/.C17[m1].\newline
\rm {\tt m1} is the position of the special upper parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C18
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over /
    \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - 
{{{{\left( -1 \right) }^{1 - r + s}} z }\over q}
   {{       \prodl_{i = 1}^{r - 1} (1 - \ai )  
       }\over
       { \prodl_{i = 1}^{s} (1 - \bi ) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \qAi\\ \let
        \over / \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ]$$
\Usage Expr/.C18[m1].\newline
\rm {\tt m1} is the position of the special upper parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C19
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
{{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z}
  {{       \prodl_{i = 1}^{s} (1 - {{\bi}\over q} )  
       }\over
       { \prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over
        / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z}
{{       \prodl_{i = 1}^{s} (1 - {{\bi}\over q} )  
          }\over { \prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\
        \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C19[m1].\newline
\rm {\tt m1} is the position of the special upper parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C20
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b q,
    \Bi\endmatrix ;q, {\displaystyle z} \right ] 
+    {{{{\left( -1 \right) }^{1 - r + s}} b z 
      }
      \over {\left( 1 - b \right)  \left( 1 - b q \right)  
        }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over{\prodl_{i = 1}^{s - 1} (1 - \bi)}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b
        {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
$$
\Usage Expr/.C20[n1].\newline
\rm {\tt n1} is the position of the special lower parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C21
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {b\over
    q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] - 
   {{{{\left( -1 \right) }^{1 - r + s}} b z 
       }\over
       {\left( 1 - b \right)  \left( 1 - {b\over q} \right)  q 
        }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 1} (1 - \bi )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b
        q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$
\Usage Expr/.C21[n1].\newline
\rm {\tt n1} is the position of the special lower parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C22
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{3 - r + s}} 
       }\over 
     {b z  }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}}
        z} \right ]
\\-   {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{3 - r + s}} 
       }\over 
     {b z  }}
{{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z}
        \right ]
\endmultline$$
\Usage Expr/.C22[n1].\newline
\rm {\tt n1} is the position of the special lower parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C23
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b q,
    \Bi\endmatrix ;q, {\displaystyle q z} \right ] 
+    {{{{\left( -1 \right) }^{1 - r + s}} z 
       }
      \over {\left( 1 - b \right)  \left( 1 - b q \right)  
        }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 1} (1 - \bi )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b
        {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
$$
\Usage Expr/.C23[n1].\newline
\rm {\tt n1} is the position of the special lower parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C24
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {b\over
    q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - 
   {{{{\left( -1 \right) }^{1 - r + s}} z 
       }\over 
     {\left( 1 - b \right)  \left( 1 - {b\over q} \right)  q 
        }}
{{    \prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 1} (1 - \bi )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b
        q, \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ]$$
\Usage Expr/.C24[n1].\newline
\rm {\tt n1} is the position of the special lower parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C25
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {z  }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}}
        z} \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       \  
       }
      \over {z  }}
{{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C25[n1].\newline
\rm {\tt n1} is the position of the special lower parameter.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C26
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - b \right)  }\over {a - b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a,
        b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}}
        \right ]+ 
   {{\left( 1 - a \right)  }\over {-a + b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a
        q, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}}
        \right ]$$
\Usage Expr/.C26[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C27
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( b - {a\over q} \right)  
       }\over 
     {1 - {a\over q}}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\
        \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ]
 + {{\left( 1 - b \right)  
       }\over 
     {1 - {a\over q}}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q,
        \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C27[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C28
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{a \left( 1 - b \right)  }\over {a - b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over /
        a, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right
        ] + {{\left( 1 - a \right)  b 
       }\over {-a + b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let
        \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C28[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C29
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( b - {a\over q} \right)  
       }\over 
     {b \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\
        \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]+ 
   {{a \left( 1 - b \right)  }\over {b \left( 1 - {a\over q} \right)  q}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over /
        {a\over q}, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle
        z} \right ]$$
\Usage Expr/.C29[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C30
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
{{\left( -1 \right) }^{1 - r + s}} \left( -b + {a\over q} \right)  z 
  {{       \prodl_{i = 1}^{r - 2} (1 - \ai )  
       }
      \over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \qAi\\ \let
        \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
+    {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\
    \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]
$$
\Usage Expr/.C30[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C31
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
       }\over 
     {\left( {a\over q} - {b\over q} \right)  z 
        }} {{\prodl_{i = 1}^{s} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {b\over q}, \Aiq\\
        \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z}
        \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
       }\over 
     {\left( {a\over q} - {b\over q} \right)  z 
        }}
{{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Aiq\\
        \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z}
        \right ]
\endmultline$$
\Usage Expr/.C31[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C32
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {b\over q},
    \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ] 
\\+   {{\left( -1 \right) }^{1 - r + s}} \left( 1 - {{a b}\over q} \right)  z 
 {{       \prodl_{i = 1}^{r - 2} (1 - \ai )  
       }\over {\prodl_{i = 1}^{s}(1 - \bi)}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a, b, a b,
        \qAi\\ \let \over / {{a b}\over q}, \qBi\endmatrix ;q, {\displaystyle
        {q^{1 - r + s}} z} \right ]
\endmultline$$
\Usage Expr/.C32[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C33
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b q, \Ai\\ \let \over
    / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] 
\\-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b q \right)  z 
       }\over 
     {q  }}
{{\prodl_{i = 1}^{r - 2} (1 - \ai )}\over
{\prodl_{i = 1}^{s} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a q, b q, a b
        {q^2}, \qAi\\ \let \over / a b q, \qBi\endmatrix ;q, {\displaystyle
        {q^{-r + s}} z} \right ]
\endmultline$$
\Usage Expr/.C33[m1,m2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special upper
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C34
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {b\over q} \right)  
       }\over
       {a - {b\over q}}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
        {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]
 + {{\left( 1 - a \right)  
       }\over 
     {-a + {b\over q}}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / b, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]$$
\Usage Expr/.C34[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C35
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( -a + b \right)  }\over {\left( 1 - {a\over q} \right)  q}} {} _{r} \phi _{s} \! \left [ \matrix \let \over /
        {a\over q}, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle q
        z} \right ]+ 
   {{\left( 1 - {b\over q} \right)  
       }
      \over {1 - {a\over q}}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C35[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C36
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( a - b \right)  
       }\over {1 - b}} 
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a,
        \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle q z} \right ]
+ {{\left( 1 - a \right)  
       }\over {1 - b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C36[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C37
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{a \left( 1 - {b\over q} \right)  
       }\over 
     {a - {b\over q}}} 
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
        {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
+ {{\left( 1 - a \right)  b 
       }\over 
     {\left( -a + {b\over q} \right)  q}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / b, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C37[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C38
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( -a + b \right)  }\over {b \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over /
        {a\over q}, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z}
        \right ]+ 
   {{a \left( 1 - {b\over q} \right)  
       }
      \over {b \left( 1 - {a\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C38[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C39
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( a - b \right)  }
      \over {a \left( 1 - b \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a,
        \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ 
   {{\left( 1 - a \right)  b }\over {a \left( 1 - b \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over /
        a q, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z}
        \right ]$$
\Usage Expr/.C39[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C40
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} 
       \left( {a\over q} - {b\over q} \right)  z 
       }
      \over {\left( 1 - b \right)  \left( 1 - {b\over q} \right)  
        }} 
{{\prodl_{i = 1}^{s - 1} (1 - \bi )}\over
{\prodl_{i = 1}^{r - 1} (1 - \ai
)}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \qAi\\ \let \over
        / b q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
+    {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let
    \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
$$
\Usage Expr/.C40[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C41
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b
    q, \Bi\endmatrix ;q, {\displaystyle z} \right ] - 
   {{{{\left( -1 \right) }^{1 - r + s}} \left( a - b \right)  z 
       }\over 
     {\left( 1 - b \right)  \left( 1 - b q \right)  
        }}
{{\prodl_{i = 1}^{r - 1} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 1} (1 - \bi )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \qAi\\ \let
        \over / b {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z}
        \right ]$$
\Usage Expr/.C41[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C42
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( -{b\over {{q^2}}} + {a\over q} \right)  z 
        }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over
        / {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z}
        \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( -{b\over {{q^2}}} + {a\over q} \right)  z 
        }}
{{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\
        \let \over / {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle
        {q^{-1 + r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C42[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C43
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let
    \over / b q, \Bi\endmatrix ;q, {\displaystyle q z} \right ] 
+    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b \right)  z 
       }\over 
     {\left( 1 - b \right)  \left( 1 - b q \right)  
        }}
{{\prodl_{i = 1}^{r - 1} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 1} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a, a b q,
        \qAi\\ \let \over / b {q^2}, a b, \qBi\endmatrix ;q, {\displaystyle
        {q^{1 - r + s}} z} \right ]
$$
\Usage Expr/.C43[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C44
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over /
    {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] 
-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b \right)  z 
       }\over 
     {\left( 1 - b \right)  \left( 1 - {b\over q} \right)  q 
        }}
{{\prodl_{i = 1}^{r - 1} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 1} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a q, a b q,
        \qAi\\ \let \over / b q, a b, \qBi\endmatrix ;q, {\displaystyle {q^{-r
        + s}} z} \right ]
$$
\Usage Expr/.C44[m1,n1].\newline
\rm {\tt m1} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C45
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {b\over q} \right)  q 
       }\over
       {a - b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
        {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]
 + {{\left( 1 - {a\over q} \right)  q 
       }
      \over {-a + b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
        {a\over q}, b, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]$$
\Usage Expr/.C45[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C46
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( -a + {b\over q} \right)  
       }\over {1 - a}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, b, \Bi\endmatrix ;q, {\displaystyle q z} \right ]+ 
   {{\left( 1 - {b\over q} \right)  
       }\over 
     {1 - a}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C46[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C47
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{a \left( 1 - {b\over q} \right)  
       }\over {a - b}
     } {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
        {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
+ {{b \left( 1 - {a\over q} \right)  
       }\over 
     {-a + b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
        {a\over q}, b, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C47[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C48
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( -a + {b\over q} \right)  q 
       }\over 
     {\left( 1 - a \right)  b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, b, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ 
   {{a \left( 1 - {b\over q} \right)  q 
       }\over 
     {\left( 1 - a \right)  b}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C48[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C49
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q,
    {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] 
\\+    {{{{\left( -1 \right) }^{1 - r + s}} \left( a - {b\over q} \right)  z 
       }\over 
     {\left( 1 - a \right)  \left( 1 - b \right)  
       \left( 1 - {b\over q} \right)  \left( 1 - a q \right)  
        }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 2} (1 - \bi )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / a
        {q^2}, b q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z}
        \right ]
\endmultline$$
\Usage Expr/.C49[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C50
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \\\longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right)  
       \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {a\over q} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( {a\over {{q^2}}} - {b\over {{q^2}}} \right)  z 
        }} {{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        {a\over {{q^2}}}, {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{-1
        + r - s}} z} \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right)  
       \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {a\over q} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( {a\over {{q^2}}} - {b\over {{q^2}}} \right)  z 
        }}
{{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over /
        {a\over q}, {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1
        + r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C50[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C51
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, b
    q, \Bi\endmatrix ;q, {\displaystyle q z} \right ] 
\\+    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b q \right)  z 
       }\over 
     {\left( 1 - a \right)  \left( 1 - b \right)  \left( 1 - a q \right)  
       \left( 1 - b q \right)   }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 2} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a b {q^2},
        \qAi\\ \let \over / a {q^2}, b {q^2}, a b q, \qBi\endmatrix ;q,
        {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline$$
\Usage Expr/.C51[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C52
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over
    q}, {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] 
\\-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {{a b}\over q} \right)  z 
       }\over 
     {\left( 1 - a \right)  \left( 1 - b \right)  
       \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right)  q 
        }}
{{\prodl_{i = 1}^{r} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 2} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a b, \qAi\\
        \let \over / a q, b q, {{a b}\over q}, \qBi\endmatrix ;q,
        {\displaystyle {q^{-r + s}} z} \right ]
\endmultline$$
\Usage Expr/.C52[n1,n2].\newline
\rm {\tt m1}, {\tt m2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C53
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, c, \Ai\\ \let \over /
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - b \right)  \left( a - {c\over q} \right)  
       }\over 
     {\left( a - b \right)  \left( 1 - {c\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, {c\over q},
        \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]+ 
   {{\left( 1 - a \right)  \left( b - {c\over q} \right)  
       }\over 
     {\left( -a + b \right)  \left( 1 - {c\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, {c\over q},
        \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C53[m1,m2,m3].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} are the positions of the special
upper parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C54
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / c,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - b \right)  \left( a - c \right)  
       }\over 
     {\left( a - b \right)  \left( 1 - c \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let
        \over / c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ 
   {{\left( 1 - a \right)  \left( b - c \right)  
       }\over 
     {\left( -a + b \right)  \left( 1 - c \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let
        \over / c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C54[m1,m2,n1].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C55
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / c,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( -b + {a\over q} \right)  \left( 1 - {c\over q} \right)  
       }
      \over {\left( 1 - {a\over q} \right)  \left( -b + {c\over q} \right) }} 
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\
        \let \over / {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    + {{\left( 1 - b \right)  \left( {a\over q} - {c\over q} \right)  
       }\over
       {\left( 1 - {a\over q} \right)  \left( b - {c\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q,
        \Ai\\ \let \over / c, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C55[m1,m2,n1].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C56
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, c,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( a - c \right)  \left( 1 - {b\over q} \right)  
       }\over 
     {\left( 1 - c \right)  \left( a - {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
        {b\over q}, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ 
   {{\left( 1 - a \right)  \left( -c + {b\over q} \right)  
       }\over 
     {\left( 1 - c \right)  \left( -a + {b\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / b, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$
\Usage Expr/.C56[m1,n1,n2].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C57
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, c,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( {a\over q} - {b\over q} \right)  \left( 1 - {c\over q} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( -{b\over q} + {c\over q} \right) }
     } 
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / b, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right
        ]
+ {{\left( 1 - {b\over q} \right)  
       \left( {a\over q} - {c\over q} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( {b\over q} - {c\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {b\over q}, c, \Bi\endmatrix ;q, {\displaystyle z} \right
        ]$$
\Usage Expr/.C57[m1,n1,n2].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C58
\Description Contiguous relation in form of a rule.
$$
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, c,
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( -c + {a\over q} \right)  \left( 1 - {b\over q} \right)  
       }\over 
     {\left( 1 - c \right)  \left( {a\over q} - {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
        {b\over q}, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]
+    {{\left( 1 - {a\over q} \right)  \left( -c + {b\over q} \right)  
       }\over
       {\left( 1 - c \right)  \left( -{a\over q} + {b\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
        {a\over q}, b, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]
$$
\Usage Expr/.C58[n1,n2,n3].\newline
\rm {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C59
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c {q^2}}\over a}, b,
   {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {{\left( b - {a\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( 1 - {{c q}\over a} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c
        q}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle q z} \right ] \\+ 
   {{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( 1 - {{c q}\over a} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c
        q}\over a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C59[m1,m2,m3,m4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper 
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C60
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over
   b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       }\over 
     {\left( b - a \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle
        {z\over q}} \right ]  \\  
   -{{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
        }\over 
      {\left( b - a \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b q,
         {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle
         {z\over q}} \right ]
\endmultline
$$
\Usage Expr/.C60[m1,m2,m3,m4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper 
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C61
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over
   b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  \left( 1 - b \right)  c 
       }\over 
     {a b \left( 1 - {c\over {a q}} \right)  
       \left( 1 - {c\over {b q}} \right)  q}}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {c\over {a q}},
        b q, {c\over {b q}}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ + 
   {{\left( 1 - {c\over q} \right)  \left( 1 - {c\over {a b q}} \right)  
       }\over 
     {\left( 1 - {c\over {a q}} \right)  \left( 1 - {c\over {b q}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}}
        {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}}  , a, {c\over {a q}},
        b, {c\over {b q}}, \Ai\\ \let \over / {{{\sqrt{c}}}\over
        {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C61[m1,m2,m3,m4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper 
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C62
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {{c
   q^2}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow 
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
    b/q, {c*q\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z}
    \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( {b\over q} - a \right)  
       \left( 1 - {c*q\over {a b}} \right)  \left( 1 - c q \right)  z 
       }}
   {{\prodl_{i = 1}^{r-4}(1 - \ai)   }\over
     {\prodl_{i = 1}^{s}(1 - \bi)}}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, {{c
        q}\over a}, b, {{c q^2}\over b}, c {q^2}, \qAi\\ \let \over / c q,
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C62[m1,m2,m3,m4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper 
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C63
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c {q^2}}\over a}, b,
   {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {a\over {d q}} \right)  
       \left( 1 - {{c q}\over {a d}} \right)  
       }\over 
     {\left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  
       \left( 1 - {a\over q} \right)  \left( 1 - {{c q}\over a} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c
        q}\over a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ + 
   {{\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       \left( -b + {a\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  d 
       \left( 1 - {a\over q} \right)  \left( 1 - {{c q}\over a} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q},
        {{c q}\over a}, b, {c\over b}, d q, {{c q}\over d}, \Ai\\ \let \over /
        d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C63[m1,m2,m3,m4,$d$].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper 
parameters, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C64
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\
   \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {c\over {a b}} \right)  \left( b - {a\over q} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {a\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over
        a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle
        q z} \right ]\\ + 
   {{\left( 1 - b \right)  \left( 1 - {c\over {b q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {a\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over
        a}, b q, \Ai\\ \let \over / {c\over {b q}}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C64[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace A,B,q*C,D%
\MATHrbrace ,%
\MATHlbrace E,A*C/D,F%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck  A, B, C q, D       %
\MATHruEck 
           %
\MATHvStrich                     %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      A C     ; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich   E, ---, F         %
\MATHvStrich 
           %
\MATHloEck       D             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.C64[3,1,4,2]
\goodbreakpoint%
                  A
             %
\MATHluEck  C, -, D, B         %
\MATHruEck 
             %
\MATHvStrich     q               %
\MATHvStrich   C        A
        D  \MATHphi  %
\MATHvStrich            ; q, q z %
\MATHvStrich  (-; q)  (---; q)
          4 3%
\MATHvStrich  A C                %
\MATHvStrich   D    1  D q    1
             %
\MATHloEck  ---, E, F          %
\MATHroEck 
                D
Out[2]= --------------------------------------------- + 
                                A
                       (C; q)  (-; q)
                             1  q    1
 
             A
        %
\MATHluEck  C, -, D q, B       %
\MATHruEck 
        %
\MATHvStrich     q               %
\MATHvStrich           A C
      \MATHphi  %
\MATHvStrich              ; q, z %
\MATHvStrich  (D; q)  (---; q)
     4 3%
\MATHvStrich   A C               %
\MATHvStrich        1  D q    1
        %
\MATHloEck   ---, E, F         %
\MATHroEck 
           D q
\MATHgroesser     -------------------------------------------
                            A
                   (C; q)  (-; q)
                         1  q    1
\goodbreakpoint%
In[3]:= \%/.pqaufl
\goodbreakpoint%
                                    A
                               %
\MATHluEck  C, -, D, B         %
\MATHruEck 
             C          A      %
\MATHvStrich     q               %
\MATHvStrich 
        (1 - -) D (1 - ---)  \MATHphi  %
\MATHvStrich            ; q, q z %
\MATHvStrich 
             D         D q  4 3%
\MATHvStrich  A C                %
\MATHvStrich 
                               %
\MATHloEck  ---, E, F          %
\MATHroEck 
                                  D
Out[3]= --------------------------------------------- + 
                                    A
                       (1 - C) (1 - -)
                                    q
 
                               A
                          %
\MATHluEck  C, -, D q, B       %
\MATHruEck 
                  A C     %
\MATHvStrich     q               %
\MATHvStrich 
     (1 - D) (1 - ---)  \MATHphi  %
\MATHvStrich              ; q, z %
\MATHvStrich 
                  D q  4 3%
\MATHvStrich   A C               %
\MATHvStrich 
                          %
\MATHloEck   ---, E, F         %
\MATHroEck 
                             D q
\MATHgroesser     -------------------------------------------
                                A
                   (1 - C) (1 - -)
                                q
\endMATH
\rm
The third, first, and fourth upper parameters in {\tt Out[1]} are {\tt
q C}, {\tt A}, and {\tt D}, respectively, the second lower parameter
in {\tt Out[1]} is $\frac {\text{\tt A C}}{\text {{\tt D}}}$. Hence {\tt C64} can be applied with
the replacements $a\to \text {{\tt q C}}$, $b\to \text {{\tt D}}$ and
$c\to \text {{\tt A C}}$.
\Seealso ContigListe, Ers, PosListe.

\Name C65
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let
   \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       }\over 
     {\left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]  \\  
   -{{\left( {b\over q}-a \right)  \left( 1 - {{c q}\over {a b}} \right)  
        }\over 
      {\left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a},
         {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q,
         {\displaystyle q z} \right ]
\endmultline
$$
\Usage Expr/.C65[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C66
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let
   \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       }\over 
     {\left( b - a \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle
        {z\over q}} \right ]  \\  
   -{{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
        }\over 
      {\left( b - a \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b q,
         \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle
         {z\over q}} \right ]
\endmultline
$$
\Usage Expr/.C66[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C67
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\
   \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  \left( 1 - b \right)  c 
       }\over 
     {a b \left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {c\over a}, b q,
        \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
        \right ]\\ + 
   {{\left( 1 - c \right)  \left( 1 - {c\over {a b}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} q,
        - {\sqrt{c}} q  , a, {c\over a}, b, \Ai\\ \let \over /
        {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C67[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C68
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let
   \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{a b \left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
       }\over 
     {c \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right)  q}}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c
        q}\over a}, {b\over q}, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]\\ - 
   {{a b \left( 1 - {c\over q} \right)  
       \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {c \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right)  q}}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}}
        {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}}  , {a\over q},
        {c\over a}, {b\over q}, \Ai\\ \let \over / {{{\sqrt{c}}}\over
        {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, {{c q}\over b},
        \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C68[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C69
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let
   \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow   
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
    {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q,
    {\displaystyle z} \right ]\\ + 
   {{{{\left( -1 \right) }^{1 - r + s}} \left( {b\over q}-a \right)  
       \left( 1 - c q \right)  \left( 1 - {{c q}\over {a b}} \right)  z 
       }\over 
     {\left( 1 - {{c q}\over b} \right)  
       \left( 1 - {{c {q^2}}\over b} \right)   }}
   {{\prodl_{i = 1}^{r-3}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-1}(1 - \bi) }}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, {{c
        q}\over a}, b, c {q^2}, \qAi\\ \let \over / {{c {q^3}}\over b}, c q,
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C69[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C70
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\
   \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow   
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over a}, b
    q, \Ai\\ \let \over / {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z}
    \right ]\\ - {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - c \right)  
       \left( 1 - {c\over {a b}} \right)  \left( b - {a\over q} \right)  z 
       }\over 
     {\left( 1 - {c\over b} \right)  \left( 1 - {c\over {b q}} \right)   }}
   {{\prodl_{i = 1}^{r-3}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-1}(1 - \bi) }}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a, {{c
        q}\over a}, b q, c q, \qAi\\ \let \over / {{c q}\over b}, c,
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C70[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C71
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\
   \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - b \right)  \left( 1 - {c\over {a d}} \right)  
       \left( 1 - {c\over {b q}} \right)  \left( 1 - {a\over {d q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {b\over d} \right)  
       \left( 1 - {a\over q} \right)  \left( 1 - {c\over {b d q}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over
        a}, b q, \Ai\\ \let \over / {c\over {b q}}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ +
     {{\left( 1 - {c\over {a b}} \right)  \left( 1 - d \right)  
       \left( -b + {a\over q} \right)  \left( 1 - {c\over {d q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {b\over d} \right)  d 
       \left( 1 - {a\over q} \right)  \left( 1 - {c\over {b d q}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q},
        {c\over a}, b, d q, {c\over d}, \Ai\\ \let \over / {c\over b}, d,
        {c\over {d q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C71[m1,m2,m3,n1,$d$].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C72
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let
   \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {{c q}\over {b d}} \right)  
       }\over 
     {\left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  
       \left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       \left( a - {b\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d 
       \left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / a, {c\over
        a}, {b\over q}, d q, {{c q}\over d}, \Ai\\ \let \over / {{c
        {q^2}}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z}
        \right ]
\endmultline
$$
\Usage Expr/.C72[m1,m2,m3,n1,$d$].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C73
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, a b, \Ai\\ \let \over
   / {{a b}\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {{a b}\over q} \right)  z 
        }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r - 3} (1 - {{\ai}\over q} )}}
{} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / a, b,
        \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}}
        z} \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {{a b}\over q} \right)  z 
        }}
{{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r - 3} (1 - {{\ai}\over q} )}}
{} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / {a\over q},
        {b\over q}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle
        {q^{r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C73[m1,m2,m3,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C74
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {{a b}\over c} \right)  c \left( -b + {a\over q} \right)  
       }\over 
     {a b \left( 1 - {c\over a} \right)  \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\
        \let \over / {{c q}\over a}, {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle q z} \right ]
\\+    {{\left( 1 - b \right)  \left( 1 - {c\over {b q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {a\over q} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q,
        \Ai\\ \let \over / {{c q}\over a}, {c\over {b q}}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline$$
\Usage Expr/.C74[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C75
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
    {{\left( 1 - a \right)  a b \left( 1 - {c\over {a q}} \right)  q 
       }\over 
     {\left( a - b \right)  c \left( 1 - {{a b q}\over c} \right) }}
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let
        \over / {c\over {a q}}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle
        {z\over q}} \right ]
\\  -{{a \left( 1 - b \right)  b \left( 1 - {c\over {b q}} \right)  q 
        }\over 
      {\left( a - b \right)  c \left( 1 - {{a b q}\over c} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let
         \over / {c\over a}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle
         {z\over q}} \right ]
\endmultline$$
\Usage Expr/.C75[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C76
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  \left( 1 - b \right)  c 
       }\over 
     {a b \left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b q, \Ai\\ \let
        \over / {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ + 
   {{\left( 1 - c \right)  \left( 1 - {c\over {a b}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} q,
        - {\sqrt{c}} q  , a, b, \Ai\\ \let \over / {\sqrt{c}},
        -{\sqrt{c}}, {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C76[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C77
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{a b \left( 1 - {c\over {a q}} \right)  \left( 1 - {c\over {b q}} \right)  
       }\over 
     {c \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {b\over
        q}, \Ai\\ \let \over / {c\over {a q}}, {c\over {b q}}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]\\ - 
   {{a b \left( 1 - {c\over {a b}} \right)  
       \left( 1 - {c\over {{q^2}}} \right)  
       }\over 
     {c \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}},
        -{\sqrt{c}}, {a\over q}, {b\over q}, \Ai\\ \let \over /
        {{{\sqrt{c}}}\over q}, -{{{\sqrt{c}}}\over q}, {c\over a}, {c\over b},
        \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C77[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C78
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {b\over q}, \Ai\\
    \let \over / {c\over {a q}}, {{c q}\over b}, \Bi\endmatrix ;q,
    {\displaystyle z} \right ] 
\\+    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - c \right)  
       \left( 1 - {c\over {a b}} \right)  \left( -a + {b\over q} \right)  z 
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {c\over {a q}} \right)  \left( 1 - {{c q}\over b} \right)  
        }}
{{\prodl_{i = 1}^{r - 2} (1 - \ai )}\over
{\prodl_{i = 1}^{s - 2} (1 - \bi )}}
{} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a q, b, c q,
        \qAi\\ \let \over / {{c q}\over a}, {{c {q^2}}\over b}, c,
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline$$
\Usage Expr/.C78[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C79
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over /
   {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - b \right)  \left( 1 - {c\over {a d}} \right)  
       \left( 1 - {c\over {b q}} \right)  \left( 1 - {a\over {d q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {b\over d} \right)  
       \left( 1 - {a\over q} \right)  \left( 1 - {c\over {b d q}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q,
        \Ai\\ \let \over / {{c q}\over a}, {c\over {b q}}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ +
     {{\left( 1 - {c\over {a b}} \right)  \left( 1 - d \right)  
       \left( -b + {a\over q} \right)  \left( 1 - {c\over {d q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {b\over d} \right)  d 
       \left( 1 - {a\over q} \right)  \left( 1 - {c\over {b d q}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q},
        b, d q, {c\over d}, \Ai\\ \let \over / {{c q}\over a}, {c\over b}, d,
        {c\over {d q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C79[m1,m2,n1,n2,$d$].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace A,B,B*F/E,D%
\MATHrbrace ,%
\MATHlbrace E,F%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck        B F          %
\MATHruEck 
           %
\MATHvStrich  A, B, ---, D       %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich         E    ; q, z %
\MATHvStrich 
        4 2%
\MATHvStrich                     %
\MATHvStrich 
           %
\MATHloEck      E, F           %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.C79[2,3,2,1,x]
\goodbreakpoint%
                                                 B  B F q
                                               %
\MATHluEck  -, -----, A, D       %
\MATHruEck 
             B F       E       F        B      %
\MATHvStrich  q    E               %
\MATHvStrich 
        (1 - ---) (1 - -) (1 - -) (1 - ---)  \MATHphi  %
\MATHvStrich                ; q, z %
\MATHvStrich 
              E        q       x       q x  4 2%
\MATHvStrich           E           %
\MATHvStrich 
                                               %
\MATHloEck      F q, -           %
\MATHroEck 
                                                          q
Out[2]= --------------------------------------------------------------- + 
                                   B       B F        E
                      (1 - F) (1 - -) (1 - ---) (1 - ---)
                                   q       E x       q x
\goodbreakpoint% 
\MATHgroesser     
 
                                                  B  B F       B F
                                                %
\MATHluEck  -, ---, q x, ---, A, D       %
\MATHruEck 
           E     B F    B       B F             %
\MATHvStrich  q   E         x              %
\MATHvStrich 
      (1 - -) (-(---) + -) (1 - ---) (1 - x)  \MATHphi  %
\MATHvStrich                        ; q, z %
\MATHvStrich 
           B      E     q       q x          6 4%
\MATHvStrich                 B F           %
\MATHvStrich 
                                                %
\MATHloEck      F q, E, x, ---           %
\MATHroEck 
                                                                 q x
\MATHgroesser      --------------------------------------------------------------------------
                                     B       B F        E
                        (1 - F) (1 - -) (1 - ---) (1 - ---) x
                                     q       E x       q x
\endMATH
\rm
The second and third upper parameters in {\tt Out[1]} are {\tt
B}, and $\frac {\text{\tt B F}}{\text {\tt E}}$, respectively, the second and first lower
parameters in {\tt Out[1]} are {\tt F} and {\tt E}, respectively. 
Hence {\tt C79} can be applied with
the replacements $a\to \text {{\tt B}}$, $b\to 
\frac {\text{\tt B F}}{\text {\tt E}}$ and
$c\to \text {{\tt B F}}$. The {\tt x} replaces $d$ in the right-hand
side expression.
\Seealso ContigListe, Ers, PosListe.

\Name C80
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( 1 - {c\over {a q}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over
        {a q}}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle q z} \right ]\\ + 
   {{\left( 1 - {b\over q} \right)  \left( 1 - {c\over {b q}} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( 1 - {c\over {a q}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over
        {a q}}, \Ai\\ \let \over / {b\over q}, {c\over {b q}}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C80[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C81
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        \Ai\\ \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]  \\  
   -{{\left( b - a \right)  \left( 1 - {c\over {a b}} \right)  
        }\over 
      {\left( 1 - b \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\
         \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle q
         z} \right ]
\endmultline
$$
\Usage Expr/.C81[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C82
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       }\over 
     {\left( {b\over q}-a \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        \Ai\\ \let \over / b, {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle {z\over q}} \right ]\\
    -{{\left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right)  
        }\over 
      {\left( {b\over q}-a \right)  \left( 1 - {{c q}\over {a b}} \right) }
      }
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\
         \let \over / {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q,
         {\displaystyle {z\over q}} \right ] 
\endmultline
$$
\Usage Expr/.C82[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C83
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  c \left( 1 - {b\over q} \right)  
       }\over 
     {a b \left( 1 - {c\over b} \right)  \left( 1 - {c\over {a q}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {c\over {a q}},
        \Ai\\ \let \over / {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ 
    + {{\left( 1 - {c\over {a b}} \right)  \left( 1 - {c\over q} \right)  
       }\over 
     {\left( 1 - {c\over b} \right)  \left( 1 - {c\over {a q}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}}
        {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}}  , a, {c\over {a q}},
        \Ai\\ \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}},
        -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, b, {{c q}\over b}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C83[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C84
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, \Ai\\
    \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
    \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( b - a \right)  
       \left( 1 - {c\over {a b}} \right)  \left( 1 - c q \right)  z 
       }
      \over {\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - b q \right)  \left( 1 - {{c q}\over b} \right)  
        }}
   {{\prodl_{i = 1}^{r-2}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-2}(1 - \bi) }}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, {{c
        q}\over a}, c {q^2}, \qAi\\ \let \over / b {q^2}, {{c {q^2}}\over b},
        c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C84[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C85
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over {a
    q}}, \Ai\\ \let \over / {b\over q}, {c\over {b q}}, \Bi\endmatrix ;q,
    {\displaystyle z} \right ]\\ - 
   {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {c\over q} \right)  z 
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {b\over q} \right)  \left( 1 - {c\over {b q}} \right)  
        }}
   {{\prodl_{i = 1}^{r-2}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-2}(1 - \bi) }}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a, {c\over
        a}, c, \qAi\\ \let \over / b q, {{c q}\over b}, {c\over q},
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C85[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C86
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {b\over q} \right)  \left( 1 - {c\over {b q}} \right)  
       \left( 1 - {a\over {d q}} \right)  
       \left( 1 - {c\over {a d q}} \right)  
       }\over 
     {\left( 1 - {a\over q} \right)  \left( 1 - {c\over {a q}} \right)  
       \left( 1 - {b\over {d q}} \right)  \left( 1 - {c\over {b d q}} \right) 
       }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over
        {a q}}, \Ai\\ \let \over / {b\over q}, {c\over {b q}}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over {a b}} \right)  \left( 1 - d \right)  
       \left( 1 - {c\over {d {q^2}}} \right)  
       \left( {a\over q} - {b\over q} \right)  
       }\over {d \left( 1 - {a\over q} \right)  
       \left( 1 - {c\over {a q}} \right)  \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {c\over {b d q}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q},
        {c\over {a q}}, d q, {c\over {d q}}, \Ai\\ \let \over / b, {c\over b},
        d, {c\over {d {q^2}}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C86[m1,m2,n1,n2,$d$].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C87
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let
   \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a},
        \Ai\\ \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
       \left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d}}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / a, {c\over
        a}, d q, {{c q}\over d}, \Ai\\ \let \over / b q, {{c q}\over b}, d,
        {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C87[m1,m2,n1,n2,$d$].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C88
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{a b}\over q}, \Ai\\
   \let \over / b, {{a b}\over {{q^2}}}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} 
       \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {b\over q} \right)  
       {q^{1 - r + s}} 
         }\over 
     {\left( 1 - {{a b}\over {{q^2}}} \right)  z 
        }} {{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}}
{} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / a, \Aiq\\
        \let \over / {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle
        {q^{-1 + r - s}} z} \right ]\\- 
   {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {{a b}\over {{q^2}}} \right)  z 
        }}
{{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}}
{} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / {a\over q},
        \Aiq\\ \let \over / {b\over q}, \Biq\endmatrix ;q, {\displaystyle
        {q^{r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C88[m1,m2,n1,n2].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C89
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {{\left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {a\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {{c q}\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle q z} \right ]\\ + 
   {{\left( 1 - {c\over b} \right)  \left( 1 - {b\over q} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {a\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {{c q}\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C89[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C90
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c
   q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       }\over {\left( 1 - b \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / {c\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
        \right ]  \\  
   -{{\left( b - a \right)  \left( 1 - {c\over {a b}} \right)  
        }\over 
      {\left( 1 - b \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over
         / {{c q}\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q,
         {\displaystyle q z} \right ]
\endmultline
$$
\Usage Expr/.C90[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C91
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow   {{\left( 1 - a \right)  \left( 1 - {c\over {a q}} \right)  
       }\over 
     {\left( 1 - {c\over {a b}} \right)  \left( {b\over q}-a \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / {c\over {a q}}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle
        {z\over q}} \right ]   \\  
   -{{\left( 1 - {c\over b} \right)  \left( 1 - {b\over q} \right)  
        }\over 
      {\left( 1 - {c\over {a b}} \right)  \left( {b\over q}-a \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over
         / {c\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q,
         {\displaystyle {z\over q}} \right ]
\endmultline
$$
\Usage Expr/.C91[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C92
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {{\left( 1 - a \right)  c \left( 1 - {b\over q} \right)  q 
       }\over 
     {a b \left( 1 - {c\over a} \right)  \left( 1 - {{c q}\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / {{c q}\over a}, {b\over q}, {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ 
    + {{\left( 1 - c \right)  \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {{c q}\over b} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} q,
        - {\sqrt{c}} q  , a, \Ai\\ \let \over / {\sqrt{c}},
        -{\sqrt{c}}, {{c q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C92[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C93
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c
   q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{a b \left( 1 - {c\over a} \right)  \left( 1 - {c\over {b q}} \right)  
       }\over 
     {\left( 1 - b \right)  c \left( 1 - {a\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {c\over a}, b q, {c\over {b q}}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{a b \left( 1 - {c\over {a b}} \right)  \left( 1 - {c\over q} \right)  
       }\over 
     {\left( 1 - b \right)  c \left( 1 - {a\over q} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}}
        {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}}  , {a\over q}, \Ai\\
        \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over
        {{\sqrt{q}}}}, {{c q}\over a}, b q, {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C93[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C94
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c
   q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow   
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over /
    {c\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
    \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( b - a \right)  
       \left( 1 - {c\over {a b}} \right)  \left( 1 - c q \right)  z 
       }\over
       {\left( 1 - b \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {c\over b} \right)  \left( 1 - b q \right)  
       \left( 1 - {{c q}\over a} \right)  \left( 1 - {{c q}\over b} \right)  
        }}
   {{\prodl_{i = 1}^{r-1}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-3}(1 - \bi) }}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, c {q^2},
        \qAi\\ \let \over / {{c {q^2}}\over a}, b {q^2}, {{c {q^2}}\over b}, c
        q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C94[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C95
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow 
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let
    \over / {{c q}\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q,
    {\displaystyle z} \right ]\\ - 
   {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - c \right)  
       \left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {{c q}\over {a b}} \right)  z 
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {c\over b} \right)  \left( 1 - {b\over q} \right)  
       \left( 1 - {{c q}\over a} \right)  \left( 1 - {{c q}\over b} \right)  
        }}
   {{\prodl_{i = 1}^{r-1}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-3}(1 - \bi) }}
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a, c q,
        \qAi\\ \let \over / {{c {q^2}}\over a}, b q, {{c {q^2}}\over b}, c,
        \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C95[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C96
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over /
   {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {{\left( 1 - {c\over b} \right)  \left( 1 - {c\over {a d}} \right)  
       \left( 1 - {b\over q} \right)  \left( 1 - {a\over {d q}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {c\over {b d}} \right)  
       \left( 1 - {a\over q} \right)  \left( 1 - {b\over {d q}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\
        \let \over / {{c q}\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ + 
   {{\left( 1 - d \right)  \left( {a\over q} - {b\over q} \right)  
       \left( 1 - {c\over {d q}} \right)  
       \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {c\over {b d}} \right)  d 
       \left( 1 - {a\over q} \right)  \left( 1 - {b\over {d q}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q}, d
        q, {c\over d}, \Ai\\ \let \over / {{c q}\over a}, b, {{c q}\over b},
        d, {c\over {d q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C96[m1,n1,n2,n3,$d$].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C97
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c
   q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over
        / {c\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
        \right ]\\ - 
   {{\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
       \left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       }\over 
     {\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d}}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / a, d q, {{c
        q}\over d}, \Ai\\ \let \over / {{c q}\over a}, b q, {{c q}\over b}, d,
        {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C97[m1,n1,n2,n3,$d$].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C98
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {{a b}\over {{q^2}}}, \Ai\\
   \let \over / a, b, {{a b}\over {{q^3}}}, \Bi\endmatrix ;q, {\displaystyle
   z} \right ] \\\longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} 
       \left( 1 - {a\over {{q^2}}} \right)  
       \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {a\over q} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {{a b}\over {{q^3}}} \right)  z 
        }} {{\prodl_{i = 1}^{s - 3} (1 - {{\bi}\over q}
)}\over
{\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}}
{} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / \Aiq\\ \let
        \over / {a\over {{q^2}}}, {b\over {{q^2}}}, \Biq\endmatrix ;q,
        {\displaystyle {q^{-1 + r - s}} z} \right ]
\\-    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right)  
       \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {a\over q} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {{a b}\over {{q^3}}} \right)  z 
        }}
{{\prodl_{i = 1}^{s - 3} (1 - {{\bi}\over q} )}\over
{\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}}
{} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / \Aiq\\ \let
        \over / {a\over q}, {b\over q}, \Biq\endmatrix ;q, {\displaystyle
        {q^{r - s}} z} \right ]
\endmultline$$
\Usage Expr/.C98[m1,n1,n2,n3].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower
parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C99
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
   {c\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \longrightarrow {{\left( {b\over q}-a \right)  
       \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - a \right)  \left( 1 - {c\over a} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, {{c q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle q z} \right ]\\ + 
   {{\left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( 1 - a \right)  \left( 1 - {c\over a} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C99[n1,n2,n3,n4].\newline
\rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C100
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
   {c\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - {a\over q} \right)  \left( 1 - {c\over {a q}} \right)  
       }\over 
     {\left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
        {a\over q}, {c\over {a q}}, b, {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle {z\over q}} \right ]  \\  
   -{{\left( 1 - {b\over q} \right)  \left( 1 - {c\over {b q}} \right)  
        }\over 
      {\left( 1 - {c\over {a b}} \right)  
        \left( -{a\over q} + {b\over q} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
         a, {c\over a}, {b\over q}, {c\over {b q}}, \Bi\endmatrix ;q,
         {\displaystyle {z\over q}} \right ]
\endmultline
$$
\Usage Expr/.C100[n1,n2,n3,n4].\newline
\rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C101
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
   {c\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{c \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right)  q 
       }\over 
     {a b \left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
        {a\over q}, {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]\\ + 
   {{\left( 1 - {c\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}}
        {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}}  , \Ai\\ \let \over /
        {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}},
        a, {{c q}\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle
        z} \right ]
\endmultline
$$
\Usage Expr/.C101[n1,n2,n3,n4].\newline
\rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C102
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {{c
   {q^2}}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
    \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over /
    {a\over q}, {{c q}\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q,
    {\displaystyle z} \right ]\\ + 
   {{{{\left( -1 \right) }^{1 - r + s}} \left( b - {a\over q} \right)  
       \left( 1 - c q \right)  \left( 1 - {{c q}\over {a b}} \right)  z 
       }\over {\left( 1 - a \right)  \left( 1 - b \right)  
       \left( 1 - {c\over b} \right)  \left( 1 - {a\over q} \right)  
       \left( 1 - b q \right)  \left( 1 - {{c q}\over a} \right)  
       \left( 1 - {{c q}\over b} \right)  
       \left( 1 - {{c {q^2}}\over a} \right)  
        }}
   {{\prodl_{i = 1}^{r}(1 - \ai)   }\over
    {\prodl_{i = 1}^{s-4}(1 - \bi) }}\\
   {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / c {q^2},
        \qAi\\ \let \over / a q, {{c {q^3}}\over a}, b {q^2}, {{c {q^2}}\over
        b}, c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]
\endmultline
$$
\Usage Expr/.C102[n1,n2,n3,n4].\newline
\rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C103
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a,
   {c\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \longrightarrow {{\left( 1 - {a\over d} \right)  
       \left( 1 - {c\over {a d}} \right)  \left( 1 - {b\over q} \right)  
       \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {{c q}\over {b d}} \right) }}
   {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a
        q, {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ + 
   {{\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       \left( a - {b\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
       }\over 
     {\left( 1 - a \right)  \left( 1 - {c\over a} \right)  d 
       \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {{c q}\over {b d}} \right) }}
   {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / d q, {{c
        q}\over d}, \Ai\\ \let \over / a q, {{c q}\over a}, b, {{c {q^2}}\over
        b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C103[n1,n2,n3,n4,$d$].\newline
\rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower
parameters, $d$ is the additional parameter at the right hand side.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C104
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over
   b}, c, \Ai\\ \let \over / {c\over q}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \\
  \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {c\over q} \right)  z 
        }}
   {{\prodl_{i = 1}^{s-1}(1 - {{\bi}\over q} ) }\over
    {\prodl_{i = 1}^{r-5}(1 - {{\ai}\over q} )}}
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q},
        {c\over {a q}}, b, {c\over b}, \Aiq\\ \let \over / \Biq\endmatrix ;q,
        {\displaystyle {q^{-1+r-s}} z} \right ]\\
    -{{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} 
        }\over 
      {\left( 1 - {c\over {a b}} \right)  
        \left( -{a\over q} + {b\over q} \right)  
        \left( 1 - {c\over q} \right)  z 
        }}
   {{\prodl_{i = 1}^{s-1}(1 - {{\bi}\over q}  ) }\over
    {\prodl_{i = 1}^{r-5}(1 - {{\ai}\over q}  )}}
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {c\over
         a}, {b\over q}, {c\over {b q}}, \Aiq\\ \let \over / \Biq\endmatrix
         ;q, {\displaystyle {q^{-1+r-s}} z} \right ]
\endmultline
$$
\Usage Expr/.C104[m1,m2,m3,m4,m5,n1].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5} and {\tt n1} are the positions of the special upper and
lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C105
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, c, \Ai\\
   \let \over / {{c q}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle
   z} \right ]\\
   \longrightarrow   {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over b} \right)  
       \left( 1 - {c\over {b q}} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {c\over q} \right)  z 
         }}
   {{\prodl_{i = 1}^{s-2}(1 - {{\bi}\over q})   }\over
    {\prodl_{i = 1}^{r-4}(1 - {{\ai}\over q})}}
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q},
        {c\over {a q}}, b, \Aiq\\ \let \over / {c\over {b q}}, \Biq\endmatrix
        ;q, {\displaystyle {q^{-1+r-s}} z} \right ]  \\  
   -{{{{\left( -1 \right) }^{1 - r + s}} 
        \left( 1 - {c\over b} \right)  \left( 1 - {c\over {b q}} \right)  
        {q^{1 - r + s}} 
            }\over 
      {\left( 1 - {c\over {a b}}   \right)
        \left( -{a\over q} + {b\over q} \right)  
        \left( 1 - {c\over q} \right)  z 
         }}
   {{\prodl_{i = 1}^{s-2}(1 - {{\bi}\over q}) }\over
    {\prodl_{i = 1}^{r-4}(1 - {{\ai}\over q})}}
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a,
         {c\over a}, {b\over q}, \Aiq\\ \let \over / {c\over b},
         \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]
\endmultline
$$
\Usage Expr/.C105[m1,m2,m3,m4,n1,n2].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2} are the positions of the special upper and
lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C106
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, c, \Ai\\ \let \over /
   {{c q}\over a}, {{c q}\over b}, {c\over q}, \Bi\endmatrix ;q,
   {\displaystyle z} \right ] \longrightarrow \\
   {{{{\left( -1 \right) }^{1 - r + s}}
       \left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
       \left( 1 - {c\over {a q}} \right)  
       \left( 1 - {c\over {b q}} \right)  
       {{ q  }^{1 - r + s}} 
      }\over 
     {\left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {c\over q} \right)  z 
      }}
    {{ \prodl_{i = 1}^{s-3}1 - {{\bi}\over q} }\over
     { \prodl_{i = 1}^{r-3}1 - {{\ai}\over q}  }}
  {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q},
        b, \Aiq\\ \let \over / {c\over a}, {c\over {b q}}, \Biq\endmatrix ;q,
        {\displaystyle {q^{-1 + r - s}} z} \right ]\\
  -{{{{ (-1) }^{1 - r + s}}
        \left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
        \left( 1 - {c\over {a q}} \right)  
        \left( 1 - {c\over {b q}} \right)  
        {{ q }^{1 - r + s}} 
       }\over 
      {\left( 1 - {c\over {a b}} \right)  
        \left( -{a\over q} + {b\over q} \right)  
        \left( 1 - {c\over q} \right)  z 
        }}
      {{ \prodl_{i = 1}^{s-3}1 - {{\bi}\over q}  }\over
       { \prodl_{i = 1}^{r-3}1 - {{\ai}\over q}  }}
        {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {b\over
         q}, \Aiq\\ \let \over / {c\over {a q}}, {c\over b}, \Biq\endmatrix
         ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]
\endmultline
$$
\Usage Expr/.C106[m1,m2,m3,n1,n2,n3].
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3} 
are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C107
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, c, \Ai\\ \let
   \over / b q, {{c q}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle
   z} \right ] \\
  \longrightarrow    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - b \right)  
       \left( 1 - {c\over b} \right)  \left( 1 - {b\over q} \right)  
       \left( 1 - {c\over {b q}} \right)  {q^{1 - r + s}} 
       }\over 
     {\left( 1 - {c\over {a b}} \right)  
       \left( -{a\over q} + {b\over q} \right)  
       \left( 1 - {c\over q} \right)  z 
        }}
   {{\prodl_{i = 1}^{s-3}(1 - {{\bi}\over q} )  }\over
    {\prodl_{i = 1}^{r-3}(1 - {{\ai}\over q} )}}
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q},
        {c\over {a q}}, \Aiq\\ \let \over / {b\over q}, {c\over {b q}},
        \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\ 
   -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - b \right)  
        \left( 1 - {c\over b} \right)  \left( 1 - {b\over q} \right)  
        \left( 1 - {c\over {b q}} \right)  {q^{1 - r + s}} 
        }\over 
      {\left( 1 - {c\over {a b}} \right)  
        \left( -{a\over q} + {b\over q} \right)  
        \left( 1 - {c\over q} \right)  z 
         }}
   {{\prodl_{i = 1}^{s-3}(1 - {{\bi}\over q} )  }\over
    {\prodl_{i = 1}^{r-3}(1 - {{\ai}\over q} )}}
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {c\over
         a}, \Aiq\\ \let \over / b, {c\over b}, \Biq\endmatrix ;q,
         {\displaystyle {q^{-1+r-s}} z} \right ]
\endmultline
$$
\Usage Expr/.C107[m1,m2,m3,n1,n2,n3].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and
lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C108
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, c, \Ai\\ \let \over / {{c
   q}\over a}, b, {{c {q^2}}\over b}, {c\over q}, \Bi\endmatrix ;q,
   {\displaystyle z} \right ] \\
  \longrightarrow    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right)  
       \left( 1 - {c\over b} \right)  \left( 1 - {b\over {{q^2}}} \right)  
       \left( 1 - {b\over q} \right)  \left( 1 - {c\over {a q}} \right)  
       {q^{1 - r + s}} \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( {b\over {{q^2}}} - {a\over q} \right)  
       \left( 1 - {c\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
       z  }}
   {{\prodl_{i = 1}^{s-4}(1 - {{\bi}\over q} )  }\over
    {\prodl_{i = 1}^{r-2}(1 - {{\ai}\over q} )}}\\
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q},
        \Aiq\\ \let \over / {c\over a}, {b\over {{q^2}}}, {c\over b},
        \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\  
   -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right)  
        \left( 1 - {c\over b} \right)  \left( 1 - {b\over {{q^2}}} \right)  
        \left( 1 - {b\over q} \right)  \left( 1 - {c\over {a q}} \right)  
        {q^{1 - r + s}} \left( 1 - {{c q}\over b} \right)  
        }\over 
      {\left( {b\over {{q^2}}} - {a\over q} \right)  
        \left( 1 - {c\over q} \right)  \left( 1 - {{c q}\over {a b}} \right)  
        z  }}
   {{\prodl_{i = 1}^{s-4}(1 - {{\bi}\over q} )  }\over
    {\prodl_{i = 1}^{r-2}(1 - {{\ai}\over q} )}}\\
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, \Aiq\\
         \let \over / {c\over {a q}}, {b\over q}, {{c q}\over b},
         \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]
\endmultline
$$
\Usage Expr/.C108[m1,m2,n1,n2,n3,n4].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper and
lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C109
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / c, \Ai\\ \let \over / a, {{c
   {q^2}}\over a}, b, {{c {q^2}}\over b}, {c\over q}, \Bi\endmatrix ;q,
   {\displaystyle z} \right ] \\
  \longrightarrow    {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right)  
       \left( 1 - {c\over b} \right)  \left( 1 - {a\over {{q^2}}} \right)  
       \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {a\over q} \right)  
       \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
       \left( 1 - {{c q}\over a} \right)  \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( -{a\over {{q^2}}} + {b\over {{q^2}}} \right)  
       \left( 1 - {c\over q} \right)  
       \left( 1 - {{c {q^2}}\over {a b}} \right)  z 
        }}
   {{\prodl_{i = 1}^{s-5}(1 - {{\bi}\over q} )  }\over
    {\prodl_{i = 1}^{r-1}(1 - {{\ai}\over q} )}}\\
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / \Aiq\\ \let
        \over / {a\over q}, {{c q}\over a}, {b\over {{q^2}}}, {c\over b},
        \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\ 
  -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right)  
        \left( 1 - {c\over b} \right)  \left( 1 - {a\over {{q^2}}} \right)  
        \left( 1 - {b\over {{q^2}}} \right)  \left( 1 - {a\over q} \right)  
        \left( 1 - {b\over q} \right)  {q^{1 - r + s}} 
        \left( 1 - {{c q}\over a} \right)  \left( 1 - {{c q}\over b} \right)  
        }\over {\left( -{a\over {{q^2}}} + {b\over {{q^2}}} \right)  
        \left( 1 - {c\over q} \right)  
        \left( 1 - {{c {q^2}}\over {a b}} \right)  z 
         }}
   {{\prodl_{i = 1}^{s-5}(1 - {{\bi}\over q} )  }\over
    {\prodl_{i = 1}^{r-1}(1 - {{\ai}\over q} )}}\\
   {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / \Aiq\\
         \let \over / {a\over {{q^2}}}, {c\over a}, {b\over q}, {{c q}\over
         b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]
\endmultline
$$
\Usage Expr/.C109[m1,n1,n2,n3,n4,n5].\newline
\rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5} are the positions of the special upper and
lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C110
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, -
   {\sqrt{c}} q  , a, {c\over a}, b, {c\over b}, \Ai\\ \let \over /
   {\sqrt{c}}, -{\sqrt{c}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
  {{\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
       }\over 
     {\left( 1 - c \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {{c
        q}\over a}, b, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{\left( 1 - a \right)  \left( 1 - b \right)  c 
       }\over 
     {a b \left( 1 - c \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q,
        {c\over a}, b q, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C110[m1,m2,m3,m4,m5,m6,n1,n2].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5}, {\tt m6} and {\tt n1}, {\tt n2} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C111
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over
   b}, d q, {{c q}\over d}, \Ai\\ \let \over / d, {c\over d}, \Bi\endmatrix
   ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  d 
       }\over 
     {\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
       \left( 1 - {c\over d} \right)  \left( 1 - d \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {{c
        q}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
        {\displaystyle z} \right ]  \\  
   -{{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
        \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d 
        }\over 
      {\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
        \left( 1 - {c\over d} \right)  \left( 1 - d \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {c\over
         a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q,
         {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C111[m1,m2,m3,m4,m5,m6,n1,n2].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5}, {\tt m6} and {\tt n1}, {\tt n2} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C112
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, -
   {\sqrt{c}} q  , a, {c\over a}, b, \Ai\\ \let \over / {\sqrt{c}},
   -{\sqrt{c}}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow
     {{\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
       }\over 
     {\left( 1 - c \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {{c
        q}\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{\left( 1 - a \right)  \left( 1 - b \right)  c 
       }\over 
     {a b \left( 1 - c \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q,
        {c\over a}, b q, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C112[m1,m2,m3,m4,m5,n1,n2,n3].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C113
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, d q, {{c
   q}\over d}, \Ai\\ \let \over / {{c q}\over b}, d, {c\over d}, \Bi\endmatrix
   ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  d 
       }\over 
     {\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
       \left( 1 - {c\over d} \right)  \left( 1 - d \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {{c
        q}\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]  \\  
   -{{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
        \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d 
        }\over 
      {\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
        \left( 1 - {c\over d} \right)  \left( 1 - d \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {c\over
         a}, b q, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q,
         {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C113[m1,m2,m3,m4,m5,n1,n2,n3].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C114
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, -
   {\sqrt{c}} q  , a, b, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}},
   {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right
   ] \longrightarrow {{\left( 1 - {c\over a} \right)  \left( 1 - {c\over b} \right)  
       }\over 
     {\left( 1 - c \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, b, \Ai\\
        \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle
        z} \right ]\\ - 
   {{\left( 1 - a \right)  \left( 1 - b \right)  c 
       }\over 
     {a b \left( 1 - c \right)  \left( 1 - {c\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, b q,
        \Ai\\ \let \over / {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C114[m1,m2,m3,m4,n1,n2,n3,n4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C115
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, -
   {\sqrt{c}} q  , a, {c\over a}, \Ai\\ \let \over / {\sqrt{c}},
   -{\sqrt{c}}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \longrightarrow {{\left( 1 - {c\over a} \right)  
       \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( 1 - c \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {{c
        q}\over a}, \Ai\\ \let \over / b, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{\left( 1 - a \right)  c \left( 1 - {b\over q} \right)  q 
       }\over 
     {a b \left( 1 - c \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q,
        {c\over a}, \Ai\\ \let \over / {b\over q}, {{c {q^2}}\over b},
        \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C115[m1,m2,m3,m4,n1,n2,n3,n4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C116
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, d q, {{c q}\over d},
   \Ai\\ \let \over / {{c q}\over a}, {{c q}\over b}, d, {c\over d},
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  
       \left( 1 - {b\over d} \right)  \left( 1 - {c\over {b d}} \right)  d 
       }\over 
     {\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
       \left( 1 - {c\over d} \right)  \left( 1 - d \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, b,
        \Ai\\ \let \over / {c\over a}, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]  \\  
   -{{\left( 1 - b \right)  \left( 1 - {c\over b} \right)  
        \left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d 
        }\over 
      {\left( a - b \right)  \left( 1 - {c\over {a b}} \right)  
        \left( 1 - {c\over d} \right)  \left( 1 - d \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, b q,
         \Ai\\ \let \over / {{c q}\over a}, {c\over b}, \Bi\endmatrix ;q,
         {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C116[m1,m2,m3,m4,n1,n2,n3,n4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C117
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, d q, {{c
   q}\over d}, \Ai\\ \let \over / b, {{c {q^2}}\over b}, d, {c\over d},
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  d 
       \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {{c q}\over {b d}} \right)  
       }\over 
     {\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       \left( a - {b\over q} \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {{c
        q}\over a}, \Ai\\ \let \over / b, {{c {q^2}}\over b}, \Bi\endmatrix
        ;q, {\displaystyle z} \right ]  \\  -{{\left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d 
        \left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right)  
        }\over 
      {\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
        \left( a - {b\over q} \right)  \left( 1 - {{c q}\over {a b}} \right) }
      }
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {c\over
         a}, \Ai\\ \let \over / {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q,
         {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C117[m1,m2,m3,m4,n1,n2,n3,n4].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C118
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, -
   {\sqrt{c}} q  , a, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c
   q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \longrightarrow {{\left( 1 - {c\over a} \right)  
       \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( 1 - c \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, \Ai\\
        \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{\left( 1 - a \right)  c \left( 1 - {b\over q} \right)  q 
       }\over 
     {a b \left( 1 - c \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, \Ai\\
        \let \over / {{c q}\over a}, {b\over q}, {{c {q^2}}\over b},
        \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C118[m1,m2,m3,n1,n2,n3,n4,n5].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C119
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / a, d q, {{c q}\over d},
   \Ai\\ \let \over / {{c q}\over a}, b, {{c {q^2}}\over b}, d, {c\over d},
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 
   {{\left( 1 - a \right)  \left( 1 - {c\over a} \right)  d 
       \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {{c q}\over {b d}} \right)  
       }\over 
     {\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       \left( a - {b\over q} \right)  \left( 1 - {{c q}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, \Ai\\
        \let \over / {c\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]  \\  -{{\left( 1 - {a\over d} \right)  \left( 1 - {c\over {a d}} \right)  d 
        \left( 1 - {b\over q} \right)  \left( 1 - {{c q}\over b} \right)  
        }\over 
      {\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
        \left( a - {b\over q} \right)  \left( 1 - {{c q}\over {a b}} \right) }
      }
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, \Ai\\
         \let \over / {{c q}\over a}, {b\over q}, {{c q}\over b},
         \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C119[m1,m2,m3,n1,n2,n3,n4,n5].\newline
\rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C120
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, -
   {\sqrt{c}} q  , \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, a, {{c
   {q^2}}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z}
   \right ] \longrightarrow {{\left( 1 - {{c q}\over a} \right)  
       \left( 1 - {{c q}\over b} \right)  
       }\over 
     {\left( 1 - c \right)  \left( 1 - {{c {q^2}}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let
        \over / a, {{c q}\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q,
        {\displaystyle z} \right ]\\ - 
   {{c \left( 1 - {a\over q} \right)  \left( 1 - {b\over q} \right)  {q^2} 
       }\over 
     {a b \left( 1 - c \right)  \left( 1 - {{c {q^2}}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let
        \over / {a\over q}, {{c {q^2}}\over a}, {b\over q}, {{c {q^2}}\over
        b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C120[m1,m2,n1,n2,n3,n4,n5,n6].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5}, {\tt n6} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name C121
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} \phi _{s} \! \left [ \matrix \let \over / d q, {{c q}\over d}, \Ai\\
   \let \over / a, {{c {q^2}}\over a}, b, {{c {q^2}}\over b}, d, {c\over d},
   \Bi\endmatrix ;q, {\displaystyle z} \right ] \\
  \longrightarrow  {{d \left( 1 - {a\over q} \right)  \left( 1 - {b\over {d q}} \right)  
       \left( 1 - {{c q}\over a} \right)  
       \left( 1 - {{c q}\over {b d}} \right)  
       }\over 
     {\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
       \left( {a\over q} - {b\over q} \right)  
       \left( 1 - {{c {q^2}}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let
        \over / {a\over q}, {{c q}\over a}, b, {{c {q^2}}\over b},
        \Bi\endmatrix ;q, {\displaystyle z} \right ]  \\  
   -{{d \left( 1 - {b\over q} \right)  \left( 1 - {a\over {d q}} \right)  
        \left( 1 - {{c q}\over b} \right)  
        \left( 1 - {{c q}\over {a d}} \right)  
        }\over 
      {\left( 1 - {c\over d} \right)  \left( 1 - d \right)  
        \left( {a\over q} - {b\over q} \right)  
        \left( 1 - {{c {q^2}}\over {a b}} \right) }}
   {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let
         \over / a, {{c {q^2}}\over a}, {b\over q}, {{c q}\over b},
         \Bi\endmatrix ;q, {\displaystyle z} \right ]
\endmultline
$$
\Usage Expr/.C121[m1,m2,n1,n2,n3,n4,n5,n6].\newline
\rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5}, {\tt n6} are the positions of the special upper
and lower parameters, respectively.
\Seealso C64, ContigListe, Ers, PosListe.

\Name ContigListe
\Description List of all contiguous relations.
\Usage ContigListe.

\Name Div
\Description Function that divides \hbox{\tt Gleichung} by \hbox{\tt Expr}.
\Usage Div[Expr].
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= Div[a\MATHhoch n]
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich      -n       \MATHvStrich 
         \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich      c
        2 1\MATHvStrich               \MATHvStrich     (-; q)
           \MATHloEck    c          \MATHroEck      a    n
Out[2]= ------------------- == -------
                 n             (c; q)
                a                    n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich      -n       \MATHvStrich 
         \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich      c
        2 1\MATHvStrich               \MATHvStrich     (-; q)
           \MATHloEck    c          \MATHroEck      a    n
Out[3]= ------------------- == -------
                 n             (c; q)
                a                    n
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Sub, 
Hoch, GlTausche, Ers.

\Name Drucke
\Description Function that directly sends an expression \hbox{\tt
Expr} in the 
  Form \hbox{\tt PrintedForm} to the printer. This function only
works for DOS-machines with a printer directly connected.
\hbox{\tt PrintedForm} 
is an optional parameter
  which can be any of the format types (\hbox{\tt InputForm},
\hbox{\tt OutputForm}, 
  \hbox{\tt TeXForm}, \dots). The default is \hbox{\tt OutputForm}.
\Usage Drucke[Expr,PrintedForm].
\Seealso TeXMat, TeX, TeXphW.

\Name Ers
\Description Function for controlled application of rules and functions.
\Usage Ers[Expr,Rules,PosList].\newline
\rm  \hbox{\tt Rules} can be a rule, a list of rules, or a function.
\hbox{\tt PosList} must 
  be a list of positions in \hbox{\tt Expr} to which \hbox{\tt Rules} should be applied. 
  For instance, if \hbox{\tt PosList=$\{\{1,2\},\{4\}\}$}, then \hbox{\tt Rules} is applied to
  \hbox{\tt Expr[[1,2]]} and \hbox{\tt Expr[[4]]} in \hbox{\tt Expr}. If
\hbox{\tt PosList=$\{2,3\}$}, then \hbox{\tt Rules} is
  applied to \hbox{\tt Expr[[2]]} and \hbox{\tt Expr[[3]]} in \hbox{\tt Expr}. The positions of 
  subexpressions can be determined by the function \hbox{\tt PosListe}.
If \hbox{\tt Ers} is applied to an equation then the new left-hand
and right-hand sides are automatically stored in the variables {\tt
LS} and {\tt RS}.
\vskip6pt
\hangafter0 \hangindent10pt\rm
There is an exceptional usage of \hbox{\tt Ers}, namely\newline
\hbox{\hskip25pt\tt Ers[Rules]}.\newline
\rm In this case the \hbox{\tt Rules} are applied to both sides of
the equation that is currently stored in \hbox{\tt Gleichung}. Again, the
new left-hand
and right-hand sides are automatically stored in the variables {\tt
LS} and {\tt RS}.
\Example 
\MATH 
In[1]:= (-1)\MATHhoch n*pq[a*q,n]*pq[c,k]/pq[1/b,m]/pq[q/d,l]
\goodbreakpoint%
            n
        (-1)  (c; q)  (a q; q)
                    k         n
Out[1]= -----------------------
             1       q
            (-; q)  (-; q)
             b    m  d    l
\goodbreakpoint%
In[2]:= PosListe[\%]
\goodbreakpoint%
              n              1
Out[2]= \MATHlbrace \MATHlbrace (-1) , \MATHlbrace \MATHlbrace 1\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace -------, \MATHlbrace \MATHlbrace 2\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace (c; q) , \MATHlbrace \MATHlbrace 3\MATHrbrace %
\MATHrbrace \MATHrbrace , 
                           1                      k
                          (-; q)
                           b    m
\goodbreakpoint%
                             1
\MATHgroesser     \MATHlbrace (a q; q) , \MATHlbrace \MATHlbrace 4\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace -------, \MATHlbrace \MATHlbrace 5\MATHrbrace %
\MATHrbrace \MATHrbrace \MATHrbrace 
             n            q
                         (-; q)
                          d    l
\goodbreakpoint%
In[3]:= Ers[\%\%,neg1,\MATHlbrace 5\MATHrbrace ]
\goodbreakpoint%
                                  1 + l
            n                    q
        (-1)  (c; q)  (a q; q)  (------; q)
                    k         n    d       -l
Out[3]= -------------------------------------
                        1
                       (-; q)
                        b    m
\goodbreakpoint%
In[4]:= Ers[\%\%\%,neg1,\MATHlbrace 2,4\MATHrbrace ]
\goodbreakpoint%
                        m
            n          q
        (-1)  (c; q)  (--; q)
                    k  b     -m
Out[4]= -----------------------
         q          1 + n
        (-; q)  (a q     ; q)
         d    l              -n
\goodbreakpoint%
In[5]:= SUM[\%,\MATHlbrace k,0,Infinity\MATHrbrace ]
\goodbreakpoint%
                                       1 + l
          \MATHinfty       n                    q
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1)  (c; q)  %
(a q; q)  (------; q)
         \MATHbackslash                k         n    d       -l
Out[5]=   \MATHgroesser   -------------------------------------
         /                   1
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck                (-; q)
         k=0                 b    m
\goodbreakpoint%
In[6]:= PosListe[\%,2]
\goodbreakpoint%
                             n
Out[6]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 2, 2\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace (-1) , \MATHlbrace \MATHlbrace 1, %
1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace k, \MATHlbrace \MATHlbrace %
2, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace \MATHinfty , \MATHlbrace %
\MATHlbrace 2, 3\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint%
         1
\MATHgroesser     \MATHlbrace -------, \MATHlbrace \MATHlbrace 1, 2\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace (c; q) , \MATHlbrace \MATHlbrace 1, %
3\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (a q; q) , \MATHlbrace %
\MATHlbrace 1, 4\MATHrbrace \MATHrbrace \MATHrbrace , 
       1                         k                      n
      (-; q)
       b    m
\goodbreakpoint%
        1 + l
       q
\MATHgroesser     \MATHlbrace (------; q)  , \MATHlbrace \MATHlbrace 1, %
5\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace 
         d       -l
\goodbreakpoint%
In[7]:= Ers[\%\%,neg1,\MATHlbrace \MATHlbrace 1,3\MATHrbrace \MATHrbrace ]
\goodbreakpoint%
                               1 + l
          \MATHinfty       n            q
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1)  %
(a q; q)  (------; q)
         \MATHbackslash                  n    d       -l
Out[7]=   \MATHgroesser   -----------------------------
         /         1          k
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck      %
(-; q)  (c q ; q)
         k=0       b    m          -k
\goodbreakpoint%
In[8]:= Ers[\%\%\%,neg1,\MATHlbrace \MATHlbrace 1,2\MATHrbrace ,\MATHlbrace %
1,5\MATHrbrace \MATHrbrace ]
\goodbreakpoint%
                                       m
          \MATHinfty       n                    q
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1)  %
(c; q)  (a q; q)  (--; q)
         \MATHbackslash                k         n  b     -m
Out[8]=   \MATHgroesser   ---------------------------------
         /                 q
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck              (-; q)
         k=0               d    l
\goodbreakpoint%
In[9]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[9]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[10]:= Ers[a-\MATHgroesser q/a]
\goodbreakpoint%
                                 n  a c
            \MATHluEck  q   -n       \MATHruEck     q  (---; q)
            \MATHvStrich  -, q         \MATHvStrich          q     n
Out[10]=  \MATHphi  \MATHvStrich  a     ; q, q \MATHvStrich  == ------------
         2 1\MATHvStrich               \MATHvStrich       n
            \MATHloEck    c          \MATHroEck      a  (c; q)
                                          n
\goodbreakpoint%
In[11]:= Gleichung
\goodbreakpoint%
                                 n  a c
            \MATHluEck  q   -n       \MATHruEck     q  (---; q)
            \MATHvStrich  -, q         \MATHvStrich          q     n
Out[11]=  \MATHphi  \MATHvStrich  a     ; q, q \MATHvStrich  == ------------
         2 1\MATHvStrich               \MATHvStrich       n
            \MATHloEck    c          \MATHroEck      a  (c; q)
                                          n
\goodbreakpoint%
In[12]:= PQ
\goodbreakpoint%
In[13]:= pq[a,4]+1/pq[b,3]
\goodbreakpoint%
                      1                                           2          3
Out[13]= ---------------------------- + (1 - a) (1 - a q) (1 - a q ) (1 - a q )
                                   2
         (1 - b) (1 - b q) (1 - b q )
\goodbreakpoint%
In[14]:= Ers[\%,Expand,\MATHlbrace 2\MATHrbrace ]
\goodbreakpoint%
                        2        2    2  2      3      2  3    3  3    2  4
Out[14]= 1 - a - a q + a  q - a q  + a  q  - a q  + 2 a  q  - a  q  + a  q  - 
 
      3  4    2  5    3  5    3  6    4  6                1
\MATHgroesser     a  q  + a  q  - a  q  - a  q  + a  q  + ----------------------------
                                                                       2
                                             (1 - b) (1 - b q) (1 - b q )
\endMATH
\Seealso PosListe, ManipulationsListe, Subst.

\Name erw1
\Description \vtab $(a;q)_n \to (a;q)_{m+n}/(aq^n;q)_m$,\\
             $(a;q)_n \to (a;q)_\infty/(aq^n;q)_\infty$.
\endvtab
\vskip6pt
\leavevmode\hskip10pt  The parameter \hbox{\tt m} has to be entered on request. To apply the second rule,
  \hbox{\tt m} has to be \hbox{\tt Infinity}.
\Usage Expr/.erw1.
\Example 
\MATH 
In[1]:= pq[a,n,q\MATHhoch 2]
\goodbreakpoint%
             2
Out[1]= (a; q )
               n
\goodbreakpoint%
In[2]:= \%/.erw1
top-extend by: m
\goodbreakpoint%
             2
        (a; q )
               m + n
Out[2]= -------------
            2 n   2
        (a q   ; q )
                    m
\goodbreakpoint%
In[3]:= pq[a,m,1/q]
\goodbreakpoint%
            1
Out[3]= (a; -)
            q m
\goodbreakpoint%
In[4]:= \%/.erw1
top-extend by: Infinity
\goodbreakpoint%
           1
        (a;-)
           q \MATHinfty 
Out[4]= -------
         a  1
        (--;-)
          m q \MATHinfty 
         q
\endMATH 
\Seealso erw2, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

\Name erw2
\Description \vtab $(a;q)_n \to (a/q^m;q)_{m+n}/(a/q^m;q)_m$,\\
             $(a;q)_\infty \to (a/q^m;q)_\infty/(a/q^m;q)_m$.
\endvtab
\vskip6pt
\leavevmode\hskip10pt  The parameter \hbox{\tt m} has to be entered on request. 
\Usage Expr/.erw2.
\Example 
\MATH 
In[1]:= pq[a,n,q\MATHhoch 2]
\goodbreakpoint%
             2
Out[1]= (a; q )
               n
\goodbreakpoint%
In[2]:= \%/.erw2
bottom-extend by: m
\goodbreakpoint%
          a     2
        (----; q )
          2 m     m + n
         q
Out[2]= ---------------
            a     2
          (----; q )
            2 m     m
           q
\goodbreakpoint%
In[3]:= pqinf[b,1/q]
\goodbreakpoint%
           1
Out[3]= (b;-)
           q \MATHinfty 
\goodbreakpoint%
In[4]:= \%/.erw2
bottom-extend by: n
\goodbreakpoint%
            n 1
        (b q ;-)
              q \MATHinfty 
Out[4]= ----------
            n  1
        (b q ; -)
               q n
\endMATH 
\Seealso erw1, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

\Name Expandq
\Description Rule that expands all the exponents in powers.
\Usage Expr/.Expandq.
\Example 
\MATH 
In[1]:= pq[a*q\MATHhoch 3,n]/.trans
\goodbreakpoint%
                              2      -2 - n
            n  n  3 n - (n - n )/2  q
Out[1]= (-1)  a  q                 (-------; q)
                                       a       n
\goodbreakpoint%
In[2]:= \%/.Expandq
\goodbreakpoint%
                             2     -2 - n
            n  n  (5 n)/2 + n /2  q
Out[2]= (-1)  a  q               (-------; q)
                                     a       n
\endMATH 
\Seealso SimplifyPQ, MinusOne, SUMExpand, Ers, PosListe.

\Name Factorialpq
\Description \hbox{\tt Factorialpq[n,k,q]} is the usual $q$-factorial, written in terms
  of $q$-factorial symbols \hbox{\tt pq}. The parameter \hbox{\tt q} is optional. It will be set 
  equal \hbox{\tt q} if it is omitted. 
\Usage Factorialpq[n,q]

   \leavevmode\hphantom{Usa}\rm or: \tt Factorialpq[n].
\Example 
\MATH 
In[1]:= Factorialpq[n]
\goodbreakpoint%
        (q; q)
              n
Out[1]= --------
               n
        (1 - q)
\goodbreakpoint%
In[2]:= Factorialpq[5]
\goodbreakpoint%
        (q; q)
              5
Out[2]= --------
               5
        (1 - q)
\endMATH 
\Seealso Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialq.

\Name Factorialq
\Description \hbox{\tt Factorialq[n,k,q]} is the usual $q$-factorial, expanded into
  a $q$-series, if possible. The parameter \hbox{\tt q} is optional. It will be set 
  equal \hbox{\tt q} if it is omitted. 
\Usage Factorialq[n,q]

   \leavevmode\hphantom{Usa}\rm or: \tt Factorialq[n].
\Example 
\MATH 
In[1]:= Factorialq[n]
\goodbreakpoint%
        \MATHluEck    \MATHruEck  
Out[1]= \MATHvStrich  n \MATHvStrich  !
        \MATHloEck    \MATHroEck  
             q
\goodbreakpoint%
In[2]:= Factorialq[5]
\goodbreakpoint%
                     2       3       4       5       6       7      8
Out[2]= 1 + 4 q + 9 q  + 15 q  + 20 q  + 22 q  + 20 q  + 15 q  + 9 q  + 

        9    10
\MATHgroesser     4 q  + q
\endMATH 
\Seealso Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialpq.

\Name Gleichung
\Description Is a variable which stores equations. The equation \hbox{\tt Gleichung}
  can be manipulated using the functions \hbox{\tt Add}, \hbox{\tt Sub}, \hbox{\tt Mal}, 
\hbox{\tt Div}, \hbox{\tt Hoch}, \hbox{\tt GlTausche}, \hbox{\tt Ers}, 
  and \hbox{\tt SUM[k,m,n]}, where \hbox{\tt m} and \hbox{\tt n} are integers 
or variables. The last command causes the equation to be 
  summed over \hbox{\tt k} from \hbox{\tt m} to \hbox{\tt n}. The
parameter \hbox{\tt k} is optional. It will be set \hbox{\tt kk}, 
\hbox{\tt ii}, \hbox{\tt jj}, \hbox{\tt ll}, \hbox{\tt mm}, 
or \hbox{\tt nn}, if it is omitted.
\Usage Gleichung.
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= Gleichung
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[3]:= Add[1]
\goodbreakpoint%
                                        n  c
               \MATHluEck               \MATHruEck         a  (-; q)
               \MATHvStrich      -n       \MATHvStrich             a    n
Out[3]= 1 +  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == 1 + ----------
            2 1\MATHvStrich               \MATHvStrich          (c; q)
               \MATHloEck    c          \MATHroEck                n
\goodbreakpoint%
In[4]:= Gleichung
\goodbreakpoint%
                                        n  c
               \MATHluEck               \MATHruEck         a  (-; q)
               \MATHvStrich      -n       \MATHvStrich             a    n
Out[4]= 1 +  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == 1 + ----------
            2 1\MATHvStrich               \MATHvStrich          (c; q)
               \MATHloEck    c          \MATHroEck                n
\goodbreakpoint%
In[5]:= LS=5
\goodbreakpoint%
Out[5]= 5
\goodbreakpoint%
In[6]:= Gleichung
\goodbreakpoint%
                  n  c
                 a  (-; q)
                     a    n
Out[6]= 5 == 1 + ----------
                  (c; q)
                        n
\goodbreakpoint%
In[7]:= SUM[n,0,m]
\goodbreakpoint%
          m         m        m   n  c
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck      %
\MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck     \MATHluEck %
\MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck a  (-; q)
         \MATHbackslash          \MATHbackslash         \MATHbackslash        a    n
Out[7]=   \MATHgroesser   5 ==   \MATHgroesser   1 +   \MATHgroesser   ----------
         /         /        /    (c; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck      %
\MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck     \MATHloEck %
\MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck        n
         n=0       n=0      n=0
\goodbreakpoint%
In[8]:= SUM[0,M]
\goodbreakpoint%
          M    m         M    m        M    m   n  c
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck %
\MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck      \MATHluEck %
\MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich %
\MATHhStrich \MATHhStrich \MATHruEck     \MATHluEck \MATHhStrich \MATHhStrich %
\MATHhStrich \MATHruEck \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck a  (-; q)
         \MATHbackslash     \MATHbackslash          \MATHbackslash     %
\MATHbackslash         \MATHbackslash     \MATHbackslash        a    n
Out[8]=   \MATHgroesser     \MATHgroesser   5 ==   \MATHgroesser     %
\MATHgroesser   1 +   \MATHgroesser     \MATHgroesser   ----------
         /    /         /    /        /    /    (c; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck %
\MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck      \MATHloEck %
\MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich %
\MATHhStrich \MATHhStrich \MATHroEck     \MATHloEck \MATHhStrich \MATHhStrich %
\MATHhStrich \MATHroEck \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHroEck        n
        kk=0  n=0      kk=0  n=0     kk=0  n=0
\goodbreakpoint%
In[9]:= Gleichung
\goodbreakpoint%
          M    m         M    m        M    m   n  c
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck %
\MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck      \MATHluEck %
\MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich %
\MATHhStrich \MATHhStrich \MATHruEck     \MATHluEck \MATHhStrich \MATHhStrich %
\MATHhStrich \MATHruEck \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck a  (-; q)
         \MATHbackslash     \MATHbackslash          \MATHbackslash     %
\MATHbackslash         \MATHbackslash     \MATHbackslash        a    n
Out[9]=   \MATHgroesser     \MATHgroesser   5 ==   \MATHgroesser     %
\MATHgroesser   1 +   \MATHgroesser     \MATHgroesser   ----------
         /    /         /    /        /    /    (c; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck %
\MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck      \MATHloEck %
\MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich %
\MATHhStrich \MATHhStrich \MATHroEck     \MATHloEck \MATHhStrich \MATHhStrich %
\MATHhStrich \MATHroEck \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHroEck        n
        kk=0  n=0      kk=0  n=0     kk=0  n=0
\endMATH 
\Seealso SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, Sub,
Hoch, GlTausche, Ers, Subst,\linebreak PQSort.

\Name GlTausche
\Description \hbox{\tt GlTausche} interchanges right-hand and left-hand sides in
    \hbox{\tt Gleichung}.
\Usage GlTausche.
\Example
\MATH
In[1]:= Sgl3201
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                      -n                c       c
           %
\MATHluEck    a, b, q           %
\MATHruEck      (-; q)  (-; q)
           %
\MATHvStrich                      %
\MATHvStrich       a    n  b    n
Out[1]=  \MATHphi  %
\MATHvStrich          1 - n; q, q %
\MATHvStrich  == -----------------
        3 2%
\MATHvStrich     a b q            %
\MATHvStrich               c
           %
\MATHloEck  c, ----------       %
\MATHroEck     (c; q)  (---; q)
                    c                       n  a b    n
\goodbreakpoint%
In[2]:= Gleichung
\goodbreakpoint%
                      -n                c       c
           %
\MATHluEck    a, b, q           %
\MATHruEck      (-; q)  (-; q)
           %
\MATHvStrich                      %
\MATHvStrich       a    n  b    n
Out[2]=  \MATHphi  %
\MATHvStrich          1 - n; q, q %
\MATHvStrich  == -----------------
        3 2%
\MATHvStrich     a b q            %
\MATHvStrich               c
           %
\MATHloEck  c, ----------       %
\MATHroEck     (c; q)  (---; q)
                    c                       n  a b    n
\goodbreakpoint%
In[3]:= GlTausche
\goodbreakpoint%
          c       c                        -n
         (-; q)  (-; q)         %
\MATHluEck    a, b, q           %
\MATHruEck 
          a    n  b    n        %
\MATHvStrich                      %
\MATHvStrich 
Out[3]= ----------------- ==  \MATHphi  %
\MATHvStrich          1 - n; q, q %
\MATHvStrich 
                  c          3 2%
\MATHvStrich     a b q            %
\MATHvStrich 
        (c; q)  (---; q)        %
\MATHloEck  c, ----------       %
\MATHroEck 
              n  a b    n                c
\goodbreakpoint%
In[4]:= Gleichung
\goodbreakpoint%
          c       c                        -n
         (-; q)  (-; q)         %
\MATHluEck    a, b, q           %
\MATHruEck 
          a    n  b    n        %
\MATHvStrich                      %
\MATHvStrich 
Out[4]= ----------------- ==  \MATHphi  %
\MATHvStrich          1 - n; q, q %
\MATHvStrich 
                  c          3 2%
\MATHvStrich     a b q            %
\MATHvStrich 
        (c; q)  (---; q)        %
\MATHloEck  c, ----------       %
\MATHroEck 
              n  a b    n                c
\endMATH
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, 
    Sub, Hoch, Ers, Subst.

\Name Hoch
\Description Function that takes \hbox{\tt Gleichung} to the
\hbox{\tt Expr}-th power.
\Usage Hoch[Expr].
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= Hoch[3]
\goodbreakpoint%
                                 3 n  c     3
           \MATHluEck               \MATHruEck      a    (-; q)
           \MATHvStrich      -n       \MATHvStrich 3          a    n
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich   == -------------
        2 1\MATHvStrich               \MATHvStrich               3
           \MATHloEck    c          \MATHroEck        (c; q)
                                        n
In[3]:= Gleichung
\goodbreakpoint%
                                 3 n  c     3
           \MATHluEck               \MATHruEck      a    (-; q)
           \MATHvStrich      -n       \MATHvStrich 3          a    n
Out[3]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich   == -------------
        2 1\MATHvStrich               \MATHvStrich               3
           \MATHloEck    c          \MATHroEck        (c; q)
                                        n
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div,
Sub, GlTausche, Ers.

\Name hypqAttributes
\Description Shows the current setup of the session. The setup can be 
  changed by the switches \hbox{\tt PQ}, \hbox{\tt phCancel}, \hbox{\tt TeX}, and \hbox{\tt TeXphW}. The 
  default-setup is shown in the following Example.
\Usage hypqAttributes.
\Example 
\MATH 
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.

\endMATH 
\Seealso PQ, phCancel, TeX, TeXphW.

\Name inv1
\Description $(a;q)_n \to (1/a;1/q)_n(-a)^nq^{\binom n2}$.
\Usage Expr/.inv1.
\Example 
\MATH 
In[1]:= pq[a\MATHhoch 2,n]
\goodbreakpoint%
          2
Out[1]= (a ; q)
               n
\goodbreakpoint%
In[2]:= \%/.inv1
\goodbreakpoint%
            n  2 n   -2  1
        (-1)  a    (a  ; -)
                         q n
Out[2]= --------------------
                   2
             (n - n )/2
            q
\endMATH 
\Seealso inv2, Ers, PosListe, ManipulationsListe.

\Name inv2
\Description $(a;q)_n \to (aq^{n-1};1/q)_n$.
\Usage Expr/.inv2.
\Example 
\MATH 
In[1]:= pq[a\MATHhoch 2,n]
\goodbreakpoint%
          2
Out[1]= (a ; q)
               n
\goodbreakpoint%
In[2]:= \%/.inv2
\goodbreakpoint%
          2  -1 + n  1
Out[2]= (a  q      ; -)
                     q n
\endMATH 
\Seealso inv1, Ers, PosListe, ManipulationsListe.

\Name LaTeX
\Description Switch that changes the output of TeXForm to be usable with
   Plain-{\TeX} and \LaTeX. By default the output of TeXForm is usable with 
   \AmSTeX.
\Usage LaTeX.
\Example 
\MATH
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AMSTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= TeXForm[ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[2]//TeXForm= 
\MATHlbrace \MATHrbrace  \MATHtief \MATHlbrace 2\MATHrbrace  %
\MATHbackslash phi \MATHtief \MATHlbrace 1\MATHrbrace  \MATHbackslash %
! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let %
\MATHbackslash over / a, b\MATHbackslash \MATHbackslash  \MATHbackslash let %
\MATHbackslash over /
 c\MATHbackslash endmatrix ;q, \MATHlbrace \MATHbackslash displaystyle %
z\MATHrbrace  \MATHbackslash right ]
\goodbreakpoint%
In[3]:= LaTeX
\goodbreakpoint%
In[4]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[5]:= TeXForm[ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[5]//TeXForm= 
\MATHlbrace \MATHrbrace  \MATHtief \MATHlbrace 2\MATHrbrace  %
\MATHbackslash phi \MATHtief \MATHlbrace 1\MATHrbrace  \MATHbackslash %
! \MATHbackslash left [ \MATHbackslash matrix \MATHlbrace  %
\MATHbackslash let \MATHbackslash over / a, b\MATHbackslash cr %
\MATHbackslash let \MATHbackslash over / c\MATHrbrace 
 ;q, \MATHlbrace \MATHbackslash displaystyle z\MATHrbrace  %
\MATHbackslash right ]
\endMATH 
\Seealso AmSTeX, AmSLaTeX, TeX, TeXMat, TeXphW.

\Name Limes
\Description Function for doing formal limits of basic hypergeometric
  expressions. If required for taking the limit, you will be asked whether
  or not the absolute value of some variable or expression is smaller than
  1. You will be offered three options, \hbox{\tt [y|n|u]}. If the
absolute value of the variable is smaller than 1 then enter {\tt y}, if it
is greater than 1 then enter {\tt n}, if you do not want to make an
explicit declaration then enter {\tt u} (for ``undetermined").
Your decision, if explicit, is stored for the rest of your MATHEMATICA session. If
  you want to change your decision later, use \hbox{\tt AbsGreater},
  \hbox{\tt AbsSmaller}, or \hbox{\tt AbsUndetermined},
  respectively.
  By default the absolute value of {\tt q} is defined to be smaller than 1.
  Also this can be changed by \hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller},
   or \hbox{\tt AbsUndetermined}, respectively.
\vskip6pt
\hangafter0 \hangindent10pt\rm
If you want to let a base $q$ tend to 1, then you should also
have the basic file \hbox{\tt hyp.m} of the MATHEMATICA package HYP
in your MAHEMATICA input directory, since in these situations the file
\hbox{\tt hyp.m} will be loaded automatically. The \hbox{\tt
\$ContextPath} (cf\. \cite{\WolfAA}) 
is set that objects of the package HYP do not
override the respective objects of the package HYPQ. The situation is
just as if you would have loaded \hbox{\tt hyp.m} first and then
\hbox{\tt hyp.q}. Concerning a
simultaneous use of the packages HYPQ and HYP confer the section
{\it Simultaneous use of\/ {\rm HYP} and\/ {\rm HYPQ}} 
at the beginning of this handbook.
\vskip6pt
\hangafter1 \hangindent10pt\rm
Hint: When using \hbox{\tt Limes} you sometimes have to do a little bit
of ``preparation". For instance, 
when applying \hbox{\tt Limes[\dots,Regel]} to expressions
of the form $(a;q)_n$, where $a$ and $n$ tend
to $\infty$ under \hbox{\tt Regel}, then the package will be unable
to do it, unless at least $1/aq^n$ tends to a definite limit. Hence,
you will sometimes have to use \hbox{\tt zerl} before being able
to apply \hbox{\tt Limes} (see the last example).
\vskip6pt
\hangafter1 \hangindent10pt\rm
\underbar{Warning}: This function uses primitive algebraic techniques to do the 
  limit. There is no check if taking the limit is actually allowed. So it
  is left to you to check the validity of a result of \hbox{\tt Limes}.
\Usage Limes[Expr, x->x0].
\Example The derivation of the identity \cite{\GaRaAA, (2.7.6)} that leads to the
Rogers--Ramanujan identities, starting from Watson's transformation
Tgl8702.
\vskip10pt
\MATH
In[1]:= Tgl8702
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
           \MATHluEck                                             -n        2  2 + n
           \MATHvStrich    a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e, q         a  q
Out[1]=  \MATHphi  \MATHvStrich                                                 ; q, %
---------
        8 7\MATHvStrich                     a q  a q  a q  a q     1 + n      b c d e
           \MATHloEck  Sqrt[a], -Sqrt[a], ---, ---, ---, ---, a q
                                 b    c    d    e
\goodbreakpoint% 
                 a q         -n               a q
       \MATHruEck        \MATHluEck  ---, d, e, q         \MATHruEck  (a q, ---; q)
       \MATHvStrich        \MATHvStrich  b c                  \MATHvStrich        %
d e    n
\MATHgroesser       \MATHvStrich  ==  \MATHphi  \MATHvStrich                %
; q, q \MATHvStrich  --------------
       \MATHvStrich     4 3\MATHvStrich  a q  a q  d e        \MATHvStrich   %
a q  a q
       \MATHroEck        \MATHloEck  ---, ---, ----       \MATHroEck  (---, %
---; q)
                  b    c      n           d    e     n
                           a q
\goodbreakpoint%
In[2]:= Limes[\%,b-\MATHgroesser Infinity]
\goodbreakpoint%
           \MATHluEck                                          -n         %
2  2 + n \MATHruEck 
           \MATHvStrich    a, Sqrt[a] q, -(Sqrt[a] q), c, d, e, q          %
a  q      \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich                                               %
; q, --------- \MATHvStrich  == 
        7 7\MATHvStrich                        a q  a q  a q     1 + n       %
c d e   \MATHvStrich 
           \MATHloEck  Sqrt[a], -Sqrt[a], 0, ---, ---, ---, a q                    %
\MATHroEck 
                                    c    d    e
\goodbreakpoint% 
                 -n
        \MATHluEck  d, e, q         \MATHruEck 
        \MATHvStrich                  \MATHvStrich             a q
      \MATHphi  \MATHvStrich  a q  d e ; q, q \MATHvStrich  (a q; q)  (---; q)
     3 2\MATHvStrich  ---, ----       \MATHvStrich          n  d e    n
        \MATHloEck   c      n       \MATHroEck 
               a q
\MATHgroesser     ------------------------------------------
                 a q       a q
                (---; q)  (---; q)
                  d     n   e     n
\goodbreakpoint%
In[3]:= Limes[\%,c-\MATHgroesser Infinity]
\goodbreakpoint%
           \MATHluEck                                        -n         2  2 + n \MATHruEck 
           \MATHvStrich     a, Sqrt[a] q, -(Sqrt[a] q), d, e, q          a  q      \MATHvStrich 
Out[3]=  \MATHphi  \MATHvStrich                                             ; q, --------- \MATHvStrich  == 
        6 7\MATHvStrich                           a q  a q     1 + n        d e    \MATHvStrich 
           \MATHloEck  Sqrt[a], -Sqrt[a], 0, 0, ---, ---, a q                    \MATHroEck 
                                       d    e
\goodbreakpoint% 
                 -n
        \MATHluEck  d, e, q         \MATHruEck 
        \MATHvStrich                  \MATHvStrich             a q
      \MATHphi  \MATHvStrich      d e  ; q, q \MATHvStrich  (a q; q)  (---; q)
     3 2\MATHvStrich   0, ----        \MATHvStrich          n  d e    n
        \MATHloEck         n        \MATHroEck 
              a q
\MATHgroesser     ------------------------------------------
                 a q       a q
                (---; q)  (---; q)
                  d     n   e     n
\goodbreakpoint%
In[4]:= Limes[\%,d-\MATHgroesser Infinity]
\goodbreakpoint%
           \MATHluEck                                     -n          %
2  2 + n \MATHruEck 
           \MATHvStrich     a, Sqrt[a] q, -(Sqrt[a] q), e, q           %
a  q      \MATHvStrich 
Out[4]=  \MATHphi  \MATHvStrich                                           ; %
q, --------- \MATHvStrich  == 
        5 7\MATHvStrich                              a q     1 + n         %
e     \MATHvStrich 
           \MATHloEck  Sqrt[a], -Sqrt[a], 0, 0, 0, ---, a q                    %
\MATHroEck 
                                          e
\goodbreakpoint% 
        \MATHluEck                1 + n \MATHruEck 
        \MATHvStrich      -n     a q      \MATHvStrich 
      \MATHphi  \MATHvStrich  e, q  ; q, -------- \MATHvStrich  (a q; q)
     2 1\MATHvStrich                e     \MATHvStrich          n
        \MATHloEck    0                 \MATHroEck 
\MATHgroesser     ------------------------------------
                   a q
                  (---; q)
                    e     n
\goodbreakpoint%
In[5]:= Limes[\%,e-\MATHgroesser Infinity]
\goodbreakpoint%
           \MATHluEck                                   -n                   %
\MATHruEck 
           \MATHvStrich      a, Sqrt[a] q, -(Sqrt[a] q), q            %
2  2 + n \MATHvStrich 
Out[5]=  \MATHphi  \MATHvStrich                                         %
; q, a  q      \MATHvStrich  == 
        4 7\MATHvStrich                                    1 + n               %
\MATHvStrich 
           \MATHloEck  Sqrt[a], -Sqrt[a], 0, 0, 0, 0, a q                    \MATHroEck 
\goodbreakpoint% 
        \MATHluEck                   \MATHruEck 
        \MATHvStrich   -n        1 + n \MATHvStrich 
\MATHgroesser      \MATHphi  \MATHvStrich  q  ; q, a q      \MATHvStrich  (a q; q)
     1 1\MATHvStrich                   \MATHvStrich          n
        \MATHloEck   0               \MATHroEck 
\goodbreakpoint%
In[6]:= Limes[\%,n-\MATHgroesser Infinity]
\goodbreakpoint%
           \MATHluEck                                             \MATHruEck 
           \MATHvStrich     a, Sqrt[a] q, -(Sqrt[a] q)         2  2 \MATHvStrich 
Out[6]=  \MATHphi  \MATHvStrich                                  %
; q, a  q  \MATHvStrich  == 
        3 7\MATHvStrich  Sqrt[a], -Sqrt[a], 0, 0, 0, 0, 0           \MATHvStrich 
           \MATHloEck                                             \MATHroEck 
\goodbreakpoint% 
                 \MATHluEck            \MATHruEck 
                 \MATHvStrich  -         \MATHvStrich 
\MATHgroesser     (a q;q)   \MATHphi  \MATHvStrich   ; q, a q \MATHvStrich 
            \MATHinfty  0 1\MATHvStrich  0         \MATHvStrich 
                 \MATHloEck            \MATHroEck 
\goodbreakpoint%
In[7]:= \%/.phSUM/.SUMRegeln
\goodbreakpoint%
          \MATHinfty 
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck                             2
         \MATHbackslash         kk  2 kk  -kk/2 + (5 kk )/2
Out[7]=   \MATHgroesser   ((-1)   a     q                  (a; q)   %
(-(Sqrt[a] q); q)   
         /                                          kk                  kk
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck 
        kk=0
\goodbreakpoint% 
\MATHgroesser          (Sqrt[a] q; q)  ) / ((-Sqrt[a]; q)   (Sqrt[a]; %
q)   (q; q)  ) == 
                        kk                  kk             kk       kk
\goodbreakpoint% 
                 \MATHinfty          2
               \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck  kk  kk
                \MATHbackslash    a   q
\MATHgroesser     (a q;q)  (  \MATHgroesser   --------)
            \MATHinfty    /   (q; q)
               \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       kk
               kk=0
\goodbreakpoint%
In[8]:= PosListe[\%,3]
\goodbreakpoint%
                                kk                  2 kk
Out[8]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 1, 2, %
2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (-1)  , %
\MATHlbrace \MATHlbrace 1, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , %
\MATHlbrace a    , \MATHlbrace \MATHlbrace 1, 1, 2\MATHrbrace \MATHrbrace %
\MATHrbrace , 
\goodbreakpoint% 
\MATHgroesser     \MATHlbrace kk, \MATHlbrace \MATHlbrace 1, 2, %
1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q, \MATHlbrace \MATHlbrace %
2, 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace a q, \MATHlbrace %
\MATHlbrace 2, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
                    2
       -kk/2 + (5 kk )/2
\MATHgroesser     \MATHlbrace q                 , \MATHlbrace \MATHlbrace 1, %
1, 3\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace \MATHinfty , %
\MATHlbrace \MATHlbrace 1, 2, 3\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
                                        1
\MATHgroesser     \MATHlbrace \MATHlbrace kk, 0, \MATHinfty \MATHrbrace , %
\MATHlbrace \MATHlbrace 2, 2, 2\MATHrbrace \MATHrbrace \MATHrbrace , %
\MATHlbrace ---------------, \MATHlbrace \MATHlbrace 1, 1, 4\MATHrbrace \MATHrbrace %
\MATHrbrace , 
                                 (-Sqrt[a]; q)
                                              kk
\goodbreakpoint% 
            1
\MATHgroesser     \MATHlbrace --------------, \MATHlbrace \MATHlbrace 1, %
1, 5\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (a; q)  , \MATHlbrace %
\MATHlbrace 1, 1, 6\MATHrbrace \MATHrbrace \MATHrbrace , 
      (Sqrt[a]; q)                         kk
                  kk
\goodbreakpoint% 
                                      2
                                kk  kk
         1                     a   q
\MATHgroesser     \MATHlbrace --------, \MATHlbrace \MATHlbrace 1, 1, %
7\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace --------, \MATHlbrace %
\MATHlbrace 2, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , 
      (q; q)                   (q; q)
            kk                       kk
\goodbreakpoint% 
\MATHgroesser     \MATHlbrace (-(Sqrt[a] q); q)  , \MATHlbrace \MATHlbrace 1, %
1, 8\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (Sqrt[a] q; q)  , %
\MATHlbrace \MATHlbrace 1, 1, 9\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace 
                       kk                               kk
\goodbreakpoint%
In[9]:= Ers[\%\%,lina1,\MATHlbrace \MATHlbrace 1,1,4\MATHrbrace ,\MATHlbrace 1,1,5\MATHrbrace %
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck                             2
         \MATHbackslash         kk  2 kk  -kk/2 + (5 kk )/2
Out[9]=   \MATHgroesser   ((-1)   a     q                  (a; q)   %
(-(Sqrt[a] q); q)   
         /                                          kk                  kk
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck 
        kk=0
\goodbreakpoint% 
\MATHgroesser          (Sqrt[a] q; q)  ) / 
                        kk
\goodbreakpoint% 
\MATHgroesser        ((1 - Sqrt[a]) (1 + Sqrt[a]) (q; q)   (-(Sqrt[a] q); q)        
                                           kk                  -1 + kk
\goodbreakpoint% 
                                                \MATHinfty          2
                                              \MATHluEck \MATHhStrich %
\MATHhStrich \MATHhStrich \MATHruEck  kk  kk
                                               \MATHbackslash    a   q
\MATHgroesser          (Sqrt[a] q; q)       ) == (a q;q)  (  \MATHgroesser   %
--------)
                        -1 + kk            \MATHinfty    /   (q; q)
                                              \MATHloEck \MATHhStrich %
\MATHhStrich \MATHhStrich \MATHroEck       kk
                                              kk=0
\goodbreakpoint%
In[10]:= PosListe[\%,3]
\goodbreakpoint%
                                 kk
Out[10]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 1, 2, %
2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (-1)  , \MATHlbrace %
\MATHlbrace 1, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
           1                           1
\MATHgroesser     \MATHlbrace -----------, \MATHlbrace \MATHlbrace 1, %
1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace -----------, %
\MATHlbrace \MATHlbrace 1, 1, 3\MATHrbrace \MATHrbrace \MATHrbrace , 
      1 - Sqrt[a]                 1 + Sqrt[a]
\goodbreakpoint% 
       2 kk
\MATHgroesser     \MATHlbrace a    , \MATHlbrace \MATHlbrace 1, 1, %
4\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace kk, \MATHlbrace \MATHlbrace %
1, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q, \MATHlbrace %
\MATHlbrace 2, 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
                                        2
                           -kk/2 + (5 kk )/2
\MATHgroesser     \MATHlbrace a q, \MATHlbrace \MATHlbrace 2, 1, %
1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q                 %
, \MATHlbrace \MATHlbrace 1, 1, 5\MATHrbrace \MATHrbrace \MATHrbrace , %
\MATHlbrace \MATHinfty , \MATHlbrace \MATHlbrace 1, 2, 3\MATHrbrace %
\MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
\MATHgroesser     \MATHlbrace \MATHlbrace kk, 0, \MATHinfty \MATHrbrace , %
\MATHlbrace \MATHlbrace 2, 2, 2\MATHrbrace \MATHrbrace \MATHrbrace , %
\MATHlbrace (a; q)  , \MATHlbrace \MATHlbrace 1, 1, 6\MATHrbrace %
\MATHrbrace \MATHrbrace , 
                                       kk
\goodbreakpoint% 
                                      2
                                kk  kk
         1                     a   q
\MATHgroesser     \MATHlbrace --------, \MATHlbrace \MATHlbrace 1, 1, %
7\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace --------, \MATHlbrace %
\MATHlbrace 2, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , 
      (q; q)                   (q; q)
            kk                       kk
\goodbreakpoint% 
                 1
\MATHgroesser     \MATHlbrace ------------------------, \MATHlbrace \MATHlbrace %
1, 1, 8\MATHrbrace \MATHrbrace \MATHrbrace , 
      (-(Sqrt[a] q); q)
                       -1 + kk
\goodbreakpoint% 
\MATHgroesser     \MATHlbrace (-(Sqrt[a] q); q)  , \MATHlbrace %
\MATHlbrace 1, 1, 9\MATHrbrace \MATHrbrace \MATHrbrace , 
                       kk
\goodbreakpoint% 
                1
\MATHgroesser     \MATHlbrace ---------------------, \MATHlbrace \MATHlbrace %
1, 1, 10\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (Sqrt[a] q; q)  %
, \MATHlbrace \MATHlbrace 1, 1, 11\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace 
      (Sqrt[a] q; q)                                       kk
                    -1 + kk
\goodbreakpoint%
In[11]:= Ers[\%\%,lina2,\MATHlbrace \MATHlbrace 1,1,9\MATHrbrace ,\MATHlbrace %
1,1,11\MATHrbrace \MATHrbrace ]
\goodbreakpoint%
           \MATHinfty 
         \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck                             2
          \MATHbackslash         kk  2 kk  -kk/2 + (5 kk )/2               kk
Out[11]=   \MATHgroesser   ((-1)   a     q                  (1 - Sqrt[a] q  ) 
          /   
         \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck 
         kk=0
\goodbreakpoint% 
                        kk
\MATHgroesser          (1 + Sqrt[a] q  ) (a; q)  ) / ((1 - Sqrt[a]) %
(1 + Sqrt[a]) (q; q)  )
                                  kk                                       kk
\goodbreakpoint% 
                      \MATHinfty          2
                    \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck  kk  kk
                     \MATHbackslash    a   q
\MATHgroesser       == (a q;q)  (  \MATHgroesser   --------)
                 \MATHinfty    /   (q; q)
                    \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHroEck       kk
                    kk=0
\goodbreakpoint%
In[12]:= Ers[\%,ExpandAll,\MATHlbrace \MATHlbrace 1,1\MATHrbrace \MATHrbrace ]
\goodbreakpoint%
                                 2
           \MATHinfty       kk  2 kk  (5 kk )/2
         \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1)   %
a     q          (a; q)
          \MATHbackslash                                  kk
Out[12]=   \MATHgroesser   --------------------------------- - 
          /    kk/2               kk/2
         \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck q     %
(q; q)   - a q     (q; q)
         kk=0             kk                 kk
\goodbreakpoint% 
                                      2
            kk  1 + 2 kk  2 kk + (5 kk )/2
        (-1)   a         q                 (a; q)
                                                 kk
\MATHgroesser        ------------------------------------------- == 
              kk/2               kk/2
             q     (q; q)   - a q     (q; q)
                         kk                 kk
\goodbreakpoint% 
                 \MATHinfty          2
               \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich %
\MATHruEck  kk  kk
                \MATHbackslash    a   q
\MATHgroesser     (a q;q)  (  \MATHgroesser   --------)
            \MATHinfty    /   (q; q)
               \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       kk
               kk=0
\goodbreakpoint%
In[13]:= Ers[\%,Factor,\MATHlbrace \MATHlbrace 1,1\MATHrbrace \MATHrbrace ]
\goodbreakpoint%
                                         2
           \MATHinfty       kk  2 kk  -kk/2 + (5 kk )/2          2 kk
         \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1)   a     q                  (-1 + a q    ) (a; q)
          \MATHbackslash                                                         kk
Out[13]=   \MATHgroesser   ------------------------------------------------------- == 
          /                      (-1 + a) (q; q)
         \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck                                   kk
         kk=0
\goodbreakpoint% 
                 \MATHinfty          2
               \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck  kk  kk
                \MATHbackslash    a   q
\MATHgroesser     (a q;q)  (  \MATHgroesser   --------)
            \MATHinfty    /   (q; q)
               \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       kk
               kk=0
\endMATH
\vskip10pt\noindent
Examples for letting the base $q$ tend to 1.
\vskip10pt
\MATH
In[1]:= Sgl3201
Do you want to set values for the equation? [y|n]: y
a=q\MATHhoch a
b=q\MATHhoch b
c=q\MATHhoch c
n=n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                                 -a + c        -b + c
           %
\MATHluEck        a   b   -n             %
\MATHruEck     (q      ; q)  (q      ; q)
           %
\MATHvStrich       q , q , q               %
\MATHvStrich                 n             n
Out[1]=  \MATHphi  %
\MATHvStrich                        ; q, q %
\MATHvStrich  == ---------------------------
        3 2%
\MATHvStrich   c   1 + a + b - c - n       %
\MATHvStrich       c        -a - b + c
           %
\MATHloEck  q , q                        %
\MATHroEck     (q ; q)  (q          ; q)
                                                      n                 n
\goodbreakpoint%
In[2]:= Limes[\%,q-\MATHgroesser 1]
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck     (-a + c)  (-b + c)
           %
\MATHvStrich        a, b, -n          %
\MATHvStrich             n         n
Out[2]=  F %
\MATHvStrich                      ; 1 %
\MATHvStrich  == -------------------
        3 2%
\MATHvStrich  c, 1 + a + b - c - n    %
\MATHvStrich     (c)  (-a - b + c)
           %
\MATHloEck                          %
\MATHroEck        n             n
\goodbreakpoint%
In[3]:= Sgl2103
Do you want to set values for the equation? [y|n]: y
a=q\MATHhoch a
b=q\MATHhoch b
c=q\MATHhoch c
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                           -a + c   -b + c
           %
\MATHluEck   a   b                 %
\MATHruEck     (q      , q      ; q)
           %
\MATHvStrich  q , q       -a - b + c %
\MATHvStrich                          \MATHinfty 
Out[3]=  \MATHphi  %
\MATHvStrich        ; q, q           %
\MATHvStrich  == ----------------------
        2 1%
\MATHvStrich     c                   %
\MATHvStrich       c   -a - b + c
           %
\MATHloEck    q                    %
\MATHroEck     (q , q          ; q)
                                                             \MATHinfty 
\goodbreakpoint%
In[4]:= Limes[\%,q-\MATHgroesser 1]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich     \MATHGamma (c) \MATHGamma (-a - b + c)
Out[4]=  F %
\MATHvStrich      ; 1 %
\MATHvStrich  == -------------------
        2 1%
\MATHvStrich   c      %
\MATHvStrich     \MATHGamma (-a + c) \MATHGamma (-b + c)
           %
\MATHloEck          %
\MATHroEck 
\endMATH
\vskip10pt\noindent
An example that should illustrate the hint in the Description of
\hbox{\tt Limes}.
\vskip10pt
\MATH
In[1]:= pq[a/q\MATHhoch n,2*n]/a\MATHhoch n*(-1)\MATHhoch n*q\MATHhoch Binomial[n+1,2]
\goodbreakpoint%
            n  (n (1 + n))/2  a
        (-1)  q              (--; q)
                               n    2 n
                              q
Out[1]= -------------------------------
                       n
                      a
\goodbreakpoint%
In[2]:= Limes[\%,n-\MATHgroesser Infinity]
\goodbreakpoint%
The expression
\goodbreakpoint%
Indeterminate
\goodbreakpoint%
was obtained.
\goodbreakpoint%
Therefore the limit n -\MATHgroesser  \MATHinfty  could not be determined.
Here is your expression:
\goodbreakpoint%
            n  (n (1 + n))/2  a
        (-1)  q              (--; q)
                               n    2 n
                              q
Out[2]= -------------------------------
                       n
                      a
\goodbreakpoint%
In[3]:= \%/.zerl
bottom-split by: n
\goodbreakpoint%
            n  (n (1 + n))/2          a
        (-1)  q              (a; q)  (--; q)
                                   n   n    n
                                      q
Out[3]= -------------------------------------
                          n
                         a
\goodbreakpoint%
In[4]:= Limes[\%,n-\MATHgroesser Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]: e
\goodbreakpoint%
                q
Out[4]= (a;q)  (-;q)
             \MATHinfty   a   \MATHinfty 
\endMATH

\Seealso AbsGreater, AbsSmaller, AbsUndetermined, MinusOne.

\Name lina1
\Description \vtab $(a;q)_n \to (1-a)(aq;q)_{n-1}$,\\
             $(a;q)_\infty \to (1-a)(aq;q)_\infty$.
\endvtab
\vskip6pt
\leavevmode\hskip10pt\Usage Expr/.lina1.
\Example 
\MATH 
In[1]:= pq[a,m,q\MATHhoch 2]
\goodbreakpoint%
             2
Out[1]= (a; q )
               m
\goodbreakpoint%
In[2]:= \%/.lina1
\goodbreakpoint%
                    2   2
Out[2]= (1 - a) (a q ; q )
                          -1 + m
\goodbreakpoint%
In[3]:= 1/pqinf[a,q\MATHhoch 2]
\goodbreakpoint%
           1
Out[3]= -------
            2
        (a;q )
              \MATHinfty 
\goodbreakpoint%
In[4]:= \%/.lina1
\goodbreakpoint%
                1
Out[4]= ------------------
                    2  2
        (1 - a) (a q ;q )
                         \MATHinfty 
\endMATH 
\Seealso lina2, linz, Ers, PosListe, ManipulationsListe.

\Name lina2
\Description $(a;q)_n \to (1-aq^{n-1})(a;q)_{n-1}$.
\Usage Expr/.lina2.
\Example 
\MATH 
In[1]:= pq[a,m,q\MATHhoch 2]
\goodbreakpoint%
             2
Out[1]= (a; q )
               m
\goodbreakpoint%
In[2]:= \%/.lina2
\goodbreakpoint%
                -2 + 2 m       2
Out[2]= (1 - a q        ) (a; q )
                                 -1 + m
\endMATH 
\Seealso lina1, linz, Ers, PosListe, ManipulationsListe.

\Name linz
\Description Rule that absorbs linear terms.
\Usage Expr/.linz.
\Example 
\MATH 
In[1]:= (1-a)*pq[a*q\MATHhoch 2,m,q\MATHhoch 2]/(1-a*q\MATHhoch (2*m-2))/pq[a,m-1,q\MATHhoch 2]
\goodbreakpoint%
                          2   2
              (1 - a) (a q ; q )
                                m
Out[1]= -------------------------------
                -2 + 2 m       2
        (1 - a q        ) (a; q )
                                 -1 + m
\goodbreakpoint%
In[2]:= \%/.linz
\goodbreakpoint%
                      2
                 (a; q )
                        1 + m
Out[2]= -------------------------------
                -2 + 2 m       2
        (1 - a q        ) (a; q )
                                 -1 + m
\goodbreakpoint%
In[3]:= \%/.linz
\goodbreakpoint%
             2
        (a; q )
               1 + m
Out[3]= ------------
               2
          (a; q )
                 m
\goodbreakpoint%
In[4]:= 1/(1-b/q)/pqinf[b]
\goodbreakpoint%
              1
Out[4]= --------------
             b
        (1 - -) (b;q)
             q       \MATHinfty 
\goodbreakpoint%
In[5]:= \%/.linz
\goodbreakpoint%
          1
Out[5]= ------
         b
        (-;q)
         q   \MATHinfty 
\endMATH 
\Seealso lina1, lina2, Ers, PosListe, ManipulationsListe.

\Name LS
\Description \hbox{\tt LS} is the left-hand side in \hbox{\tt Gleichung}.
\Usage LS.
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= LS
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich      -n       \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich 
        2 1\MATHvStrich               \MATHvStrich 
           \MATHloEck    c          \MATHroEck 
\goodbreakpoint%
In[3]:= Add[1]
\goodbreakpoint%
                                        n  c
               \MATHluEck               \MATHruEck         a  (-; q)
               \MATHvStrich      -n       \MATHvStrich             a    n
Out[3]= 1 +  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == 1 + ----------
            2 1\MATHvStrich               \MATHvStrich          (c; q)
               \MATHloEck    c          \MATHroEck                n
\goodbreakpoint%
In[4]:= LS
\goodbreakpoint%
               \MATHluEck               \MATHruEck 
               \MATHvStrich      -n       \MATHvStrich 
Out[4]= 1 +  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich 
            2 1\MATHvStrich               \MATHvStrich 
               \MATHloEck    c          \MATHroEck 
\goodbreakpoint%
In[5]:= LS=pq[a,m]
\goodbreakpoint%
Out[5]= (a; q)
              m
\goodbreakpoint%
In[6]:= Gleichung
\goodbreakpoint%
                        n  c
                       a  (-; q)
                           a    n
Out[6]= (a; q)  == 1 + ----------
              m         (c; q)
                              n
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, RS, Mal, Add, Div, Sub, 
Hoch, GlTausche, Ers,\linebreak Subst.

\Name Mal
\Description Function that multiplies \hbox{\tt Gleichung} by \hbox{\tt Expr}.
\Usage Mal[Expr].
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= Mal[pq[c,n,q]]
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich      -n       \MATHvStrich              n  c
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  (c; q)  == a  (-; q)
        2 1\MATHvStrich               \MATHvStrich        n        a    n
           \MATHloEck    c          \MATHroEck
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich      -n       \MATHvStrich              n  c
Out[3]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  (c; q)  == a  (-; q)
        2 1\MATHvStrich               \MATHvStrich        n        a    n
           \MATHloEck    c          \MATHroEck
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Add, Div, Sub, 
Hoch, GlTausche, Ers.

\Name ManipulationsListe
\Description Gives a list of all available rules for manipulating finite
  and infinite $q$-factorial symbols.
\Usage ManipulationsListe.

\Name MinusOne
\Description: Rule for getting rid of expressions of the form
$(-1)^N$ where $N$ is an even or odd integer.
\Usage Expr/.MinusOne.
\MATH
In[1]:= pq[a,2*n]
\goodbreakpoint%
Out[1]= (a; q)
              2 n
\goodbreakpoint%
In[2]:= \%/.trans
\goodbreakpoint%
                       1 - 2 n
            2 n  2 n  q
        (-1)    a    (--------; q)
                         a        2 n
Out[2]= -----------------------------
                          2
                (2 n - 4 n )/2
               q
\goodbreakpoint%
In[3]:= \%/.MinusOne
Is 2 n even, odd, or neither of both?
[e|o|n]: e
\goodbreakpoint%
               1 - 2 n
         2 n  q
        a    (--------; q)
                 a        2 n
Out[3]= ---------------------
                      2
            (2 n - 4 n )/2
           q
\endMATH
\Seealso SimplifyPQ, Expandq, SUMExpand.

\Name Multinomialpq
\Description: \hbox{\tt Multinomialq[n1,n2,\dots,q]} is the 
$q$-multinomial coefficient $\left[\smallmatrix {\sum_i n_i}
\\{n_1,n_2,\dots}\endsmallmatrix\right]_q$, 
written in terms of $q$-factorial symbols {\tt pq}. 
\Usage Multinomialpq[n1,n2,\dots,q].
\MATH
In[1]:= Multinomialpq[a,b,c,s]
\goodbreakpoint%
            (s; s)
                  a + b + c
Out[1]= -----------------------
        (s; s)  (s; s)  (s; s)
              a       b       c
\goodbreakpoint%
In[2]:= Multinomialpq[2,3,1,q]
\goodbreakpoint%
                (q; q)
                      6
Out[2]= -----------------------
        (q; q)  (q; q)  (q; q)
              1       2       3
\endMATH
\Seealso Multinomialq, Binomialq, Binomialpq, Factorialq, Factorialpq.

\Name Multinomialq
\Description: \hbox{\tt Multinomialq[n1,n2,\dots,q]} is the 
$q$-multinomial coefficient $\left[\smallmatrix {\sum_i n_i}
\\{n_1,n_2,\dots}\endsmallmatrix\right]_q$, 
expanded into a $q$-series, if possible.
\Usage Multinomialq[n1,n2,\dots,q].
\MATH
In[1]:= Multinomialq[2,3,1,s]
\goodbreakpoint%
                     2      3      4      5      6      7      8      9
Out[1]= 1 + 2 s + 4 s  + 6 s  + 8 s  + 9 s  + 9 s  + 8 s  + 6 s  + 4 s  + 
 
        10    11
\MATHgroesser     2 s   + s
\goodbreakpoint%
In[2]:= Multinomialq[a,b,c,q]
\goodbreakpoint%
        %
\MATHluEck            %
\MATHruEck  
        %
\MATHvStrich  a + b + c %
\MATHvStrich  
Out[2]= %
\MATHvStrich            %
\MATHvStrich  
        %
\MATHvStrich   a, b, c  %
\MATHvStrich  
        %
\MATHloEck            %
\MATHroEck  
                     q
\endMATH
\Seealso Multinomialpq, Binomialq, Binomialpq, Factorialq, Factorialpq.

\Name neg1
\Description $(a;q)_n \to 1/(aq^n;q)_{-n}$.
\Usage Expr/.neg1.
\Example 
\MATH 
In[1]:= pq[a\MATHhoch 2,-n]
\goodbreakpoint%
          2
Out[1]= (a ; q)
               -n
\goodbreakpoint%
In[2]:= \%/.neg1
\goodbreakpoint%
            1
Out[2]= ------------
          2  -n
        (a  q  ; q)
                   n
\endMATH 
\Seealso neg2, Ers, PosListe, ManipulationsListe.

\Name neg2
\Description $(a;q)_n \to q^{\binom {n+1}2}/((-q/a)^n(q/a;q)_{-n})$.
\Usage Expr/.neg2.
\Example 
\MATH 
In[1]:= pq[a\MATHhoch 2,-n]
\goodbreakpoint%
          2
Out[1]= (a ; q)
               -n
\goodbreakpoint%
In[2]:= \%/.neg2
\goodbreakpoint%
                        2
                 n/2 + n /2
            n   q
Out[2]= (-1)  --------------
               2 n  q
              a    (--; q)
                     2    n
                    a
\endMATH 
\Seealso neg1, Ers, PosListe, ManipulationsListe.

\Name Ph
\Description \hbox{\tt 
Ph[List1A,List1B,q1,List2A,List2B,q2,\dots,ListkA,ListkB,qk,z]}
is the multibasic basic hypergeometric series with upper parameters 
\hbox{\tt List}1A and lower parameters \hbox{\tt List}1B for base
\hbox{\tt q}1,\dots , with upper 
parameters \hbox{\tt List}kA and lower parameters \hbox{\tt List}kB for base 
\hbox{\tt q}k, and argument \hbox{\tt z}.
\Usage Ph[List1A,List1B,q1,List2A,List2B,q2,\dots,ListkA,ListkB,qk,z].
\Example
\MATH
In[1]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,%
\MATHlbrace e,f,g%
\MATHrbrace ,%
\MATHlbrace E,F,G%
\MATHrbrace ,p,z]
\goodbreakpoint%
         %
\MATHluEck                          %
\MATHruEck 
         %
\MATHvStrich  a, b: e, f, g          %
\MATHvStrich 
Out[1]= \MATHphi %
\MATHvStrich                ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   c: E, F, G            %
\MATHvStrich 
         %
\MATHloEck                          %
\MATHroEck 
\endMATH
\Seealso SListe, TListe, SUMRegeln, SUMph, PhSUM, ph, ps, pq, pqinf, 
phCancel, phOrdne, phPerm,\linebreak 
phTausche, PQ, phFormat.

\Name ph
\Description \hbox{\tt ph[List1,List2,q,z]} is the basic hypergeometric series with 
  upper parameters \hbox{\tt List}1, lower parameters \hbox{\tt List}2, base
\hbox{\tt q}, and argument \hbox{\tt z}.
\Usage ph[List1,List2,q,z].
\Example 
\MATH 
In[1]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck             \MATHruEck 
           \MATHvStrich  a, b       \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich      ; q, z \MATHvStrich 
        2 1\MATHvStrich   c         \MATHvStrich 
           \MATHloEck             \MATHroEck 
\goodbreakpoint%
In[2]:= ph[\MATHlbrace a,b,c\MATHrbrace ,\MATHlbrace d,e,0\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck                \MATHruEck 
           \MATHvStrich  a, b, c       \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich         ; q, z \MATHvStrich 
        3 3\MATHvStrich  d, e, 0       \MATHvStrich 
           \MATHloEck                \MATHroEck
\endMATH 
\Seealso SListe, TListe, SUMRegeln, SUMph, phSUM, Ph, ph, pq, pqinf, phCancel, phOrdne,
phPerm, phTausche,
PQ, phFormat.\NoBlackBoxes\par\BlackBoxes

\Name phCancel
\Description Switch that activates automatic cancelling of the upper and 
  lower parameters in \hbox{\tt ph[]}, \hbox{\tt Ph[]} and \hbox{\tt ps[]}, 
or makes it inactive, respectively. By default
  automatic cancelling is active.
\Usage phCancel.
\Example 
\MATH 
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck          \MATHruEck 
           \MATHvStrich  b       \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich   ; q, z \MATHvStrich 
        1 1\MATHvStrich  c       \MATHvStrich 
           \MATHloEck          \MATHroEck 
\goodbreakpoint%
In[3]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,n,q\MATHhoch 2]
\goodbreakpoint%
             2
        (b; q )
               n
Out[3]= --------
             2
        (c; q )
               n
\goodbreakpoint%
In[4]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ]
\goodbreakpoint%
        (b; q)
              \MATHinfty 
Out[4]= -------
        (c; q)
              \MATHinfty 
\goodbreakpoint%
In[5]:= phCancel
\goodbreakpoint%
In[6]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph inactive.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[7]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck             \MATHruEck 
           \MATHvStrich  a, b       \MATHvStrich 
Out[7]=  \MATHphi  \MATHvStrich      ; q, z \MATHvStrich 
        2 2\MATHvStrich  a, c       \MATHvStrich 
           \MATHloEck             \MATHroEck 
\goodbreakpoint%
In[8]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,n,q\MATHhoch 2]
\goodbreakpoint%
             2
        (b; q )
               n
Out[8]= --------
             2
        (c; q )
               n
\goodbreakpoint%
In[9]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ]
\goodbreakpoint%
        (b; q)
              \MATHinfty 
Out[9]= -------
        (c; q)
              \MATHinfty 
\goodbreakpoint%
In[10]:= phCancel
\goodbreakpoint%
In[11]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\endMATH 
\Seealso ph, W, hypqAttributes.

\Name PhEinf
\Description Rule that inactivates automatic cancelling in \hbox{\tt
Ph[]} and then
adds a parameter which has to be entered on request, together with
the information to which base it belongs, to the upper and lower 
 parameters of \hbox{\tt Ph[]}.
\Usage Expr/.PhEinf.
\Example
\MATH
In[1]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,%
\MATHlbrace e,f,g%
\MATHrbrace ,%
\MATHlbrace E,F,G%
\MATHrbrace ,p,z]
\goodbreakpoint%
         %
\MATHluEck                          %
\MATHruEck 
         %
\MATHvStrich  a, b: e, f, g          %
\MATHvStrich 
Out[1]= \MATHphi %
\MATHvStrich                ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   c: E, F, G            %
\MATHvStrich 
         %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.PhEinf
Add the parameter: A
to the parameters belonging to the i-th base, where i=1
\goodbreakpoint%
         %
\MATHluEck                             %
\MATHruEck 
         %
\MATHvStrich  A, a, b: e, f, g          %
\MATHvStrich 
Out[2]= \MATHphi %
\MATHvStrich                   ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   A, c: E, F, G            %
\MATHvStrich 
         %
\MATHloEck                             %
\MATHroEck 
\goodbreakpoint%
In[3]:= \%/.PhEinf
Add the parameter: B
to the parameters belonging to the i-th base, where i=2
\goodbreakpoint%
         %
\MATHluEck                                %
\MATHruEck 
         %
\MATHvStrich  A, a, b: B, e, f, g          %
\MATHvStrich 
Out[3]= \MATHphi %
\MATHvStrich                      ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   A, c: B, E, F, G            %
\MATHvStrich 
         %
\MATHloEck                                %
\MATHroEck 
\endMATH
\Seealso phCancel, phOrdne, PhPerm, PQSort, SUMRegeln, SUMUmkehr, Ers, 
PosListe.

\Name phEinf
\Description Rule that inactivates automatic cancelling in \hbox{\tt
ph[]} and then
  adds a parameter which has to be entered on request to the upper and
  lower parameters of \hbox{\tt ph[]}.
\Usage Expr/.phEinf.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace b,c,q%
\MATHrbrace ,%
\MATHlbrace q*a/c,q*a/b%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck  b, c, q        %
\MATHruEck 
           %
\MATHvStrich                 %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich  a q  a q; q, z %
\MATHvStrich 
        3 2%
\MATHvStrich  ---, ---       %
\MATHvStrich 
           %
\MATHloEck   c    b        %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.phEinf
Add the parameter: a
\goodbreakpoint%
           %
\MATHluEck  a, b, c, q        %
\MATHruEck 
           %
\MATHvStrich                    %
\MATHvStrich 
Out[2]=  \MATHphi  %
\MATHvStrich     a q  a q; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich  a, ---, ---       %
\MATHvStrich 
           %
\MATHloEck      c    b        %
\MATHroEck 
\endMATH
\Seealso phOrdne, phPerm, phTausche, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe.

\Name phFormat
\Description Switch that activates basic hypergeometric output, or makes 
  it inactive, respectively. By default basic hypergeometric output is 
  active.
\Usage phFormat.
\Example
\MATH
In[1]:= pq[a,n]/pqinf[b,q\MATHhoch 2]*ph[\MATHlbrace c,d\MATHrbrace ,\MATHlbrace c*d\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck             \MATHruEck 
           \MATHvStrich  c, d       \MATHvStrich 
         \MATHphi  \MATHvStrich      ; q, z \MATHvStrich  (a; q)
        2 1\MATHvStrich  c d        \MATHvStrich        n
           \MATHloEck             \MATHroEck 
Out[1]= -------------------------
                     2
                 (b;q )
                       \MATHinfty 
\goodbreakpoint%
In[2]:= Tgl5402
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%

Format::toobig: Expression too big for output.
                 Enter "phFormat" and retry.

\goodbreakpoint%
In[3]:= phFormat
\goodbreakpoint%
In[4]:= \%\%
\goodbreakpoint%
                                1 - n   1 - n   1 - n
             -n                q       q       q        2  2  2    -2 + 2 n
Out[4]= ph[\MATHlbrace q  , b, c, d, e\MATHrbrace , \MATHlbrace ------, ------, ------, b  c  d  e q        \MATHrbrace , 
                                 b       c       d
\goodbreakpoint% 
                    1 - 2 n          3/2 - n
                   q                q
\MATHgroesser      q, q] == ph[\MATHlbrace --------, -----------------------, 
                    b c d    Sqrt[b] Sqrt[c] Sqrt[d]
\goodbreakpoint% 
                  3/2 - n            1 - n   1 - n   1 - n
                 q                  q       q       q        -n/2    -n/2
\MATHgroesser        -(-----------------------), ------, ------, ------, q    , -q    , 
          Sqrt[b] Sqrt[c] Sqrt[d]    c d     b d     b c
\goodbreakpoint% 
                                      3 - 3 n
         1/2 - n/2    1/2 - n/2      q
\MATHgroesser        q         , -q         , e, ----------\MATHrbrace , 
                                     2  2  2
                                    b  c  d  e
\goodbreakpoint% 
                1/2 - n                    1/2 - n            1 - n   1 - n
               q                          q                  q       q
\MATHgroesser       \MATHlbrace -----------------------, -(-----------------------), ------, ------, 
        Sqrt[b] Sqrt[c] Sqrt[d]    Sqrt[b] Sqrt[c] Sqrt[d]     b       c
\goodbreakpoint% 
         1 - n   2 - (3 n)/2     2 - (3 n)/2    3/2 - (3 n)/2
        q       q               q              q
\MATHgroesser        ------, ------------, -(------------), --------------, 
          d        b c d           b c d           b c d
\goodbreakpoint% 
           3/2 - (3 n)/2    2 - 2 n
          q                q                  -1 + n
\MATHgroesser        -(--------------), --------, b c d e q      \MATHrbrace , q, q] 
              b c d        b c d e
\goodbreakpoint% 
           2 - 2 n   3 - 3 n     2 - 2 n    3 - 3 n
          q         q           q          q
\MATHgroesser      pq[\MATHlbrace --------, --------\MATHrbrace , \MATHlbrace --------, ----------\MATHrbrace , n, q]
          b c d e    2  2  2     b c d     2  2  2
                    b  c  d               b  c  d  e
\goodbreakpoint%
In[5]:= \%1
\goodbreakpoint%
        ph[\MATHlbrace c, d\MATHrbrace , \MATHlbrace c d\MATHrbrace , q, z] pq[a, n, q]
Out[5]= -----------------------------------
                              2
                   pqinf[b, q ]
\goodbreakpoint%
In[6]:= phFormat
\goodbreakpoint%
In[7]:= \%1
\goodbreakpoint%
           \MATHluEck             \MATHruEck 
           \MATHvStrich  c, d       \MATHvStrich 
         \MATHphi  \MATHvStrich      ; q, z \MATHvStrich  (a; q)
        2 1\MATHvStrich  c d        \MATHvStrich        n
           \MATHloEck             \MATHroEck 
Out[7]= -------------------------
                     2
                 (b;q )
                       \MATHinfty 
\endMATH
\Seealso ph.

\Name Phinv
\Description Rule that transforms a multibasic hypergeometric
series\linebreak
\hbox{\tt
Ph[List1A,List1B,q1,List2A,List2B,q2,\dots,ListkA,ListkB,qk,z]} 
with bases \hbox{\tt q}1,\dots, into a multibasic hypergeometric series
\hbox{\tt Ph[\dots,1/q1,,\dots,1/q2,\dots,\dots,1/qk,z]} with bases
1/\hbox{\tt q}1,\dots
\Usage Expr/.Phinv.
\Example
\MATH
In[1]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,%
\MATHlbrace e,f,g%
\MATHrbrace ,%
\MATHlbrace E,F,G%
\MATHrbrace ,p,z]
\goodbreakpoint%
         %
\MATHluEck                          %
\MATHruEck 
         %
\MATHvStrich  a, b: e, f, g          %
\MATHvStrich 
Out[1]= \MATHphi %
\MATHvStrich                ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   c: E, F, G            %
\MATHvStrich 
         %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.Phinv
\goodbreakpoint%
           1  1   1  1  1
         %
\MATHluEck  -, -: -, -, -                    %
\MATHruEck 
         %
\MATHvStrich  a  b   e  f  g  1  1  a b e f g z %
\MATHvStrich 
Out[2]= \MATHphi %
\MATHvStrich                ; -, -; ----------- %
\MATHvStrich 
         %
\MATHvStrich   1   1  1  1    q  p   c E F G q  %
\MATHvStrich 
         %
\MATHloEck   -: -, -, -                      %
\MATHroEck 
            c   E  F  G
\goodbreakpoint%
In[3]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,0%
\MATHrbrace ,q,%
\MATHlbrace e,f,g%
\MATHrbrace ,%
\MATHlbrace E,F%
\MATHrbrace ,p,z]
\goodbreakpoint%
         %
\MATHluEck                          %
\MATHruEck 
         %
\MATHvStrich  a, b: e, f, g          %
\MATHvStrich 
Out[3]= \MATHphi %
\MATHvStrich                ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   c, 0: E, F            %
\MATHvStrich 
         %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.Phinv
\goodbreakpoint%
           1  1      1  1  1
         %
\MATHluEck  -, -, 0: -, -, -                    %
\MATHruEck 
         %
\MATHvStrich  a  b      e  f  g  1  1  a b e f g z %
\MATHvStrich 
Out[4]= \MATHphi %
\MATHvStrich                   ; -, -; ----------- %
\MATHvStrich 
         %
\MATHvStrich     1   1  1        q  p    c E F q   %
\MATHvStrich 
         %
\MATHloEck     -: -, -, 0                       %
\MATHroEck 
              c   E  F
\endMATH
\Seealso SUMRegeln, Ers, PosListe, phinv, psinv.

\Name phinv
\Description Rule that transforms a basic hypergeometric series
    \hbox{\tt ph[List1,List2,q,z]} with base {\tt q} into a basic hypergeometric series
    \hbox{\tt ph[\dots,1/q,..]} with base \hbox{\tt 1/q}.
\Usage Expr/.phinv.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2 1%
\MATHvStrich   c         %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.phinv
\goodbreakpoint%
             1  1
           %
\MATHluEck  -, -           %
\MATHruEck 
           %
\MATHvStrich  a  b  1  a b z %
\MATHvStrich 
Out[2]=  \MATHphi  %
\MATHvStrich      ; -, ----- %
\MATHvStrich 
        2 1%
\MATHvStrich   1    q   c q  %
\MATHvStrich 
           %
\MATHloEck   -             %
\MATHroEck 
              c
\goodbreakpoint%
In[3]:= ph[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,0%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[3]=  \MATHphi  %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3 2%
\MATHvStrich   d, 0         %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.phinv
\goodbreakpoint%
             1  1  1
           %
\MATHluEck  -, -, -             %
\MATHruEck 
           %
\MATHvStrich  a  b  c  1  a b c z %
\MATHvStrich 
Out[4]=  \MATHphi  %
\MATHvStrich         ; -, ------- %
\MATHvStrich 
        3 1%
\MATHvStrich     1     q    d q   %
\MATHvStrich 
           %
\MATHloEck     -                %
\MATHroEck 
                d
\endMATH
\Seealso SUMRegeln, Ers, PosListe.

\Name phOrdne
\Description Rule that tries to order the parameters of a basic 
  hypergeometric series in ``well-poised" order. If there is an
  upper parameter of the form $q^{-n}$, where $n$ might be a
nonnegative integer, then it
  is put at the very last place in the upper list. If the parameters
  could be paired such that the product of each pair equals 
\hbox{\tt A}$q$, however
\hbox{\tt A} is
  missing in the upper parameters, then you have to add \hbox{\tt A} to the upper
  and lower parameters by \hbox{\tt phEinf} before applying \hbox{\tt
phOrdne}.
\Usage Expr/.phOrdne.
\Example 
\MATH 
In[1]:= ph[\MATHlbrace q\MATHhoch -n,b,q*Sqrt[a],a,-q*Sqrt[a]\MATHrbrace %
,\MATHlbrace a*q/b,-Sqrt[a],Sqrt[a],
        a*q\MATHhoch (1+n)\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck   -n                                      \MATHruEck 
           \MATHvStrich  q  , b, Sqrt[a] q, a, -(Sqrt[a] q)       \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich                                    ; q, z \MATHvStrich 
        5 4\MATHvStrich   a q                        1 + n        \MATHvStrich 
           \MATHloEck   ---, -Sqrt[a], Sqrt[a], a q             \MATHroEck 
               b
\goodbreakpoint%
In[2]:= \%/.phOrdne
\goodbreakpoint%
           \MATHluEck                                  -n       \MATHruEck 
           \MATHvStrich  a, Sqrt[a] q, -(Sqrt[a] q), b, q         \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich                                    ; q, z \MATHvStrich 
        5 4\MATHvStrich                      a q     1 + n        \MATHvStrich 
           \MATHloEck   Sqrt[a], -Sqrt[a], ---, a q             \MATHroEck 
                                  b
\goodbreakpoint%
In[3]:= ph[\MATHlbrace q\MATHhoch -n,b,c\MATHrbrace ,\MATHlbrace d,e\MATHrbrace %
,q\MATHhoch 2,z]
\goodbreakpoint%
           \MATHluEck                   \MATHruEck 
           \MATHvStrich   -n         2    \MATHvStrich 
Out[3]=  \MATHphi  \MATHvStrich  q  , b, c; q , z \MATHvStrich 
        3 2\MATHvStrich                   \MATHvStrich 
           \MATHloEck    d, e           \MATHroEck 
\goodbreakpoint%
In[4]:= \%/.phOrdne
\goodbreakpoint%
           \MATHluEck                   \MATHruEck 
           \MATHvStrich         -n   2    \MATHvStrich 
Out[4]=  \MATHphi  \MATHvStrich  b, c, q  ; q , z \MATHvStrich 
        3 2\MATHvStrich                   \MATHvStrich 
           \MATHloEck    e, d           \MATHroEck
In[5]:= ph[%
\MATHlbrace b,c,q%
\MATHrbrace ,%
\MATHlbrace a*q/c,a*q/b%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck  b, c, q        %
\MATHruEck 
           %
\MATHvStrich                 %
\MATHvStrich 
Out[5]=  \MATHphi  %
\MATHvStrich  a q  a q; q, z %
\MATHvStrich 
        3 2%
\MATHvStrich  ---, ---       %
\MATHvStrich 
           %
\MATHloEck   c    b        %
\MATHroEck 
\goodbreakpoint%
In[6]:= \%/.phOrdne
\goodbreakpoint%
           %
\MATHluEck  b, c, q        %
\MATHruEck 
           %
\MATHvStrich                 %
\MATHvStrich 
Out[6]=  \MATHphi  %
\MATHvStrich  a q  a q; q, z %
\MATHvStrich 
        3 2%
\MATHvStrich  ---, ---       %
\MATHvStrich 
           %
\MATHloEck   c    b        %
\MATHroEck 
\goodbreakpoint%
In[7]:= \%/.phEinf
Add the parameter: a
\goodbreakpoint%
           %
\MATHluEck  a, b, c, q        %
\MATHruEck 
           %
\MATHvStrich                    %
\MATHvStrich 
Out[7]=  \MATHphi  %
\MATHvStrich     a q  a q; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich  a, ---, ---       %
\MATHvStrich 
           %
\MATHloEck      c    b        %
\MATHroEck 
\goodbreakpoint%
In[8]:= \%/.phOrdne
\goodbreakpoint%
           %
\MATHluEck  a, b, c, q        %
\MATHruEck 
           %
\MATHvStrich                    %
\MATHvStrich 
Out[8]=  \MATHphi  %
\MATHvStrich  a q  a q   ; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich  ---, ---, a       %
\MATHvStrich 
           %
\MATHloEck   b    c           %
\MATHroEck 
\goodbreakpoint%
In[9]:= phCancel
\goodbreakpoint%
In[10]:= \%\%/.phOrdne
\goodbreakpoint%
            %
\MATHluEck  b, c, q        %
\MATHruEck 
            %
\MATHvStrich                 %
\MATHvStrich 
Out[10]=  \MATHphi  %
\MATHvStrich  a q  a q; q, z %
\MATHvStrich 
         3 2%
\MATHvStrich  ---, ---       %
\MATHvStrich 
            %
\MATHloEck   b    c        %
\MATHroEck 
\endMATH 
\Seealso phEinf, phPerm, phTausche, ph, W, PQSort, Ers, PosListe.

\Name phPerm
\Description Rule for permuting parameters in basic hypergeometric series.
\Usage Expr/.phPerm[<Permutation>,x].\newline
\rm  {\tt x} can be {\tt u}, {\tt l}, {\tt b}. u causes a permutation of upper parameters,
{\tt l} causes a 
  permutation of lower parameters, {\tt b} causes a simultaneous permutation 
  of respective upper and lower parameters. \hbox{\tt Permutation} must be a 
  sequence of positive numbers forming a permutation. Under the options
{\tt u} and {\tt l} the effect is that
  the new parameter at position {\tt i} is the old parameter from position
  \hbox{\tt Permutation[i]}. However, the behaviour of \hbox{\tt
FPerm} under the option {\tt b} is special. The option {\tt b} is
especially designed for the permutation of parameters of {\it
well-poised} series. Hence, the first upper parameter is not moved,
whereas the new {\it upper} parameter at position {\tt i+1} is the 
old upper parameter from position \hbox{\tt Permutation[i]+1}, and
the new {\it lower} parameter at position {\tt i} is the old lower
parameter from position \hbox{\tt Permutation[i]}.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,b,c,d%
\MATHrbrace ,%
\MATHlbrace e,f,g%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck                   %
\MATHruEck 
           %
\MATHvStrich  a, b, c, d       %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich            ; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich   e, f, g         %
\MATHvStrich 
           %
\MATHloEck                   %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%1/.phPerm[3,2,1,u]
\goodbreakpoint%
           %
\MATHluEck                   %
\MATHruEck 
           %
\MATHvStrich  c, b, a, d       %
\MATHvStrich 
Out[2]=  \MATHphi  %
\MATHvStrich            ; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich   e, f, g         %
\MATHvStrich 
           %
\MATHloEck                   %
\MATHroEck 
\goodbreakpoint%
In[3]:= \%1/.phPerm[3,2,1,l]
\goodbreakpoint%
           %
\MATHluEck                   %
\MATHruEck 
           %
\MATHvStrich  a, b, c, d       %
\MATHvStrich 
Out[3]=  \MATHphi  %
\MATHvStrich            ; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich   g, f, e         %
\MATHvStrich 
           %
\MATHloEck                   %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%1/.phPerm[2,1,b]
\goodbreakpoint%
           %
\MATHluEck                   %
\MATHruEck 
           %
\MATHvStrich  a, c, b, d       %
\MATHvStrich 
Out[4]=  \MATHphi  %
\MATHvStrich            ; q, z %
\MATHvStrich 
        4 3%
\MATHvStrich   f, e, g         %
\MATHvStrich 
           %
\MATHloEck                   %
\MATHroEck 
\endMATH
\Seealso phTausche, phOrdne, ph, W, PQSort, Ers, PosListe.

\Name Phph
\Description Rule that transforms a \hbox{\tt Ph[]} into ``ordinary" 
hypergeometric notation, if possible.
\Usage Expr/.Phph.
\Example
\MATH
In[1]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,%
\MATHlbrace e,f,g%
\MATHrbrace ,%
\MATHlbrace E,F,G%
\MATHrbrace ,q\MATHhoch 2,z]
\goodbreakpoint%
         %
\MATHluEck                           %
\MATHruEck 
         %
\MATHvStrich  a, b: e, f, g      2    %
\MATHvStrich 
Out[1]= \MATHphi %
\MATHvStrich                ; q, q ; z %
\MATHvStrich 
         %
\MATHvStrich   c: E, F, G             %
\MATHvStrich 
         %
\MATHloEck                           %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.Phph
\goodbreakpoint%
           %
\MATHluEck  
           %
\MATHvStrich  
Out[2]=  \MATHphi  %
\MATHvStrich  
        8 7%
\MATHvStrich  
           %
\MATHloEck  
 
                                                                          %
\MATHruEck 
      a, b, Sqrt[e], -Sqrt[e], Sqrt[f], -Sqrt[f], Sqrt[g], -Sqrt[g]       %
\MATHvStrich 
\MATHgroesser                                                                   ; q, z %
\MATHvStrich 
       c, Sqrt[E], -Sqrt[E], Sqrt[F], -Sqrt[F], Sqrt[G], -Sqrt[G]         %
\MATHvStrich 
                                                                          %
\MATHroEck 
\endMATH
\Seealso Ph, ph, phPh, Ers, PosListe.

\Name phPh
\Description Rule that transforms a \hbox{\tt ph[]} into multibasic notation.
\Usage Expr/.phPh.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,d,e\MATHhoch (1/2),-e\MATHhoch (1/2),f\MATHhoch (1/2),-f\MATHhoch (1/2),g\MATHhoch (1/2),-g\MATHhoch (1/2)%
\MATHrbrace ,
        %
\MATHlbrace c,E\MATHhoch (1/2),-E\MATHhoch (1/2),F\MATHhoch (1/2),-F\MATHhoch (1/2),G\MATHhoch (1/2),-G\MATHhoch (1/2)%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck  
           %
\MATHvStrich  
Out[1]=  \MATHphi  %
\MATHvStrich  
        8 7%
\MATHvStrich  
           %
\MATHloEck  
 
                                                                          %
\MATHruEck 
      a, d, Sqrt[e], -Sqrt[e], Sqrt[f], -Sqrt[f], Sqrt[g], -Sqrt[g]       %
\MATHvStrich 
\MATHgroesser                                                                   ; q, z %
\MATHvStrich 
       c, Sqrt[E], -Sqrt[E], Sqrt[F], -Sqrt[F], Sqrt[G], -Sqrt[G]         %
\MATHvStrich 
                                                                          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.phPh
\goodbreakpoint%
         %
\MATHluEck                           %
\MATHruEck 
         %
\MATHvStrich  a, d: e, f, g      2    %
\MATHvStrich 
Out[2]= \MATHphi %
\MATHvStrich                ; q, q ; z %
\MATHvStrich 
         %
\MATHvStrich   c: E, F, G             %
\MATHvStrich 
         %
\MATHloEck                           %
\MATHroEck 
\endMATH
\Seealso ph, Ph, Phph, Ers, PosListe.

\Name phps
\Description Rule that transforms a ph[] into a difference of a
\hbox{\tt ps[]} and a \hbox{\tt ph[]}.
\Usage Expr/.phps.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2 1%
\MATHvStrich   c         %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.phps
\goodbreakpoint%
                2
               q                     c
             %
\MATHluEck  --, q, q           %
\MATHruEck  (-, 1; q)
             %
\MATHvStrich  c             c q  %
\MATHvStrich   q       1
           \MATHphi  %
\MATHvStrich          ; q, ----- %
\MATHvStrich  ----------
          3 2%
\MATHvStrich    2   2      a b z %
\MATHvStrich   a  b
             %
\MATHloEck   q   q             %
\MATHroEck  (-, -; q)
                --, --               q  q    1        %
\MATHluEck             %
\MATHruEck 
                a   b                                 %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[2]= -(------------------------------------) +  ps %
\MATHvStrich      ; q, z %
\MATHvStrich 
                           z                      2  2%
\MATHvStrich  c, q       %
\MATHvStrich 
                                                      %
\MATHloEck             %
\MATHroEck 
In[3]:= \%/.pqaufl
\goodbreakpoint%
            %
\MATHluEck             %
\MATHruEck 
            %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[3]=  ps %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2  2%
\MATHvStrich  c, q       %
\MATHvStrich 
            %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[4]:= ph[%
\MATHlbrace a,b,q%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, b, q       %
\MATHvStrich 
Out[4]=  \MATHphi  %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3 2%
\MATHvStrich   c, d         %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[5]:= \%/.phps
\goodbreakpoint%
                2   2
               q   q                  c  d
             %
\MATHluEck  --, --, q           %
\MATHruEck  (-, -; q)
             %
\MATHvStrich  c   d          c d  %
\MATHvStrich   q  q    1
           \MATHphi  %
\MATHvStrich           ; q, ----- %
\MATHvStrich  ----------
          3 2%
\MATHvStrich    2   2       a b z %
\MATHvStrich   a  b
             %
\MATHloEck   q   q              %
\MATHroEck  (-, -; q)
                --, --                q  q    1        %
\MATHluEck             %
\MATHruEck 
                a   b                                  %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[5]= -(-------------------------------------) +  ps %
\MATHvStrich      ; q, z %
\MATHvStrich 
                            z                      2  2%
\MATHvStrich  c, d       %
\MATHvStrich 
                                                       %
\MATHloEck             %
\MATHroEck 
\endMATH
\Seealso ph, ps, psph, Ers, PosListe.

\Name PhSUM
\Description Rule that transforms a \hbox{\tt Ph[]} into a \hbox{\tt
SUM[]}.
\Usage Expr/.PhSUM.
\Example
\MATH
In[1]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,%
\MATHlbrace e,f,g%
\MATHrbrace ,%
\MATHlbrace E,F,G%
\MATHrbrace ,p,z]
\goodbreakpoint%
         %
\MATHluEck                          %
\MATHruEck 
         %
\MATHvStrich  a, b: e, f, g          %
\MATHvStrich 
Out[1]= \MATHphi %
\MATHvStrich                ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich   c: E, F, G            %
\MATHvStrich 
         %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.PhSUM
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
          \MATHinfty    k
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q)  (b; q)  (e; p)  (f; p)  (g; p)
         \MATHbackslash             k       k       k       k       k
Out[2]=   \MATHgroesser   ------------------------------------------
         /    (c; q)  (E; p)  (F; p)  (G; p)  (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck        k       k       k       k       k
         k=0
\goodbreakpoint%
In[3]:= Ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,%
\MATHlbrace e,f,p\MATHhoch -n%
\MATHrbrace ,%
\MATHlbrace E,F,G%
\MATHrbrace ,p,z]
\goodbreakpoint%
         %
\MATHluEck                            %
\MATHruEck 
         %
\MATHvStrich                -n          %
\MATHvStrich 
Out[3]= \MATHphi %
\MATHvStrich  a, b: e, f, p  ; q, p; z %
\MATHvStrich 
         %
\MATHvStrich                            %
\MATHvStrich 
         %
\MATHloEck    c: E, F, G             %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.PhSUM
Is n a nonnegative integer?
[y|n]:  y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
          n   k                                   -n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q)  (b; q)  (e; p)  (f; p)  (p  ; p)
         \MATHbackslash             k       k       k       k         k
Out[4]=   \MATHgroesser   --------------------------------------------
         /     (c; q)  (E; p)  (F; p)  (G; p)  (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck         k       k       k       k       k
         k=0
\endMATH
\Seealso Ph, SUM, SUMPh, Ers, PosListe.

\Name phSUM
\Description Rule that transforms a \hbox{\tt ph[]} into a \hbox{\tt
SUM[]}.
\Usage Expr/.phSUM.
\Example 
\MATH 
In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q\MATHhoch 2,z]
\goodbreakpoint%
           %
\MATHluEck              %
\MATHruEck 
           %
\MATHvStrich  a, b   2    %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      ; q , z %
\MATHvStrich 
        2 1%
\MATHvStrich   c          %
\MATHvStrich 
           %
\MATHloEck              %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.phSUM
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
          \MATHinfty    k      2        2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q )  (b; q )
         \MATHbackslash              k        k
Out[2]=   \MATHgroesser   --------------------
         /         2     2   2
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck  (c; q )  (q ; q )
         k=0         k         k
\goodbreakpoint%
In[3]:= ph[%
\MATHlbrace a,q\MATHhoch (-2*n)%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q\MATHhoch 2,z]
\goodbreakpoint%
           %
\MATHluEck                  %
\MATHruEck 
           %
\MATHvStrich      -2 n   2    %
\MATHvStrich 
Out[3]=  \MATHphi  %
\MATHvStrich  a, q    ; q , z %
\MATHvStrich 
        2 1%
\MATHvStrich                  %
\MATHvStrich 
           %
\MATHloEck     c            %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.phSUM
Is n a nonnegative integer?
[y|n]:  y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: j
\goodbreakpoint%
          n   j      2     -2 n   2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q )  (q    ; q )
         \MATHbackslash              j            j
Out[4]=   \MATHgroesser   ------------------------
         /           2     2   2
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    (c; q )  (q ; q )
         j=0           j         j
\goodbreakpoint%
In[5]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[5]=  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2 2%
\MATHvStrich  c, d       %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[6]:= \%/.phSUM
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
                            2
          \MATHinfty       k  -k/2 + k /2  k
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (-1)  q            z  (a; q)  (b; q)
         \MATHbackslash                                k       k
Out[6]=   \MATHgroesser   -------------------------------------
         /          (c; q)  (d; q)  (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck              k       k       k
         k=0
\endMATH 
\Seealso ph, SUM, SUMph, Ers, PosListe.

\Name phTausche
\Description Rule for reordering parameters in basic hypergeometric series.
\Usage Expr/.phTausche[n1,n2,x].\newline
\rm  \hbox{\tt x} can be \hbox{\tt u}, \hbox{\tt l}, \hbox{\tt b}.
\hbox{\tt u} causes a reordering of upper parameters, \hbox{\tt l} causes a 
  reordering of lower parameters, \hbox{\tt b} causes a simultaneous reordering 
  of respective upper and lower parameters. \hbox{\tt n1} is the position of the
  parameter to be reordered, \hbox{\tt n2} is the new position.
\Example 
\MATH 
In[1]:= ph[\MATHlbrace q\MATHhoch -n,b,q*Sqrt[a],a,-q*Sqrt[a]\MATHrbrace %
,\MATHlbrace a*q/b,-Sqrt[a],Sqrt[a],
        a*q\MATHhoch (1+n)\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck   -n                                      \MATHruEck 
           \MATHvStrich  q  , b, Sqrt[a] q, a, -(Sqrt[a] q)       \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich                                    ; q, z \MATHvStrich 
        5 4\MATHvStrich   a q                        1 + n        \MATHvStrich 
           \MATHloEck   ---, -Sqrt[a], Sqrt[a], a q             \MATHroEck 
               b
\goodbreakpoint%
In[2]:= \%/.phTausche[1,3,u]
\goodbreakpoint%
           \MATHluEck                 -n                        \MATHruEck 
           \MATHvStrich  b, Sqrt[a] q, q  , a, -(Sqrt[a] q)       \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich                                    ; q, z \MATHvStrich 
        5 4\MATHvStrich   a q                        1 + n        \MATHvStrich 
           \MATHloEck   ---, -Sqrt[a], Sqrt[a], a q             \MATHroEck 
               b
\goodbreakpoint%
In[3]:= \%/.phTausche[4,2,l]
\goodbreakpoint%
           \MATHluEck                 -n                        \MATHruEck 
           \MATHvStrich  b, Sqrt[a] q, q  , a, -(Sqrt[a] q)       \MATHvStrich 
Out[3]=  \MATHphi  \MATHvStrich                                    ; q, z \MATHvStrich 
        5 4\MATHvStrich   a q     1 + n                           \MATHvStrich 
           \MATHloEck   ---, a q     , -Sqrt[a], Sqrt[a]        \MATHroEck 
               b
\goodbreakpoint%
In[4]:= \%/.phTausche[1,4,b]
\goodbreakpoint%
           \MATHluEck      -n                                   \MATHruEck 
           \MATHvStrich  b, q  , a, -(Sqrt[a] q), Sqrt[a] q       \MATHvStrich 
Out[4]=  \MATHphi  \MATHvStrich                                    ; q, z \MATHvStrich 
        5 4\MATHvStrich      1 + n                     a q        \MATHvStrich 
           \MATHloEck   a q     , -Sqrt[a], Sqrt[a], ---        \MATHroEck 
                                            b
\endMATH 
\Seealso phPerm, phOrdne, ph, W, PQSort, Ers, PosListe.

\Name PosListe
\Description Function that provides a list of subexpressions of
\hbox{\tt Expr}
together with the respective positions in \hbox{\tt Expr}. This helps to use 
  controlled application of rules or functions by means of \hbox{\tt Ers}.
\Usage PosListe[Expr].
\Example 
\MATH 
In[1]:= pq[a,n]/pq[q,n]*SUM[pq[b,k]/pq[c,k+1]*q\MATHhoch k,\MATHlbrace k,0,Infinity\MATHrbrace ]
\goodbreakpoint%
           \MATHinfty    k
         \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q  (b; q)
          \MATHbackslash             k
        (  \MATHgroesser   -----------) (a; q)
          /   (c; q)             n
         \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       1 + k
          k=0
Out[1]= --------------------------
                 (q; q)
                       n
\goodbreakpoint%
In[2]:= PosListe[\%]
\goodbreakpoint%
            \MATHinfty    k
          \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q  (b; q)
           \MATHbackslash             k                                 1
Out[2]= \MATHlbrace \MATHlbrace   \MATHgroesser   -----------, \MATHlbrace %
\MATHlbrace 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (a; q) , %
\MATHlbrace \MATHlbrace 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace %
-------, \MATHlbrace \MATHlbrace 3\MATHrbrace \MATHrbrace \MATHrbrace %
\MATHrbrace 
           /   (c; q)                      n           (q; q)
          \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       %
1 + k                                   n
           k=0
\goodbreakpoint%
In[3]:= PosListe[\%\%,2]
\goodbreakpoint%
Out[3]= \MATHlbrace \MATHlbrace -1, \MATHlbrace \MATHlbrace 3, 2\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace a, \MATHlbrace \MATHlbrace 2, 1\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace n, \MATHlbrace \MATHlbrace 2, 2\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace q, \MATHlbrace \MATHlbrace 2, 3\MATHrbrace %
\MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
                              k
                             q  (b; q)
                                      k
\MATHgroesser     \MATHlbrace \MATHlbrace k, 0, \MATHinfty \MATHrbrace , %
\MATHlbrace \MATHlbrace 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace %
-----------, \MATHlbrace \MATHlbrace 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , %
\MATHlbrace (q; q) , \MATHlbrace \MATHlbrace 3, 1\MATHrbrace \MATHrbrace %
\MATHrbrace \MATHrbrace 
                             (c; q)                         n
                                   1 + k
\goodbreakpoint%
In[4]:= PosListe[\%\%\%,3]
\goodbreakpoint%
Out[4]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 1, 2, %
2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace k, \MATHlbrace \MATHlbrace %
1, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace n, \MATHlbrace %
\MATHlbrace 3, 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
                                    k
\MATHgroesser     \MATHlbrace q, \MATHlbrace \MATHlbrace 3, 1, 1\MATHrbrace %
, \MATHlbrace 3, 1, 3\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q , %
\MATHlbrace \MATHlbrace 1, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace %
\MATHinfty , \MATHlbrace \MATHlbrace 1, 2, 3\MATHrbrace \MATHrbrace \MATHrbrace , 
\goodbreakpoint% 
                                   1
\MATHgroesser     \MATHlbrace (b; q) , \MATHlbrace \MATHlbrace 1, 1, 2\MATHrbrace %
\MATHrbrace \MATHrbrace , \MATHlbrace -----------, \MATHlbrace \MATHlbrace 1, %
1, 3\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace 
            k                 (c; q)
                                    1 + k
\endMATH 
\Seealso Ers, Subst.

\Name pq
\Description \hbox{\tt pq[x,n,q]} is the $q$-factorial symbol $(x;q)_n$. 
  \hbox{\tt pq[List1,List2,n,q]} is also provided as the usual abbreviation for the
  quotient of $q$-factorial symbols (see \cite{\GaRaAA, (1.2.41)}). 
  In both cases the parameter \hbox{\tt q} is optional. It will be set equal
\hbox{\tt q} if it
  is omitted.
\Usage pq[x,n,q]

   \leavevmode\hphantom{Usa}\rm or: \tt pq[x,n]

   \leavevmode\hphantom{Usa}\rm or: \tt pq[List1,List2,n,q]

   \leavevmode\hphantom{Usa}\rm or: \tt pq[List1,List2,n].
\Example 
\MATH 
In[1]:= pq[a,n]
\goodbreakpoint%
Out[1]= (a; q)
              n
\goodbreakpoint%
In[2]:= pq[a,n,q\MATHhoch 2]
\goodbreakpoint%
             2
Out[2]= (a; q )
               n
\goodbreakpoint%
In[3]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,2*m]
\goodbreakpoint%
        (a, b; q)
                 2 m
Out[3]= ------------
        (c, d; q)
                 2 m
\goodbreakpoint%
In[4]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,2*m,1/q]
\goodbreakpoint%
               1
        (a, b; -)
               q 2 m
Out[4]= ------------
               1
        (c, d; -)
               q 2 m
\endMATH 
\Seealso pqinf, Binomialq, Binomialpq, Multinomialpq, Multinomialq,
Factorialq, Factorialpq, PQ, pqaufl, pqzerl, pqzus,\linebreak phFormat.

\Name PQ
\Description Is a switch that activates automatic evaluating of $q$-factorial
symbols \hbox{\tt pq} and basic hypergeometric series \hbox{\tt ph},
\hbox{\tt Ph}, \hbox{\tt ps}, or makes it inactive, 
respectively. By default automatic evaluating is inactive.
\Usage PQ.
\Example 
\MATH 
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= pq[a,5]
\goodbreakpoint%
Out[2]= (a; q)
              5
\goodbreakpoint%
In[3]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck             \MATHruEck 
           \MATHvStrich  a, b       \MATHvStrich 
Out[3]=  \MATHphi  \MATHvStrich      ; q, z \MATHvStrich 
        2 1\MATHvStrich   c         \MATHvStrich 
           \MATHloEck             \MATHroEck 
\goodbreakpoint%
In[4]:= ph[\MATHlbrace q\MATHhoch -n,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich   -n          \MATHvStrich 
Out[4]=  \MATHphi  \MATHvStrich  q  , b; q, z \MATHvStrich 
        2 1\MATHvStrich               \MATHvStrich 
           \MATHloEck    c          \MATHroEck 
\goodbreakpoint%
In[5]:= ph[\MATHlbrace q\MATHhoch -3,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich   -3          \MATHvStrich 
Out[5]=  \MATHphi  \MATHvStrich  q  , b; q, z \MATHvStrich 
        2 1\MATHvStrich               \MATHvStrich 
           \MATHloEck    c          \MATHroEck 
\goodbreakpoint%
In[6]:= PQ
\goodbreakpoint%
In[7]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is active.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[8]:= pq[a,5]
\goodbreakpoint%
                                  2          3          4
Out[8]= (1 - a) (1 - a q) (1 - a q ) (1 - a q ) (1 - a q )
\goodbreakpoint%
In[9]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
          \MATHinfty    k
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q)  (b; q)
         \MATHbackslash             k       k
Out[9]=   \MATHgroesser   ------------------
         /    (c; q)  (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck        k       k
         k=0
\goodbreakpoint%
In[10]:= ph[%
\MATHlbrace q\MATHhoch -n,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
Is n a nonnegative integer?
[y|n]:  y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: j
\goodbreakpoint%
           n   j           -n
         %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (b; q)  (q  ; q)
          \MATHbackslash             j         j
Out[10]=   \MATHgroesser   --------------------
          /     (c; q)  (q; q)
         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck         j       j
          j=0
\goodbreakpoint%
In[11]:= ph[%
\MATHlbrace q\MATHhoch -3,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: s
\goodbreakpoint%
                           -3                    -3        -2             2
             (1 - b) (1 - q  ) z   (1 - b) (1 - q  ) (1 - q  ) (1 - b q) z
Out[11]= 1 + ------------------- + ---------------------------------------- + 
               (1 - c) (1 - q)                                        2
                                      (1 - c) (1 - q) (1 - c q) (1 - q )
 
                   -3        -2       1                    2   3
     (1 - b) (1 - q  ) (1 - q  ) (1 - -) (1 - b q) (1 - b q ) z
                                      q
\MATHgroesser     -----------------------------------------------------------
                                       2          2        3
       (1 - c) (1 - q) (1 - c q) (1 - q ) (1 - c q ) (1 - q )
\goodbreakpoint%
In[12]:= PQ
\goodbreakpoint%
In[13]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\endMATH 
\Seealso ph, Ph, ps, pq, hypqAttributes.

\Name pqaufl
\Description Rule that writes $(x;q)_n$ as the defining product
    $\prod _{i=0} ^{n-1}(1-xq^{i})$, if $n$ is an integer.
\Usage Expr/.pqaufl.
\Example
\MATH
In[1]:= pq[a,-3]/pq[b,2]*pq[c,1]
\goodbreakpoint%
        (a; q)   (c; q)
              -3       1
Out[1]= ----------------
            (b; q)
                  2
\goodbreakpoint%
In[2]:= \%/.pqaufl
\goodbreakpoint%
                           1 - c
Out[2]= -------------------------------------------
                     a        a        a
        (1 - b) (1 - --) (1 - --) (1 - -) (1 - b q)
                      3        2       q
                     q        q
\goodbreakpoint%
In[3]:= ph[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[3]=  \MATHphi  %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f       %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.C01
\goodbreakpoint%
                 %
\MATHluEck                            %
\MATHruEck 
                 %
\MATHvStrich  q, a q, b q, c q          %
\MATHvStrich 
            z  \MATHphi  %
\MATHvStrich                   ; q, q z %
\MATHvStrich  (a; q)  (b; q)  (c; q)
              4 4%
\MATHvStrich   2                        %
\MATHvStrich        1       1       1
                 %
\MATHloEck  q , d q, e q, f q         %
\MATHroEck 
Out[4]= 1 - ----------------------------------------------------------
                         (d; q)  (e; q)  (f; q)  (q; q)
                               1       1       1       1
\goodbreakpoint%
In[5]:= \%/.pqaufl
\goodbreakpoint%
                                         %
\MATHluEck                            %
\MATHruEck 
                                         %
\MATHvStrich  q, a q, b q, c q          %
\MATHvStrich 
            (1 - a) (1 - b) (1 - c) z  \MATHphi  %
\MATHvStrich                   ; q, q z %
\MATHvStrich 
                                      4 4%
\MATHvStrich   2                        %
\MATHvStrich 
                                         %
\MATHloEck  q , d q, e q, f q         %
\MATHroEck 
Out[5]= 1 - ----------------------------------------------------------
                         (1 - d) (1 - e) (1 - f) (1 - q)
\endMATH
\Seealso pqzerl, pqzus, pq, pqinfzerl, pqinfzus, Ers, PosListe.

\Name pqinf
\Description \hbox{\tt pqinf[x,q]} is the infinite $q$-factorial symbol $(x;q)_\infty$.
  \hbox{\tt pqinf[List1,List2,q]} is also provided as the usual abbreviation for the
  quotient of infinite $q$-factorial symbols (see \cite{\GaRaAA,
(1.2.42)}). 
  In both cases the parameter \hbox{\tt q} is optional. If it is omitted it is set 
  equal \hbox{\tt q}.
\Usage pqinf[x,q]

   \leavevmode\hphantom{Usa}\rm or: \tt pqinf[x]

   \leavevmode\hphantom{Usa}\rm or: \tt pqinf[List1,List2,q]

   \leavevmode\hphantom{Usa}\rm or: \tt pqinf[List1,List2].
\Example 
\MATH 
In[1]:= pqinf[a\MATHhoch 2*q]
\goodbreakpoint%
          2
Out[1]= (a  q;q)
                \MATHinfty 
\goodbreakpoint%
In[2]:= pqinf[a\MATHhoch 2*q,q\MATHhoch 3]
\goodbreakpoint%
          2    3
Out[2]= (a  q;q )
                 \MATHinfty 
\goodbreakpoint%
In[3]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ]
\goodbreakpoint%
        (a, b; q)
                 \MATHinfty 
Out[3]= ----------
        (c, d; q)
                 \MATHinfty 
\goodbreakpoint%
In[4]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,1/q]
\goodbreakpoint%
               1
        (a, b; -)
               q \MATHinfty 
Out[4]= ----------
               1
        (c, d; -)
               q \MATHinfty 
\endMATH 
\Seealso pq, PQ, pqinfzerl, pqinfzus, phFormat.

\Name pqinfzerl
\Description Rule that splits \hbox{\tt pqinf[List1,List2,q]} into a quotient of 
  products of infinite $q$-factorial symbols.
\Usage Expr/.pqinfzerl.
\Example 
\MATH 
In[1]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ]
\goodbreakpoint%
        (a, b; q)
                 \MATHinfty 
Out[1]= ----------
        (c, d; q)
                 \MATHinfty 
\goodbreakpoint%
In[2]:= \%/.pqinfzerl
\goodbreakpoint%
        (a;q)  (b;q)
             \MATHinfty       \MATHinfty 
Out[2]= -------------
        (c;q)  (d;q)
             \MATHinfty       \MATHinfty 
\endMATH 
\Seealso pqaufl, pqzerl, pqzus, pqinf, pqinfzus, Ers, PosListe.

\Name pqinfzus
\Description Rule that collects several infinite $q$-factorial symbols 
  \hbox{\tt pqinf[x$_{\text {\tt i}}$,q]} to an expression\linebreak
\hbox{\tt pqinf[List1,List2,q]}. The parameter 
  \hbox{\tt q} is optional. It is set equal \hbox{\tt q} if it is omitted.
\Usage Expr/.pqinfzus[q]

   \leavevmode\hphantom{Usa}\rm or: \tt Expr/.pqinfzus[].
\Example 
\MATH 
In[1]:= pqinf[a]*pqinf[b]/pqinf[c]/pqinf[d]
\goodbreakpoint%
        (a;q)  (b;q)
             \MATHinfty       \MATHinfty 
Out[1]= -------------
        (c;q)  (d;q)
             \MATHinfty       \MATHinfty 
\goodbreakpoint%
In[2]:= \%/.pqinfzus[]
\goodbreakpoint%
        (a, b; q)
                 \MATHinfty 
Out[2]= ----------
        (c, d; q)
                 \MATHinfty 
\goodbreakpoint%
In[3]:= pqinf[a,q]*pqinf[b,q\MATHhoch 2]/pqinf[c,q\MATHhoch 2]/pqinf[d,q]
\goodbreakpoint%
                   2
        (a;q)  (b;q )
             \MATHinfty        \MATHinfty 
Out[3]= --------------
            2
        (c;q )  (d;q)
              \MATHinfty       \MATHinfty 
\goodbreakpoint%
In[4]:= \%/.pqinfzus[]
\goodbreakpoint%
                (a; q)
            2         \MATHinfty 
        (b;q )  -------
              \MATHinfty  (d; q)
                      \MATHinfty 
Out[4]= ---------------
                2
            (c;q )
                  \MATHinfty 
\goodbreakpoint%
In[5]:= \%/.pqinfzus[q\MATHhoch 2]
\goodbreakpoint%
                     2
        (a; q)  (b; q )
              \MATHinfty         \MATHinfty 
Out[5]= ------- --------
        (d; q)       2
              \MATHinfty  (c; q )
                       \MATHinfty 
\endMATH 
\Seealso pqaufl, pqzerl, pqzus, pqinf, pqinfzerl, Ers, PosListe.

\Name PQSort
\Description Rule that orders the parameters of basic hypergeometric 
   series \hbox{\tt ph[List1,List2,q,z]}, \hbox{\tt Ph[\dots]},
\hbox{\tt ps[\dots]}, \hbox{\tt W[\dots]}, of ``multiple" upper $q$-factorials 
   \hbox{\tt pq[List1,List2,n,q]}, and of ``multiple" infinite $q$-factorials 
   \hbox{\tt pqinf[List1,List2,q]} in a standard order. For instance, 
   this function can be used for a quick test if two expressions agree.
It is recommended to apply \hbox{\tt pqinfzus[q]} and \hbox{\tt pqzus[n,q]}
first.
\Usage Expr/.PQSort.
\Example
\MATH
In[1]:= pq[%
\MATHlbrace c,c/a,b%
\MATHrbrace ,%
\MATHlbrace a/b/c,b*q%
\MATHrbrace ,n,q\MATHhoch 2]*pqinf[%
\MATHlbrace b/a,c%
\MATHrbrace ,%
\MATHlbrace a*b*c,1/b%
\MATHrbrace ,q]*
        ph[%
\MATHlbrace b/c,b*a,a%
\MATHrbrace ,%
\MATHlbrace b,a%
\MATHrbrace ,1/q,z]
\goodbreakpoint%
           b                                   c      2
          (-, c; q)       %
\MATHluEck  b            %
\MATHruEck  (c, -, b; q )
           a       \MATHinfty       %
\MATHvStrich  -, a b  1    %
\MATHvStrich      a        n
Out[1]= --------------  \MATHphi  %
\MATHvStrich  c     ; -, z %
\MATHvStrich  ---------------
                1      2 1%
\MATHvStrich          q    %
\MATHvStrich    a         2
        (a b c, -; q)     %
\MATHloEck    b          %
\MATHroEck  (---, b q; q )
                b    \MATHinfty                       b c          n
\goodbreakpoint%
In[2]:= \%/.PQSort
\goodbreakpoint%
           b                                      c   2
          (-, c; q)       %
\MATHluEck       b       %
\MATHruEck  (b, c, -; q )
           a       \MATHinfty       %
\MATHvStrich  a b, -  1    %
\MATHvStrich         a     n
Out[2]= --------------  \MATHphi  %
\MATHvStrich       c; -, z %
\MATHvStrich  ---------------
         1             2 1%
\MATHvStrich          q    %
\MATHvStrich    a         2
        (-, a b c; q)     %
\MATHloEck    b          %
\MATHroEck  (---, b q; q )
         b           \MATHinfty                       b c          n
\endMATH
\Seealso SimplifyPQ, SUMExpand, phEinf, phOrdne, phPerm, 
   phTausche, ph, Ph, ps, W, pq, pqinf.

\Name pqzerl
\Description Rule that splits \hbox{\tt pq[List1,List2,n,q]} into a quotient of 
products of $q$-factorial symbols.
\Usage Expr/.pqzerl.
\Example 
\MATH 
In[1]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,m,q\MATHhoch 2]
\goodbreakpoint%
                2
        (a, b; q )
                  m
Out[1]= -----------
                2
        (c, d; q )
                  m
\goodbreakpoint%
In[2]:= \%/.pqzerl
\goodbreakpoint%
             2        2
        (a; q )  (b; q )
               m        m
Out[2]= -----------------
             2        2
        (c; q )  (d; q )
               m        m
\endMATH 
\Seealso pqaufl, pqzus, pq, pqinfzerl, pqinfzus, Ers, PosListe.

\Name pqzus
\Description Rule that collects several $q$-factorial symbols
\hbox{\tt pq[x$_{\text {\tt 
i}}$,n,q]} to
an expression \hbox{\tt pq[List1,List2,n,q]}. The parameter \hbox{\tt q} is optional. It is set
equal \hbox{\tt q} if it is omitted.
\Usage Expr/.pqzus[n,q]

 \leavevmode\hphantom{Usa}\rm or: \tt Expr/.pqzus[n].
\Example 
\MATH 
In[1]:= pq[a,n]*pq[b,m,q\MATHhoch 2]/pq[c,n]/pq[d,m,q\MATHhoch 2]
\goodbreakpoint%
                     2
        (a; q)  (b; q )
              n        m
Out[1]= ----------------
                     2
        (c; q)  (d; q )
              n        m
\goodbreakpoint%
In[2]:= \%/.pqzus[n,q]
\goodbreakpoint%
                 (a; q)
             2         n
        (b; q )  -------
               m (c; q)
                       n
Out[2]= ----------------
                 2
            (d; q )
                   m
\goodbreakpoint%
In[3]:= \%/.pqzus[m,q\MATHhoch 2]
\goodbreakpoint%
                     2
        (a; q)  (b; q )
              n        m
Out[3]= ------- --------
        (c; q)       2
              n (d; q )
                       m
\endMATH 
\Seealso pqaufl, pqzerl, pq, pqinfzus, pqinfzerl, Ers, PosListe.

\Name ps
\Description \hbox{\tt ps[List1,List2,q,z]} is the bilateral basic 
hypergeometric series with upper parameters \hbox{\tt List}1, 
lower parameters \hbox{\tt List}2, base \hbox{\tt q}, and argument
\hbox{\tt z}.
\Usage ps[List1,List2,q,z].
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  d, e, f       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\endMATH
\Seealso SListe, TListe, SUMRegeln, SUMps, psSUM, ph, Ph, pq, pqinf, 
phCancel, phOrdne, phPerm,\linebreak
phTausche, PQ, phFormat.

\Name psEinf
\Description Rule that inactivates automatic cancelling in \hbox{\tt
ps[]} and then
adds a parameter which has to be entered on request to the upper and
lower parameters of \hbox{\tt ps[]}.
\Usage Expr/.psEinf.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  d, e, f       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.psEinf
Add the parameter: A
\goodbreakpoint%
            %
\MATHluEck                   %
\MATHruEck 
            %
\MATHvStrich  A, a, b, c       %
\MATHvStrich 
Out[2]=  ps %
\MATHvStrich            ; q, z %
\MATHvStrich 
        4  4%
\MATHvStrich  A, d, e, f       %
\MATHvStrich 
            %
\MATHloEck                   %
\MATHroEck 
\endMATH
\Seealso phCancel, psOrdne, psPerm, PQSort, SUMRegeln, SUMUmkehr, Ers, 
PosListe.

\Name psinv
\Description Rule that transforms a bilateral basic hypergeometric series
\hbox{\tt ps[List1,List2,q,z]} with base \hbox{\tt q} 
into a bilateral basic hypergeometric  series 
\hbox{\tt ps[\dots,1/q,..]} with base 1/\hbox{\tt q}.
\Usage Expr/.psinv.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck             %
\MATHruEck 
            %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2  2%
\MATHvStrich  c, d       %
\MATHvStrich 
            %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.psinv
\goodbreakpoint%
              1  1
            %
\MATHluEck  -, -           %
\MATHruEck 
            %
\MATHvStrich  a  b  1  a b z %
\MATHvStrich 
Out[2]=  ps %
\MATHvStrich      ; -, ----- %
\MATHvStrich 
        2  2%
\MATHvStrich  1  1  q   c d  %
\MATHvStrich 
            %
\MATHloEck  -, -           %
\MATHroEck 
              c  d
\goodbreakpoint%
In[3]:= ps[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d,0%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich   a, b         %
\MATHvStrich 
Out[3]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        2  3%
\MATHvStrich  c, d, 0       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.psinv
\goodbreakpoint%
              1  1
            %
\MATHluEck  -, -, 0           %
\MATHruEck 
            %
\MATHvStrich  a  b     1  a b z %
\MATHvStrich 
Out[4]=  ps %
\MATHvStrich         ; -, ----- %
\MATHvStrich 
        3  2%
\MATHvStrich   1  1    q   c d  %
\MATHvStrich 
            %
\MATHloEck   -, -             %
\MATHroEck 
               c  d
\endMATH
\Seealso SUMRegeln, Ers, PosListe, phinv, Phinv.

\Name psOrdne
\Description Rule that tries to order the parameters of a bilateral basic 
hypergeometric series in ``well-poised" order. If there is an
upper parameter of the form $q^{-n}$, where $n$ is a nonnegative integer, 
then it is put at the very last place in the upper list. 
\Usage Expr/.psOrdne.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace q\MATHhoch -n,b,q*Sqrt[a],-q*Sqrt[a]%
\MATHrbrace ,%
\MATHlbrace a*q/b,-Sqrt[a],a*q\MATHhoch (1+n),
        Sqrt[a]%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck   -n                                    %
\MATHruEck 
            %
\MATHvStrich  q  , b, Sqrt[a] q, -(Sqrt[a] q)        %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich                                  ; q, z %
\MATHvStrich 
        4  4%
\MATHvStrich  a q               1 + n                %
\MATHvStrich 
            %
\MATHloEck  ---, -Sqrt[a], a q     , Sqrt[a]       %
\MATHroEck 
               b
\goodbreakpoint%
In[2]:= \%/.psOrdne
\goodbreakpoint%
            %
\MATHluEck                               -n        %
\MATHruEck 
            %
\MATHvStrich  Sqrt[a] q, -(Sqrt[a] q), b, q          %
\MATHvStrich 
Out[2]=  ps %
\MATHvStrich                                  ; q, z %
\MATHvStrich 
        4  4%
\MATHvStrich                     a q     1 + n       %
\MATHvStrich 
            %
\MATHloEck  Sqrt[a], -Sqrt[a], ---, a q            %
\MATHroEck 
                                  b
\endMATH
\Seealso psEinf, psPerm, ps, PQSort, Ers, PosListe.

\Name psPerm
\Description Rule for permuting parameters in bilateral basic 
hypergeometric series.
\Usage Expr/.psPerm[$\langle$Permutation$\rangle$,x].\newline
\rm  {\tt x} can be {\tt u}, {\tt l}, {\tt b}. {\tt u} causes a permutation of upper parameters,
{\tt l} causes a 
  permutation of lower parameters, {\tt b} causes a simultaneous permutation 
  of respective upper and lower parameters. \hbox{\tt Permutation} must be a 
  sequence of positive numbers forming a permutation. The effect is that
  the new parameter at position {\tt i} is the old parameter from position
  \hbox{\tt Permutation[i]}.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace e,f,g%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  e, f, g       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%1/.psPerm[3,2,1,u]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  c, b, a       %
\MATHvStrich 
Out[2]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  e, f, g       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[3]:= \%1/.psPerm[3,2,1,l]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[3]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  g, f, e       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%1/.psPerm[3,2,1,b]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  c, b, a       %
\MATHvStrich 
Out[4]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  g, f, e       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\endMATH
\Seealso psOrdne, ps, PQSort, Ers, PosListe, phPerm, PhPerm.

\Name psph
\Description Rule that transforms a \hbox{\tt ps[]} into a sum of two
\hbox{\tt ph[]}'s. 
\vskip6pt
\leavevmode\hphantom{Description: }
$$\multline 
\hskip2cm
{}_r \psi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; 
q,z\right]
\to\sum _{n=-\infty} ^{m-1}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}}
{\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom
n2}\right)^{s-r+1}z^n\\
+\sum _{n=m} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}}
{\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom
n2}\right)^{s-r+1}z^n.
\endmultline$$
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.psph.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace e,f,g%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  e, f, g       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.psph
Split at: 0
\goodbreakpoint%
           %
\MATHluEck                   %
\MATHruEck 
           %
\MATHvStrich  a, b, c, q       %
\MATHvStrich 
Out[2]=  \MATHphi  %
\MATHvStrich            ; q, z %
\MATHvStrich  + 
        4 3%
\MATHvStrich   e, f, g         %
\MATHvStrich 
           %
\MATHloEck                   %
\MATHroEck 
 
           2   2   2
          q   q   q
        %
\MATHluEck  --, --, --, q             %
\MATHruEck  (a, b, c; q)
        %
\MATHvStrich  e   f   g          e f g  %
\MATHvStrich              -1
      \MATHphi  %
\MATHvStrich               ; q, ------- %
\MATHvStrich  --------------
     4 3%
\MATHvStrich    2   2   2       a b c z %
\MATHvStrich  (e, f, g; q)
        %
\MATHloEck   q   q   q                %
\MATHroEck              -1
           --, --, --
           a   b   c
\MATHgroesser     -----------------------------------------------
                            z
\goodbreakpoint%
In[3]:= \%1/.psph
Split at: 4
\goodbreakpoint%
                 1     1     1
                ----, ----, ----, q
              %
\MATHluEck     2     2     2                %
\MATHruEck  (a, b, c; q)
         3    %
\MATHvStrich  e q   f q   g q          e f g  %
\MATHvStrich              3
Out[3]= z   \MATHphi  %
\MATHvStrich                     ; q, ------- %
\MATHvStrich  ------------- + 
           4 3%
\MATHvStrich    1     1     1         a b c z %
\MATHvStrich  (e, f, g; q)
              %
\MATHloEck   ----, ----, ----               %
\MATHroEck              3
                    2     2     2
                 a q   b q   c q
 
           %
\MATHluEck     4     4     4          %
\MATHruEck  (a, b, c; q)
      4    %
\MATHvStrich  a q , b q , c q , q       %
\MATHvStrich              4
\MATHgroesser     z   \MATHphi  %
\MATHvStrich                     ; q, z %
\MATHvStrich  -------------
        4 3%
\MATHvStrich      4     4     4         %
\MATHvStrich  (e, f, g; q)
           %
\MATHloEck   e q , f q , g q          %
\MATHroEck              4
\endMATH
\Seealso ps, ph, phps, Ers, PosListe.

\Name psShift
\Description Rule that shifts the summation index in a bilateral basic
hypergeometric series. 
\vskip6pt
\leavevmode\hphantom{Description: }
$\dsize {}_r \psi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; 
q,z\right]
\to z^m\left((-1)^mq^{\binom m2}\right)^{s-r}
\frac {\prod _{i=1} ^{r}(a_i;q)_m} {\prod _{i=1} ^{s}(b_i;q)_m}
{}_r \psi_s\!\left[\matrix a_1q^m,\dots,a_rq^m\\ b_1q^m,\dots,b_sq^m\endmatrix; 
q,zq^{m(s-r)}\right]$.
\vskip6pt
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.psShift.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace e,f,g%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck                %
\MATHruEck 
            %
\MATHvStrich  a, b, c       %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich         ; q, z %
\MATHvStrich 
        3  3%
\MATHvStrich  e, f, g       %
\MATHvStrich 
            %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.psShift
shift by: 5
\goodbreakpoint%
           (a, b, c; q)      %
\MATHluEck     5     5     5       %
\MATHruEck 
         5             5     %
\MATHvStrich  a q , b q , c q        %
\MATHvStrich 
Out[2]= z  -------------  ps %
\MATHvStrich                  ; q, z %
\MATHvStrich 
           (e, f, g; q)  3  3%
\MATHvStrich     5     5     5       %
\MATHvStrich 
                       5     %
\MATHloEck  e q , f q , g q        %
\MATHroEck 
\endMATH
\Seealso psEinf, psPerm, ps, PSort, Ers, PosListe.

\Name psSUM
\Description Rule that transforms a \hbox{\tt ps[]} into a \hbox{\tt
SUM[]}.
\Usage Expr/.psSUM.
\Example
\MATH
In[1]:= ps[%
\MATHlbrace a,b,c,q\MATHhoch -n%
\MATHrbrace ,%
\MATHlbrace e,f,q\MATHhoch m%
\MATHrbrace ,q,z]
\goodbreakpoint%
            %
\MATHluEck            -n       %
\MATHruEck 
            %
\MATHvStrich  a, b, c, q         %
\MATHvStrich 
Out[1]=  ps %
\MATHvStrich              ; q, z %
\MATHvStrich 
        4  3%
\MATHvStrich           m         %
\MATHvStrich 
            %
\MATHloEck    e, f, q          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.psSUM
Is n a nonnegative integer?
[y|n]:  n
Is m a nonnegative integer?
[y|n]:  n
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
                     2
          \MATHinfty    k/2 - k /2  k                           -n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck q           z  (a; q)  (b; q)  (c; q)  (q  ; q)
         \MATHbackslash                         k       k       k         k
Out[2]=   \MATHgroesser   ------------------------------------------------
         /                k                   m
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck          (-1)  (e; q)  (f; q)  (q ; q)
        k=-\MATHinfty                       k       k        k
\goodbreakpoint%
In[3]:= \%1/.psSUM
Is n a nonnegative integer?
[y|n]:  y
Is m a nonnegative integer?
[y|n]:  y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index: k
\goodbreakpoint%
                       2
           n    k/2 - k /2  k                           -n
         %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck  q           z  (a; q)  (b; q)  (c; q)  (q  ; q)
          \MATHbackslash                          k       k       k         k
Out[3]=    \MATHgroesser    ------------------------------------------------
          /                 k                   m
         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck           (-1)  (e; q)  (f; q)  (q ; q)
        k=1 - m                     k       k        k
\endMATH
\Seealso ps, SUM, SUMps, Ers, PosListe.

\Name RS
\Description \hbox{\tt RS} is the right-hand side in \hbox{\tt Gleichung}.
\Usage RS.
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= RS
\goodbreakpoint%
         n  c
        a  (-; q)
            a    n
Out[2]= ----------
         (c; q)
               n
\goodbreakpoint%
In[3]:= Add[1]
\goodbreakpoint%
                                        n  c
               \MATHluEck               \MATHruEck         a  (-; q)
               \MATHvStrich      -n       \MATHvStrich             a    n
Out[3]= 1 +  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == 1 + %
----------
            2 1\MATHvStrich               \MATHvStrich          (c; q)
               \MATHloEck    c          \MATHroEck                n
\goodbreakpoint%
In[4]:= RS
\goodbreakpoint%
             n  c
            a  (-; q)
                a    n
Out[4]= 1 + ----------
             (c; q)
                   n
\goodbreakpoint%
In[5]:= RS=1/pq[q,m]
\goodbreakpoint%
           1
Out[5]= -------
        (q; q)
              m
\goodbreakpoint%
In[6]:= Gleichung
\goodbreakpoint%
               \MATHluEck               \MATHruEck 
               \MATHvStrich      -n       \MATHvStrich        1
Out[6]= 1 +  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == -------
            2 1\MATHvStrich               \MATHvStrich     (q; q)
               \MATHloEck    c          \MATHroEck           m
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, Mal, Add, Div, Sub, 
Hoch, GlTausche, Ers,\linebreak Subst.

\Name S1001
\Description Summation formula (\cite{\GaRaAA}, (1.3.2); Appendix (II.3)) in form of a rule.
$$
{}_1\phi _0\!\left [ \matrix \let\over/ a\\ \let\over/  -\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ a z;q)_\infty}}
{{( \let\over/  z;q)_\infty}}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S0110
\Description Summation formula (\cite{\GaRaAA}, (1.6.1); Appendix
(II.28)) in form of a rule.
$$
{}_0\psi _1\!\left [ \matrix -\\ 0\endmatrix ;q,z\right ] \longrightarrow 
(q,q/z,z;q)_\infty
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S1101
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.6(ii); Appendix (II.5)) in form of a rule.
$$
{}_1\phi _1\!\left [ \matrix \let\over/ a\\ \let\over/  c\endmatrix ;q,{c\over a}\right ] \longrightarrow 
  \frac {{( \let\over/ {c\over a};q)_\infty}}
{{( \let\over/  c;q)_\infty}}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S1102
\Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9);
$b\to\infty$) in form of a rule.
$$
{}_1\phi _1\!\left [ \matrix \let\over/ a\\ \let\over/  0\endmatrix ;q,-q\right ] \longrightarrow 
   {{( \let\over/ -q;q)_\infty}}
{{( \let\over/  aq;q^2)_\infty}}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S1110
\Description Summation formula (\cite{\GaRaAA}, (5.2.1); Appendix
(II.29)) in form of a rule.
$$
{} _{1} \psi _{1} \! \left [ \matrix \let \over / a\\ \let \over / b\endmatrix
   ;q, {\displaystyle z} \right ] \longrightarrow 
  { {(\let \over / q, {b\over a}, a z, {q\over {a z}} ;q) _\infty} \over
   {(\let \over / b, {q\over a}, z, {b\over {a z}} ;q) _\infty} }
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2101
\Description Summation formula (\cite{\GaRaAA}, (1.5.3); Appendix (II.6)) in form of a rule.
$$
{}_2\phi _1\!\left [ \matrix \let\over/ a,{q^{-n}}\\ \let\over/  c\endmatrix ;q,q\right ] \longrightarrow 
  {{{a^n} {(\let\over/ {c\over a};q)}_{n}}\over {{(\let\over/ c;q)}_{n}}}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2102
\Description Summation formula (\cite{\GaRaAA}, (1.5.2); Appendix (II.7)) in form of a rule.
$$
{}_2\phi _1\!\left [ \matrix \let\over/ a,{q^{-n}}\\ \let\over/  c\endmatrix ;q,{{c {q^n}}\over
   a}\right ] \longrightarrow {{{(\let\over/ {c\over a};q)}_{n}}\over {{(\let\over/ c;q)}_{n}}}
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2103
\Description Summation formula (\cite{\GaRaAA}, (1.5.1); Appendix (II.8)) in form of a rule.
$$
{}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,{c\over {a b}}\right ] \longrightarrow 
  \frac {{( \let\over/ {c\over a},{c\over b};q)_\infty}}
{{( \let\over/  c,{c\over {a b}};q)_\infty}}
$$
\Example 
\MATH 
In[1]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace q\MATHhoch 2*a*b\MATHrbrace %
,q,q\MATHhoch 2]
\goodbreakpoint%
           \MATHluEck                \MATHruEck 
           \MATHvStrich   a, b       2 \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich        ; q, q  \MATHvStrich 
        2 1\MATHvStrich       2        \MATHvStrich 
           \MATHloEck  a b q         \MATHroEck 
\goodbreakpoint%
In[2]:= \%/.S2103
\goodbreakpoint%
            2     2
        (b q , a q ; q)
                       \MATHinfty 
Out[2]= ----------------
              2   2
        (a b q , q ; q)
                       \MATHinfty 
\endMATH 
\Seealso S3201, SListe, SumListe, Ers, PosListe.

\Name S2104
\Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9)) in form of a rule.
$$
{}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  {{a q}\over b}\endmatrix ;q,-{q\over
   b}\right ] \longrightarrow \frac {{( \let\over/ -q;q)_\infty}}
{{( \let\over/  -{q\over b},{{a q}\over
    b};q)_\infty}} {(\let\over/ a q,{{a {q^2}}\over {{b^2}}};{q^2})}_{\infty}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2105
\Description Summation formula (\cite{\GaRaAA}, Ex~1.6(i)) 
in form of a rule.
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / {q^{-n}}, {q^{1 - n}}\\ \let
   \over / b^2 q\endmatrix ;q^2, {\displaystyle q^2} \right ] \longrightarrow  
q^{-\binom n2}
  {{({\let \over / b^2}; q^2) _{n} }\over 
    {  ({\let \over / b^2}; q) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2106
\Description Summation formula (\cite{\GaRaAA}, Ex~1.7) 
in form of a rule.
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, a q\\ \let \over /
   a\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow  
  {{\left( 1 + a z \right)  ({\let \over / a^2 q z}; q) _{\infty} }\over 
    {({\let \over / z}; q) _{\infty} }}
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2107
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.8) 
in form of a rule.
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, {{a^2}\over b}\\
   \let \over / b\endmatrix ;q^2, {\displaystyle {{b q}\over {a^2}}}
   \right ] \longrightarrow  \frac {1} {2}
   {{\left( { {(\let \over / -{b\over a} ;q) _\infty} \over {(\let
         \over / -a ;q) _\infty} } + 
        { {(\let \over / {b\over a} ;q) _\infty} \over {(\let \over / a ;q)
         _\infty} } \right)  { {(\let \over / a^2, q ;q^2) _\infty} \over
       {(\let \over / b, {{b q}\over {a^2}} ;q^2) _\infty} }}}
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2161
\Description Summation formula (\cite{\GaRaAA}, (2.10.13); Appendix (II.23)) in form of a rule.
$$
{}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,q\right ] 
 \longrightarrow 
  \frac {{( \let\over/ {q\over c},{{a b q}\over c};q)_\infty}}
{{( \let\over/  {{a q}\over c},{{b
   q}\over c};q)_\infty}} -
   \frac {{( \let\over/ {q\over c},a,b;q)_\infty}}
{{( \let\over/  {c\over q},{{a q}\over c},{{b
     q}\over c};q)_\infty}} 
    {}_2\phi _1\!\left [ \matrix \let\over/ {{a q}\over c},{{b q}\over c}\\ \let\over/  {{{q^2}}\over
     c}\endmatrix ;q,q\right ]
$$
 

\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2201
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(i); Appendix (II.10)) in form of a rule.
$$
{}_2\phi _2\!\left [ \matrix \let\over/ a,{q\over a}\\ \let\over/  -q,b\endmatrix ;q,-b\right ] \longrightarrow 
  {{(\let\over/ a b,{{b q}\over a};{q^2})}_{\infty} \over {{(\let\over/ b;q)}_{\infty}}} 
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2202
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(ii); Appendix (II.11)) in form of a rule.
$$
{}_2\phi _2\!\left [ \matrix \let\over/ {a^2},{b^2}\\ \let\over/  a b {\sqrt{q}},- a b
   {\sqrt{q}}  \endmatrix ;q,-q\right ] \longrightarrow 
  \frac {{( \let\over/ {a^2} q,{b^2} q;{q^2})_\infty}}
{{( \let\over/  q,{a^2} {b^2} q;{q^2})_\infty}}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2210
\Description Summation formula (\cite{\GaRaAA}, (5.3.4); Appendix
(II.30)) in form of a rule.
$$
{} _{2} \psi _{2} \! \left [ \matrix \let \over / b, c\\ \let \over /
   {{a q}\over b}, {{a q}\over c}\endmatrix ;q, {\displaystyle -{{a q}\over
   {b c}}} \right ] \longrightarrow { {(\let \over / {{a q}\over {b c}} ;q) _\infty}
    \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {q\over b}, {q\over
    c}, -{{a q}\over {b c}} ;q) _\infty} } 
   ({\let \over / {{a {q^2}}\over {{b^2}}}, {{a {q^2}}\over {{c^2}}}, {q^2},
    a q, {q\over a}}; {q^2}) _{\infty} 
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3201
\Description Summation formula (\cite{\GaRaAA}, (1.7.2); Appendix (II.12)) in form of a rule.
$$
{}_3\phi _2\!\left [ \matrix \let\over/ a,b,{q^{-n}}\\ \let\over/  c,{{a b {q^{1 - n}}}\over
   c}\endmatrix ;q,q\right ] \longrightarrow {{{(\let\over/ {c\over a};q)}_{n} {(\let\over/ {c\over b};q)}_{n}}\over 
    {{(\let\over/ c;q)}_{n} {(\let\over/ {c\over {a b}};q)}_{n}}}
$$
where $n$ is a nonnegative integer.
\Example 
\MATH 
In[1]:= ph[\MATHlbrace a,b,q\MATHhoch -n\MATHrbrace ,\MATHlbrace %
a*b/q\MATHhoch n,q\MATHrbrace ,q,q]
\goodbreakpoint%
                    -n
           \MATHluEck  a, b, q         \MATHruEck 
           \MATHvStrich                  \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich   a b     ; q, q \MATHvStrich 
        3 2\MATHvStrich   ---, q         \MATHvStrich 
           \MATHloEck    n             \MATHroEck 
              q
\goodbreakpoint%
In[2]:= \%/.S3201
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
          a        b
         (--; q)  (--; q)
           n    n   n    n
          q        q
Out[2]= -------------------
          -n       a b
        (q  ; q)  (---; q)
                n   n     n
                   q
\goodbreakpoint%
In[3]:= \%\%/.S3201
Is n a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
                    -n
           \MATHluEck  a, b, q         \MATHruEck 
           \MATHvStrich                  \MATHvStrich 
Out[3]=  \MATHphi  \MATHvStrich   a b     ; q, q \MATHvStrich 
        3 2\MATHvStrich   ---, q         \MATHvStrich 
           \MATHloEck    n             \MATHroEck 
              q
\goodbreakpoint%
In[4]:= ph[\MATHlbrace a,q\MATHhoch -m,q\MATHhoch -n\MATHrbrace ,\MATHlbrace %
a/q\MATHhoch (n+m),q\MATHrbrace ,q,q]
\goodbreakpoint%
           \MATHluEck      -m   -n        \MATHruEck 
           \MATHvStrich  a, q  , q          \MATHvStrich 
Out[4]=  \MATHphi  \MATHvStrich              ; q, q \MATHvStrich 
        3 2\MATHvStrich     -m - n          \MATHvStrich 
           \MATHloEck  a q      , q       \MATHroEck 
\goodbreakpoint%
In[5]:= \%/.S3201
Is n a nonnegative integer?
[y|n]:  n
Is m a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
          a         -m - n
         (--; q)  (q      ; q)
           m    m             m
          q
Out[5]= -------------------------
          -m          -m - n
        (q  ; q)  (a q      ; q)
                m               m
\endMATH 
\Seealso S2103, SListe, SumListe, Ers, PosListe.

\Name S3202
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.7; Appendix (II.15)) in form of a rule.
$$
{}_3\phi _2\!\left [ \matrix \let\over/ {q^{-2 n}},b,c\\ \let\over/  {{{q^{1 - 2 n}}}\over b},{{{q^{1
   - 2 n}}}\over c}\endmatrix ;q,{{{q^{2 - n}}}\over {b c}}\right ] \longrightarrow 
  {{{(\let\over/ b;q)}_{n} {(\let\over/ c;q)}_{n} {(\let\over/ b c;q)}_{2 n} {(\let\over/ q;q)}_{2 n}}\over 
    {{(\let\over/ b;q)}_{2 n} {(\let\over/ c;q)}_{2 n} {(\let\over/ b c;q)}_{n} {(\let\over/ q;q)}_{n}}}
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3203
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.1) 
in form of a rule.
$$
{} _{3} W _{2} ({\displaystyle a; -}; q, {\displaystyle t}) \longrightarrow  
  \left( 1 - a q t^2 \right)  
   { {(\let \over / a q^2 t ;q) _\infty} \over {(\let \over / t ;q) _\infty}
    }
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3204
\Description Summation formula (\cite{\GaRaAA}, Ex.~3.9) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, { \lambda } q, b\\ \let
   \over / { \lambda }, {{{{{ \lambda }}^2} q}\over b}\endmatrix ;q, {\displaystyle
   {{{{{ \lambda }}^2}}\over {a b^2}}} \right ] \longrightarrow  
  {{\left( 1 - { \lambda } + {{{{ \lambda } \over b}
            \left( 1 - {{{ \lambda }}\over a} \right) }} \right)}
\over {\left( 1 - { \lambda } \right)  \left( 1 + {{{ \lambda }}\over b} \right)
}}
{{ { {(\let \over / {{{{{ \lambda }}^2}}\over {b^2}}, {{{{{ \lambda }}^2}
       q}\over {a b}} ;q) _\infty} \over {(\let \over / {{{{{ \lambda }}^2}
       q}\over b}, {{{{{ \lambda }}^2}}\over {a b^2}} ;q) _\infty} }} 
    }
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3261
\Description Summation formula (\cite{\GaRaAA}, (2.10.12); Appendix (II.24)) in form of a rule.
$$\multline 
{}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/  e,{{a b c q}\over e}\endmatrix ;q,q\right
    ] 
    \\\longrightarrow \frac {{( \let\over/ {q\over e},{{b c q}\over e},{{a c q}\over
   e},{{a b q}\over e};q)_\infty}}
{{( \let\over/  {{a q}\over e},{{b q}\over e},{{c q}\over e},{{a b c
   q}\over e};q)_\infty}} - \frac {{( \let\over/ {q\over e},a,b,c,{{a b c {q^2}}\over
     {{e^2}}};q)_\infty}}
{{( \let\over/  {e\over q},{{a q}\over e},{{b q}\over e},{{c q}\over e},{{a b
     c q}\over e};q)_\infty}} 
    {}_3\phi _2\!\left [ \matrix \let\over/ {{a q}\over e},{{b q}\over e},{{c q}\over
     e}\\ \let\over/  {{{q^2}}\over e},{{a b c {q^2}}\over {{e^2}}}\endmatrix ;q,q\right ]
\endmultline$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3310
\Description Summation formula (\cite{\GaRaAA}, Ex.~5.18(i); Appendix
(II.31)) in form of a rule.
$$
{} _{3} \psi _{3} \! \left [ \matrix \let \over / b, c, d\\ \let \over /
   {q\over b}, {q\over c}, {q\over d}\endmatrix ;q, {\displaystyle {q\over
   {b c d}}} \right ] \longrightarrow { {(\let \over / q, {q\over {b c}}, {q\over
   {b d}}, {q\over {c d}} ;q) _\infty} \over {(\let \over / {q\over b},
   {q\over c}, {q\over d}, {q\over {b c d}} ;q) _\infty} }
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4301
\Description Summation formula (\cite{\GaRaAA}, (2.7.2); Appendix (II.13)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a,- {\sqrt{a}} q  ,b,c\\ \let\over/ 
   -{\sqrt{a}},{{a q}\over b},{{a q}\over c}\endmatrix ;q,{{{\sqrt{a}} q}\over
   {b c}}\right ] \longrightarrow \frac {{( \let\over/ a q,{{{\sqrt{a}} q}\over
   b},{{{\sqrt{a}} q}\over c},{{a q}\over {b c}};q)_\infty}}
{{( \let\over/  {{a q}\over b},{{a q}\over
   c},{\sqrt{a}} q,{{{\sqrt{a}} q}\over {b c}};q)_\infty}}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4302
\Description Summation formula (\cite{\GaRaAA}, (2.7.2), terminating form; Appendix (II.14)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a,- {\sqrt{a}} q  ,b,{q^{-n}}\\ \let\over/ 
   -{\sqrt{a}},{{a q}\over b},a {q^{1 + n}}\endmatrix ;q,{{{\sqrt{a}} {q^{1 +
   n}}}\over b}\right ] \longrightarrow {{{(\let\over/ a q;q)}_{n} {(\let\over/ {{{\sqrt{a}} q}\over b};q)}_{n}}\over 
    {{(\let\over/ {\sqrt{a}} q;q)}_{n} {(\let\over/ {{a q}\over b};q)}_{n}}}
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4303
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.8; Appendix (II.17)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a {q^n},c,-c,{q^{-n}}\\ \let\over/  {\sqrt{a}}
   {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}}  ,{c^2}\endmatrix
   ;q,q\right ] \longrightarrow 
\cases 0&\text {if $n$ is odd}\\ \displaystyle c^n {{(\let\over/ q;{q^2})}_{{n\over 2}} 
       {(\let\over/ {{a q}\over {{c^2}}};{q^2})}_{{n\over 2}}}\over 
     {\displaystyle{(\let\over/ a q;{q^2})}_{{n\over 2}} {(\let\over/ {c^2} q;{q^2})}_{{n\over 2}}}&
\text {if $n$ is even}\endcases
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4304
\Description Summation formula (Andrews [1976a] in \cite{\GaRaAA}, Appendix
(II.19), corrected) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ {q^{1 + n}},c,-c,{q^{-n}}\\ \let\over/  e,{{{c^2} q}\over
   e},-q\endmatrix ;q,q\right ] \longrightarrow 
  q^{n(n+1)/2}\, {  {(\let\over/ {e\over {{q^n}}},e {q^{1 + n}},{{{c^2} {q^{1 - n}}}\over e},{{{c^2} {q^{2
    + n}}}\over e};{q^2})}_{\infty}\over {{(\let\over/ e,{{{c^2} q}
   \over e};q)}_{\infty}}}
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4305
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.6) 
in form of a rule.
$$
{} _{4} \phi _{3} \! \left [ \matrix \let \over / {q^{-n}}, b, c, -{{{q^{1 -
   n}}}\over {b c}}\\ \let \over / {{{q^{1 - n}}}\over b}, {{{q^{1 - n}}}\over
   c}, - b c  \endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
\cases 0&\text {if $n$ is odd}\\ \displaystyle
  {{({\let \over / b c}; q) _{n}}\over {({\let \over / b, c}; q) _{n}}}  
   {{({\let \over / q, b^2, c^2}; q^2) _{\frac {n} {2}}}\over {({\let \over / b^2
    c^2}; q^2) _{\frac {n} {2}}}} &\text {if $n$ is even}\endcases
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4306
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.14(i)) 
in form of a rule.
$$
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, b,
   {q^{-n}}\\ \let \over / {\sqrt{a}}, {{a q}\over b}, b^2 {q^{1 -
   n}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
  {{({\let \over / {a\over {b^2}}, {1\over b}, -{{{\sqrt{a}} q}\over b}}; q)
   _{n}}\over {({\let \over / {b^{-2}}, {{a q}\over b}, -{{{\sqrt{a}}}\over
   b}}; q) _{n}}} 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4307
\Description Summation formula (\cite{\GaRaAA}, (2.7.2)) 
in form of a rule.
$$
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, b, c\\ \let
   \over / {\sqrt{a}}, {{a q}\over b}, {{a q}\over c}\endmatrix ;q,
   {\displaystyle -{{{\sqrt{a}} q}\over {b c}}} \right ] \longrightarrow 
  { {(\let \over / a q, {{a q}\over {b c}}, -{{{\sqrt{a}} q}\over b},
   -{{{\sqrt{a}} q}\over c} ;q) _\infty} \over {(\let \over / {{a q}\over b},
   {{a q}\over c}, - {\sqrt{a}} q , -{{{\sqrt{a}} q}\over {b c}}
   ;q) _\infty} }
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4308
\Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to
\sqrt{aq}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA},
(2.6.2)) in form of a rule.
$$
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, - {\sqrt{a}} q
    , b, {q^{-n}}\\ \let \over / -{\sqrt{a}}, {{a q}\over b}, {b^2}
   {q^{1 - n}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow 
  {{({\let \over / {a\over {{b^2}}}, {1\over b}, {{{\sqrt{a}} q}\over b}}; q)
   _{n}} \over {({\let \over / {b^{-2}}, {{a q}\over
   b}, {{{\sqrt{a}}}\over b}}; q) _{n}}} 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4361
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.9) 
in form of a rule.\NoBlackBoxes
$$\multline
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, -b, {{a q}\over
    {c^2}}\\ \let \over / {{a q}\over c}, -{{a q}\over c}, b^2\endmatrix
    ;q, {\displaystyle q} \right ] 
 \longrightarrow  
 { {(\let \over / {q\over {b^2}}, -q ;q) _\infty} \over {(\let \over / {{a
    q}\over {b^2}}, - a q   ;q) _\infty} } 
   { {(\let \over / a^2 q^2, {{a q^2}\over {b^2}}, {{a q^2}\over
    {c^2}}, {{a^2 q^2}\over {b^2 c^2}} ;q^2) _\infty} \over {(\let
    \over / {{q^2}\over {b^2}}, {{a^2 q^2}\over {c^2}}, a q^2, {{a
    q^2}\over {b^2 c^2}} ;q^2) _\infty} }\\
-   { {(\let \over / {q\over {b^2}}, a, b, -b, {{a q}\over {c^2}}, {{a
     q^2}\over {b^2 c}}, -{{a q^2}\over {b^2 c}} ;q) _\infty} \over
     {(\let \over / {{b^2}\over q}, {q\over b}, -{q\over b}, {{a q}\over
     {b^2}}, {{a q}\over c}, -{{a q}\over c}, {{a q^2}\over {b^2 c^2}}
     ;q) _\infty} }
 {} _{4} \phi _{3} \! \left [ \matrix \let \over / {q\over
     b}, -{q\over b}, {{a q}\over {b^2}}, {{a q^2}\over {b^2 c^2}}\\
     \let \over / {{a q^2}\over {b^2 c}}, -{{a q^2}\over {b^2 c}},
     {{q^2}\over {b^2}}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4410
\Description Summation formula (\cite{\GaRaAA}, (5.3.3); Appendix
(II.32)) in form of a rule.
$$
{} _{4} \psi _{4} \! \left [ \matrix \let \over / - {\sqrt{a}} q
    , b, c, d\\ \let \over / -{\sqrt{a}}, {{a q}\over b}, {{a q}\over
   c}, {{a q}\over d}\endmatrix ;q, {\displaystyle {{{a^{{3\over
   2}}} q}\over {b c d}}} \right ] \longrightarrow 
  { {(\let \over / a q, {{a q}\over {b c}}, {{a q}\over {b d}},
   {{a q}\over {c d}}, {{{\sqrt{a}} q}\over b}, {{{\sqrt{a}} q}\over c},
   {{{\sqrt{a}} q}\over d}, q, {q\over a} ;q) _\infty} \over {(\let \over /
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {q\over b}, {q\over c},
   {q\over d}, {\sqrt{a}} q, {q\over {{\sqrt{a}}}}, {{{a^{{3\over
   2}}} q}\over {b c d}} ;q) _\infty} }
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S5401
\Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to
\sqrt{aq}$) in form of a rule.
$$
{} _{5} W _{4} ({\displaystyle a; b, c}; q, {\displaystyle {{{\sqrt{a q}}
   }\over {b  c}}}) \longrightarrow { {(\let \over / a q, {{a q}\over {b c}},
   {{{\sqrt{a q}} }\over b}, {{{\sqrt{a q}} }\over c} ;q)
   _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {\sqrt{a q}}
   , {{{\sqrt{a q}} }\over {b c}} ;q) _\infty} }
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S5402
\Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to
q\sqrt{a}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA},
(2.6.2)) 
in form of a rule.
$$
{} _{5} \phi _{4} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, -
   {\sqrt{a}} q  , b, {q^{-n}}\\ \let \over / {\sqrt{a}}, -{\sqrt{a}},
   {{a q}\over b}, {b^2} {q^{2 - n}}\endmatrix ;q, {\displaystyle q} \right ] 
   \longrightarrow   
   {{({\let \over / {a\over {{b^2} q}}, {1\over {b q}}}; q) _{n}} \over
    {({\let \over / {1\over {{b^2} q}}, {{a q}\over b}}; q) _{n}}} 
  {{({\let \over / {{a q}\over {{b^2}}}}; {q^2}) _{n}} \over {({\let \over
    / {a\over {{b^2} q}}}; {q^2}) _{n}}}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S5501
\Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to\infty$) 
in form of a rule.
$$
{} _{5} \phi _{5} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, -
   {\sqrt{a}} q , b, c\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a
   q}\over b}, {{a q}\over c}, 0\endmatrix ;q, {\displaystyle {{a q}\over {b
   c}}} \right ] \longrightarrow { {(\let \over / a q, {{a q}\over {b c}} ;q) _\infty} \over
   {(\let \over / {{a q}\over b}, {{a q}\over c} ;q) _\infty} }
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S6501
\Description Summation formula (\cite{\GaRaAA}, (2.7.1); Appendix (II.20)) in form of a rule.
\BlackBoxes
$$
{}_6W _5(a;b,c,d;q,{{a q}\over {b c d}}) \longrightarrow 
  \frac {{( \let\over/ a q,{{a q}\over {b c}},{{a q}\over {b d}},{{a
   q}\over {c d}};q)_\infty}}
{{( \let\over/  {{a q}\over b},{{a q}\over c},{{a q}\over d},{{a q}\over
   {b c d}};q)_\infty}}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S6502
\Description Summation formula (\cite{\GaRaAA}, (2.4.2); Appendix (II.21)) in form of a rule.
$$
{}_6W _5(a;b,c,{q^{-n}};q,{{a {q^{1 + n}}}\over {b c}}) \longrightarrow 
  {{{(\let\over/ a q;q)}_{n} {(\let\over/ {{a q}\over {b c}};q)}_{n}}\over 
    {{(\let\over/ {{a q}\over b};q)}_{n} {(\let\over/ {{a q}\over c};q)}_{n}}}
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S6610
\Description Summation formula (\cite{\GaRaAA}, (5.3.1); Appendix
(II.33)) in form of a rule.
$$
{} _{6} \psi _{6} \! \left [ \matrix \let \over / {\sqrt{a}} q, -
   {\sqrt{a}} q  , b, c, d, e\\ \let \over / {\sqrt{a}}, -{\sqrt{a}},
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over
   e}\endmatrix ;q, {\displaystyle {{{a^2} q}\over {b c d e}}} \right ] 
\longrightarrow 
  { {(\let \over / a q, {{a q}\over {b c}}, {{a q}\over {b d}},
   {{a q}\over {b e}}, {{a q}\over {c d}}, {{a q}\over {c e}},
   {{a q}\over {d e}}, q, {q\over a} ;q) _\infty} \over {(\let \over /
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {q\over
   b}, {q\over c}, {q\over d}, {q\over e}, {{{a^2} q}\over {b c d e}} ;q)
   _\infty} }
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S8701
\Description Summation formula (\cite{\GaRaAA}, (2.6.2); Appendix (II.22)) in form of a rule.
$$
{}_8\phi _7\!\left [ \matrix \let\over/ a,{\sqrt{a}} q,- {\sqrt{a}} q 
   ,b,c,d,{a^2 q^{1+n} \over b c d },{q^{-n}}\\ \let\over/  {\sqrt{a}},-{\sqrt{a}},
   {{a q}\over b},{{a q}\over
   c},{{a q}\over d},{{b c d}\over {a {q^n}}},a {q^{1 + n}}\endmatrix
   ;q,q\right ] \longrightarrow {{{(\let\over/ a q;q)}_{n} {(\let\over/ {{a q}\over {b c}};q)}_{n} 
      {(\let\over/ {{a q}\over {b d}};q)}_{n} {(\let\over/ {{a q}\over {c d}};q)}_{n}}\over 
    {{(\let\over/ {{a q}\over b};q)}_{n} {(\let\over/ {{a q}\over c};q)}_{n} {(\let\over/ {{a q}\over d};q)}_{n} 
      {(\let\over/ {{a q}\over {b c d}};q)}_{n}}}
$$
where $n$ is a nonnegative integer.
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S8702
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(i); Appendix (II.16)) in form of a rule.
$$
{}_8W _7(-{{c {\sqrt{a b}}}\over
{{\sqrt{q}}}};a,b,c,-c,-{{{\sqrt{a b q}}
    }\over c};q,{{c {\sqrt{q}}}\over {{\sqrt{a b}}
   }}) \longrightarrow 
\frac {{( \let\over/ - c {\sqrt{a b q}}
    ,-{{c  {\sqrt{q}}}\over {{\sqrt{a b}} }};q)_\infty}}
{{( \let\over/ 
    -{{c {\sqrt{b q}} }\over {{\sqrt{a}}}},-{{c {\sqrt{a q}} 
    }\over {{\sqrt{b}}}};q)_\infty}} 
   \frac {{( \let\over/ a q,b q,{{{c^2} q}\over a},{{{c^2} q}\over b};{q^2})_\infty}}
{{( \let\over/ 
    q,a b q,{c^2} q,{{{c^2} q}\over {a b}};{q^2})_\infty}}
$$
 

\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S8703
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(ii); Appendix (II.18)) in form of a rule.
$$
{}_8W _7(-c;a,{q\over a},c,-d,-{q\over d};q,c) \longrightarrow 
  \frac {{( \let\over/ -c,- c q  ;q)_\infty}}
{{( \let\over/  c d,{{c q}\over
    d},- a c  ,-{{c q}\over a};q)_\infty}} 
   {(\let\over/ a c d,{{a c q}\over d},{{c d q}\over a},{{c {q^2}}\over
    {a d}};{q^2})}_{\infty}
$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S8704
\Description Summation formula (\cite{\GaRaAA}, Ex.~3.10) 
in form of a rule.
$$
{} _{8} W _{7} ({\displaystyle -\lambda ; {\sqrt{\lambda }} q, -
   {\sqrt{\lambda }} q  , a, b, -b}; q, {\displaystyle {{
   \lambda}\over {a {b^2}}}})
 \longrightarrow  { {(\let \over / { \lambda } q, {{{ \lambda }}\over a},
- {
   \lambda } q  , {{{ \lambda }}\over {b^2}}, {{{ \lambda } q}\over {a b}},
   -{{{ \lambda } q}\over {a b}} ;q) _\infty} \over {(\let \over / { \lambda },
   {{{ \lambda } q}\over a}, -{{{ \lambda } q}\over a}, {{{ \lambda } q}\over b},
   -{{{ \lambda } q}\over b}, {{{ \lambda }}\over {a b^2}} ;q) _\infty} }
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S8761
\Description Summation formula (\cite{\GaRaAA}, (2.11.7); Appendix (II.25)) in form of a rule.
$$\multline 
{}_8W _7(a;b,c,d,e,{{{a^2} q}\over {b c d e}};q,q)  \longrightarrow 
  \frac {{( \let\over/ a q,{b\over a},{{a q}\over {c d}},{{a q}\over {c
   e}},{{b d e}\over a},{{a q}\over {d e}},{{b c e}\over a},{{b c d}\over a};q)_\infty}}
{{( \let\over/ 
   {{a q}\over c},{{a q}\over d},{{a q}\over e},{{b c d e}\over a},{{b c}\over
   a},{{b d}\over a},{{b e}\over a},{{a q}\over {c d e}};q)_\infty}}
\\ - \frac {{( \let\over/ a q,{b\over a},c,d,e,{{{a^2} q}\over {b c d
     e}},{{b q}\over c},{{b q}\over d},{{b q}\over e},{{{b^2} c d e}\over
     {{a^2}}};q)_\infty}}
{{( \let\over/  {a\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{b c
     d e}\over a},{{b c}\over a},{{b d}\over a},{{b e}\over a},{{a q}\over {c
     d e}},{{{b^2} q}\over a};q)_\infty}} \\
    {}_8W _7({{{b^2}}\over a};b,{{b c}\over a},{{b d}\over a},{{b e}\over
     a},{{a q}\over {c d e}};q,q)
\endmultline$$
 
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S10901
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.12) 
in form of a rule.
$$
{} _{10} W _{9} ({\displaystyle a; {\sqrt{b}}, -{\sqrt{b}}, {\sqrt{b q}}
   , - {\sqrt{b q}}   , {a\over b}, {{a^2
   {q^{1 + n}}}\over b}, {q^{-n}}}; q, {\displaystyle q}) \longrightarrow  
  {{({\let \over / a q, {{a^2 q}\over {b^2}}}; q) _{n}}\over {({\let \over
   / {{a q}\over b}, {{a^2 q}\over b}}; q) _{n}}} 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name SchreibeZahl
\Description Variable that counts the number of expressions already written
  by using \hbox{\tt TeXMat}. Can be reset by defining a new value.
\Usage SchreibeZahl=n\MATHtief\ Integer.
\Example 
\MATH 
In[1]:= SchreibeZahl
\goodbreakpoint%
Out[1]= 0
\goodbreakpoint%
In[2]:= TeXMat[pq[a,n],file]
\goodbreakpoint%
In[3]:= !type file.m
A[1]:=
pq[a, n, q]
\goodbreakpoint%
In[3]:= !type file.tex
A[1]:=
(\MATHlbrace \MATHbackslash let \MATHbackslash %
over / a\MATHrbrace ; q) \MATHtief \MATHlbrace n\MATHrbrace  
\goodbreakpoint%
In[3]:= SchreibeZahl=4
\goodbreakpoint%
Out[3]= 4
\goodbreakpoint%
In[4]:= TeXMat[pq[a,n],file]
\goodbreakpoint%
In[5]:= !type file.m
A[1]:=
pq[a, n, q]
A[5]:=
pq[a, n, q]
\goodbreakpoint%
In[5]:= !type file.tex
A[1]:=
(\MATHlbrace \MATHbackslash let \MATHbackslash %
over / a\MATHrbrace ; q) \MATHtief \MATHlbrace n\MATHrbrace  
A[5]:=
(\MATHlbrace \MATHbackslash let \MATHbackslash %
over / a\MATHrbrace ; q) \MATHtief \MATHlbrace n\MATHrbrace  
\endMATH 
\Seealso TeXMat.

\Name Sgl0110
\Description Summation formula (\cite{\GaRaAA}, (1.6.1); Appendix (II.28)) in form of an equation.
It is the same summation as that in \hbox{\tt S0110}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl1001
\Description Summation formula (\cite{\GaRaAA}, (1.3.2); Appendix (II.3)) in form of an equation.
It is the same summation as that in \hbox{\tt S1001}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl1101
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.6(ii); Appendix (II.5)) in form of an equation.
It is the same summation as that in \hbox{\tt S1101}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl1102
\Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9);
$b\to\infty$) in form of an equation.
It is the same summation as that in \hbox{\tt S1102}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl1110
\Description Summation formula (\cite{\GaRaAA}, (5.2.1); Appendix (II.29)) in form of an equation.
It is the same summation as that in \hbox{\tt S1110}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2101
\Description Summation formula (\cite{\GaRaAA}, (1.5.3); Appendix (II.6)) in form of an equation.
It is the same summation as that in \hbox{\tt S2101}.
\Example 
\MATH 
In[1]:= \MATHlbrace a,c,q\MATHrbrace 
\goodbreakpoint%
Out[1]= \MATHlbrace a, c, q\MATHrbrace 
\goodbreakpoint%
In[2]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[3]:= Sgl2101
Do you want to set values for the equation? [y|n]: y
a=a\MATHhoch 2
c=1/b
n=n
Do you want to set a value for q in the equation? [y|n]: y
q=q\MATHhoch 2
\goodbreakpoint%
                                     2 n   1     2
                                    a    (----; q )
           \MATHluEck   2   -2 n         \MATHruEck            2       n
           \MATHvStrich  a , q       2   2 \MATHvStrich           a  b
Out[3]=  \MATHphi  \MATHvStrich           ; q , q  \MATHvStrich  == %
----------------
        2 1\MATHvStrich      1             \MATHvStrich          1   2
           \MATHloEck      -             \MATHroEck         (-; q )
                 b                       b     n
\goodbreakpoint%
In[4]:= \MATHlbrace a,c,q\MATHrbrace 
\goodbreakpoint%
Out[4]= \MATHlbrace a, c, q\MATHrbrace 
\goodbreakpoint%
In[5]:= a=b\MATHhoch 2
\goodbreakpoint%
         2
Out[5]= b
\goodbreakpoint%
In[6]:= q=p
\goodbreakpoint%
Out[6]= p
\goodbreakpoint%
In[7]:= Sgl2101
Some variables have a value. Should the variables 
\MATHlbrace a, c, n\MATHrbrace  be cleared? Do you want to set       
values for the equation (v)? [y|n|yv|nv]: nv
a=a
c=c
n=n
q has a value. Should q be cleared?        
Do you want to set a value for q in the equation (v)? [y|n|yv|nv]: yv
q=p\MATHhoch 2
\goodbreakpoint%
                                     2 n  c    2
                                    b    (--; p )
           \MATHluEck                    \MATHruEck            2     n
           \MATHvStrich   2   -2 n   2   2 \MATHvStrich           b
Out[7]=  \MATHphi  \MATHvStrich  b , p    ; p , p  \MATHvStrich  == %
--------------
        2 1\MATHvStrich                    \MATHvStrich             2
           \MATHloEck      c             \MATHroEck        (c; p )
                                              n
\goodbreakpoint%
In[8]:= \MATHlbrace a,c,q\MATHrbrace 
\goodbreakpoint%
          2
Out[8]= \MATHlbrace b , c, q\MATHrbrace 
\goodbreakpoint%
In[9]:= Sgl2101
Some variables have a value. Should the variables 
\MATHlbrace a, c, n\MATHrbrace  be cleared? Do you want to set       
values for the equation (v)? [y|n|yv|nv]: y
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[9]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[10]:= \MATHlbrace a,c,q\MATHrbrace 
\goodbreakpoint%
Out[10]= \MATHlbrace a, c, q\MATHrbrace 
\endMATH 
\Seealso SumListe\$gl, Gleichung.

\Name Sgl2102
\Description Summation formula (\cite{\GaRaAA}, (1.5.2); Appendix (II.7)) in form of an equation.
It is the same summation as that in \hbox{\tt S2102}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2103
\Description Summation formula (\cite{\GaRaAA}, (1.5.1); Appendix (II.8)) in form of an equation.
It is the same summation as that in \hbox{\tt S2103}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2104
\Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9)) in form of an equation.
It is the same summation as that in \hbox{\tt S2104}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2105
\Description Summation formula (\cite{\GaRaAA}, Ex~1.6(i)) 
in form of an equation.
It is the same summation as that in \hbox{\tt S2105}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2106
\Description Summation formula (\cite{\GaRaAA}, Ex~1.7) 
in form of an equation.
It is the same summation as that in \hbox{\tt S2106}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2107
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.8) 
in form of an equation.
It is the same summation as that in \hbox{\tt S2107}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2161
\Description Summation formula (\cite{\GaRaAA}, (2.10.13); Appendix (II.23)) in form of an equation.
It is the same summation as that in \hbox{\tt S2161}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2201
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(i); Appendix (II.10)) in form of an equation.
It is the same summation as that in \hbox{\tt S2201}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2202
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(ii); Appendix (II.11)) in form of an equation.
It is the same summation as that in \hbox{\tt S2202}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2210
\Description Summation formula (\cite{\GaRaAA}, (5.3.4); Appendix (II.30)) in form of an equation.
It is the same summation as that in \hbox{\tt S2210}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3201
\Description Summation formula (\cite{\GaRaAA}, (1.7.2); Appendix (II.12)) in form of an equation.
It is the same summation as that in \hbox{\tt S3201}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3202
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.7; Appendix (II.15)) in form of an equation.
It is the same summation as that in \hbox{\tt S3202}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3203
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.1) 
in form of an equation.
It is the same summation as that in \hbox{\tt S3203}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3204
\Description Summation formula (\cite{\GaRaAA}, Ex.~3.9) 
in form of an equation.
It is the same summation as that in \hbox{\tt S3204}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3261
\Description Summation formula (\cite{\GaRaAA}, (2.10.12); Appendix (II.24)) in form of an equation.
It is the same summation as that in \hbox{\tt S3261}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3310
\Description Summation formula (\cite{\GaRaAA}, Ex.~5.18(i); Appendix (II.31)) in form of an equation.
It is the same summation as that in \hbox{\tt S3310}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4301
\Description Summation formula (\cite{\GaRaAA}, (2.7.2); Appendix (II.13)) in form of an equation.
It is the same summation as that in \hbox{\tt S4301}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4302
\Description Summation formula (\cite{\GaRaAA}, (2.7.2), terminating form; Appendix (II.14)) in form of an equation.
It is the same summation as that in \hbox{\tt S4302}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4303
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.8; Appendix (II.17)) in form of an equation.
It is the same summation as that in \hbox{\tt S4303}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4304
\Description Summation formula (Andrews [1976a] in \cite{\GaRaAA}, Appendix (II.19)) in form of an equation.
It is the same summation as that in \hbox{\tt S4304}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4305
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.6) 
in form of an equation.
It is the same summation as that in \hbox{\tt S4305}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4306
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.14(i)) 
in form of an equation.
It is the same summation as that in \hbox{\tt S4306}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4307
\Description Summation formula (\cite{\GaRaAA}, (2.7.2)) 
in form of an equation.
It is the same summation as that in \hbox{\tt S4307}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4308
\Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to
\sqrt{aq}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA},
(2.6.2)) in form of an equation.
It is the same summation as that in \hbox{\tt S4308}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4361
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.9) 
in form of an equation.
It is the same summation as that in \hbox{\tt S4361}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4410
\Description Summation formula (\cite{\GaRaAA}, (5.3.3); Appendix (II.32)) in form of an equation.
It is the same summation as that in \hbox{\tt S4410}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl5401
\Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to \sqrt{aq}$) 
in form of an equation.
It is the same summation as that in \hbox{\tt S5401}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl5402
\Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to
q\sqrt{a}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA},
(2.6.2)) in form of an equation.
It is the same summation as that in \hbox{\tt S5402}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl5501
\Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to\infty$) 
in form of an equation.
It is the same summation as that in \hbox{\tt S5501}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl6501
\Description Summation formula (\cite{\GaRaAA}, (2.7.1); Appendix (II.20)) in form of an equation.
It is the same summation as that in \hbox{\tt S6501}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl6502
\Description Summation formula (\cite{\GaRaAA}, (2.4.2); Appendix (II.21)) in form of an equation.
It is the same summation as that in \hbox{\tt S6502}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl6610
\Description Summation formula (\cite{\GaRaAA}, (5.3.1); Appendix (II.33)) in form of an equation.
It is the same summation as that in \hbox{\tt S6610}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl8701
\Description Summation formula (\cite{\GaRaAA}, (2.6.2); Appendix (II.22)) in form of an equation.
It is the same summation as that in \hbox{\tt S8701}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl8702
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(i); Appendix (II.16)) in form of an equation.
It is the same summation as that in \hbox{\tt S8702}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl8703
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(ii); Appendix (II.18)) in form of an equation.
It is the same summation as that in \hbox{\tt S8703}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl8704
\Description Summation formula (\cite{\GaRaAA}, Ex.~3.10) 
in form of an equation.
It is the same summation as that in \hbox{\tt S8704}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl8761
\Description Summation formula (\cite{\GaRaAA}, (2.11.7); Appendix (II.25)) in form of an equation.
It is the same summation as that in \hbox{\tt S8761}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl10901
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.12) 
in form of an equation.
It is the same summation as that in \hbox{\tt S10901}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name SimplifyPQ
\Description Rule that simplifies arguments in \hbox{\tt pq},
\hbox{\tt pqinf}, \hbox{\tt ph}, \hbox{\tt Ph}, \hbox{\tt ps},
\hbox{\tt W}, \hbox{\tt SUM}, and
    expands exponents in powers.
\Usage Expr/.SimplifyPQ.
\Example
\MATH
In[1]:= pq[a\MATHhoch 2+a,(k-l)*2]/pqinf[1-f\MATHhoch 2,p]*ph[%
\MATHlbrace a\MATHhoch (1/2-(1-s)/2),-n%
\MATHrbrace ,%
\MATHlbrace a\MATHhoch (s/2)%
\MATHrbrace ,q,
        (t-1)/(1-t)]
\goodbreakpoint%
           %
\MATHluEck   1/2 + (-1 + s)/2                %
\MATHruEck 
           %
\MATHvStrich  a                , -n     -1 + t %
\MATHvStrich        2
         \MATHphi  %
\MATHvStrich                       ; q, ------ %
\MATHvStrich  (a + a ; q)
        2 1%
\MATHvStrich           s/2              1 - t  %
\MATHvStrich             2 (k - l)
           %
\MATHloEck          a                        %
\MATHroEck 
Out[1]= ------------------------------------------------------------
                                      2
                                (1 - f ;p)
                                          \MATHinfty 
\goodbreakpoint%
In[2]:= \%/.SimplifyPQ
\goodbreakpoint%
           %
\MATHluEck            %
\MATHruEck 
           %
\MATHvStrich  -n        %
\MATHvStrich 
         \MATHphi  %
\MATHvStrich    ; q, -1 %
\MATHvStrich  (a (1 + a); q)
        1 0%
\MATHvStrich  -         %
\MATHvStrich                2 k - 2 l
           %
\MATHloEck            %
\MATHroEck 
Out[2]= ----------------------------------------
                  ((1 - f) (1 + f);p)
                                     \MATHinfty 
\goodbreakpoint%
In[3]:= Simplify[\%1]
\goodbreakpoint%
           %
\MATHluEck            %
\MATHruEck 
           %
\MATHvStrich  -n        %
\MATHvStrich        2
         \MATHphi  %
\MATHvStrich    ; q, -1 %
\MATHvStrich  (a + a ; q)
        1 0%
\MATHvStrich  -         %
\MATHvStrich             2 (k - l)
           %
\MATHloEck            %
\MATHroEck 
Out[3]= -------------------------------------
                           2
                     (1 - f ;p)
                               \MATHinfty 
\endMATH
\Seealso Expandq, MinusOne, SUMExpand, PQSort.

\Name SListe
\Description Rule that gives for a basic hypergeometric series a list of 
   applicable summation formulas. Each entry of this list has the format 
   \hbox{\tt $\{$ArgumentPermutations,S$\langle$number$\rangle$$\}$}, 
where \hbox{\tt
ArgumentPermutations} is a 
   sequence of reorderings of the parameters of the basic hypergeometric
   series (given in terms of \hbox{\tt phPerm} and \hbox{\tt
phTausche}) and \hbox{\tt S$\langle$number$\rangle$} 
   is the name of the summation in form of a rule which can be applied 
   subsequently. You should be aware that \hbox{\tt SListe} automatically applies
   \hbox{\tt phOrdne} before checking which summation could be applied.
\vskip6pt
\hangafter1 \hangindent10pt\rm
\underbar{Important Note}: If the value returned by \hbox{\tt SListe}
is the empty set this does {\it not} mean that no summation can be
applied.
You always must remember that the list of summations included in
this package is a list of {\it basic} summations. There are numerous
special cases of these summations which are not contained in
this list as a separate summation. The examples below should
illustrate these remarks.
\Usage Expr/.SListe.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q\MATHhoch 2,c/a/b]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, b   2   c  %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      ; q , --- %
\MATHvStrich 
        2 1%
\MATHvStrich   c        a b %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.SListe
\goodbreakpoint%
Be sure to apply "phOrdne" before using the following information!
\goodbreakpoint%
Out[2]= %
\MATHlbrace %
\MATHlbrace S2103%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[3]:= ph[%
\MATHlbrace q\MATHhoch -n,-c,c,q\MATHhoch (1+n)%
\MATHrbrace ,%
\MATHlbrace -q,e,c\MATHhoch 2*q/e%
\MATHrbrace ,q,q]
\goodbreakpoint%
              -n          1 + n
           %
\MATHluEck  q  , -c, c, q            %
\MATHruEck 
           %
\MATHvStrich                           %
\MATHvStrich 
Out[3]=  \MATHphi  %
\MATHvStrich             2      ; q, q %
\MATHvStrich 
        4 3%
\MATHvStrich            c  q           %
\MATHvStrich 
           %
\MATHloEck     -q, e, ----           %
\MATHroEck 
                        e
\goodbreakpoint%
In[4]:= \%/.SListe
Is n a nonnegative integer?
[y|n]:  y
Is -1 - n a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
Be sure to apply "phOrdne" before using the following information!
\goodbreakpoint%
Out[4]= %
\MATHlbrace %
\MATHlbrace phPerm[3,1,2,4,u], phTausche[1,3,l], S4304%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[5]:= \%3/.phOrdne/.phPerm[3,1,2,4,u]/.phTausche[1,3,l]/.S4304
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
                        2  1 - n   2  2 + n
              1        c  q       c  q       e      1 + n   2
Out[5]= ------------- (---------, ---------, --, e q     ; q )
          2                e          e       n               \MATHinfty 
         c  q                                q
        (----, e; q)
          e         \MATHinfty 
\endMATH
\vskip10pt\noindent
Now we consider two examples illustrating the note above. Though none of the 
implemented summations can be applied, both series are special cases of a 
$q$-analogue of Dixon's sum. This fact is also observed by using this
package.
\vskip10pt
\MATH
In[6]:= ph[%
\MATHlbrace a,-q*Sqrt[a],q\MATHhoch -n%
\MATHrbrace ,%
\MATHlbrace -Sqrt[a],q*a*q\MATHhoch n%
\MATHrbrace ,q,-Sqrt[q]*q\MATHhoch n]
\goodbreakpoint%
           %
\MATHluEck                    -n               %
\MATHruEck 
           %
\MATHvStrich  a, -(Sqrt[a] q), q         1/2 + n %
\MATHvStrich 
Out[6]=  \MATHphi  %
\MATHvStrich                      ; q, -q        %
\MATHvStrich 
        3 2%
\MATHvStrich                1 + n                %
\MATHvStrich 
           %
\MATHloEck   -Sqrt[a], a q                     %
\MATHroEck 
\goodbreakpoint%
In[7]:= \%/.SListe
\goodbreakpoint%
Out[7]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[8]:= Sgl4301
Do you want to set values for the equation? [y|n]: y
a=a
b=q\MATHhoch -n
c=-Sqrt[a*q]
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
           %
\MATHluEck                    -n               %
\MATHruEck 
           %
\MATHvStrich  a, -(Sqrt[a] q), q         1/2 + n %
\MATHvStrich 
Out[8]=  \MATHphi  %
\MATHvStrich                      ; q, -q        %
\MATHvStrich  == 
        3 2%
\MATHvStrich                1 + n                %
\MATHvStrich 
           %
\MATHloEck   -Sqrt[a], a q                     %
\MATHroEck 
 
                    1 + n                       1/2 + n
     (a q, Sqrt[a] q     , -Sqrt[q], -(Sqrt[a] q       ); q)
                                                            \MATHinfty 
\MATHgroesser     --------------------------------------------------------
         1 + n                                   1/2 + n
     (a q     , -(Sqrt[a] Sqrt[q]), Sqrt[a] q, -q       ; q)
                                                            \MATHinfty 
\goodbreakpoint%
In[9]:= ph[%
\MATHlbrace -Sqrt[a]*q,q,c%
\MATHrbrace ,%
\MATHlbrace -Sqrt[a],a*q/c%
\MATHrbrace ,q,Sqrt[a]/c]
\goodbreakpoint%
           %
\MATHluEck  -(Sqrt[a] q), q, c             %
\MATHruEck 
           %
\MATHvStrich                         Sqrt[a] %
\MATHvStrich 
Out[9]=  \MATHphi  %
\MATHvStrich              a q   ; q, ------- %
\MATHvStrich 
        3 2%
\MATHvStrich    -Sqrt[a], ---           c    %
\MATHvStrich 
           %
\MATHloEck               c                 %
\MATHroEck 
\goodbreakpoint%
In[10]:= \%/.SListe
\goodbreakpoint%
Out[10]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[11]:= \%\%/.phEinf
Add the parameter: a
\goodbreakpoint%
            %
\MATHluEck  a, -(Sqrt[a] q), q, c             %
\MATHruEck 
            %
\MATHvStrich                            Sqrt[a] %
\MATHvStrich 
Out[11]=  \MATHphi  %
\MATHvStrich                 a q   ; q, ------- %
\MATHvStrich 
         4 3%
\MATHvStrich    a, -Sqrt[a], ---           c    %
\MATHvStrich 
            %
\MATHloEck                  c                 %
\MATHroEck 
\goodbreakpoint%
In[12]:= \%/.SListe
\goodbreakpoint%
Be sure to apply "phOrdne" before using the following information!
\goodbreakpoint%
Out[12]= %
\MATHlbrace %
\MATHlbrace S4301%
\MATHrbrace %
\MATHrbrace 
\endMATH
\Seealso TListe, phPerm, phTausche, SumListe.

\Name Sub
\Description Function that subtracts \hbox{\tt Expr} from \hbox{\tt Gleichung}.
\Usage Sub[Expr].
\Example 
\MATH 
In[1]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
                                n  c
           \MATHluEck               \MATHruEck     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich         a    n
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  == ----------
        2 1\MATHvStrich               \MATHvStrich      (c; q)
           \MATHloEck    c          \MATHroEck            n
\goodbreakpoint%
In[2]:= Sub[pq[a,n]/pq[c/a,n]]
\goodbreakpoint%
                                                       n  c
           \MATHluEck               \MATHruEck    (a; q)       (a; q)     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich          n            n        a    n
Out[2]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  - ------- == -(-------) + ----------
        2 1\MATHvStrich               \MATHvStrich     c            c          (c; q)
           \MATHloEck    c          \MATHroEck    (-; q)       (-; q)            n
                               a    n       a    n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
                                                       n  c
           \MATHluEck               \MATHruEck    (a; q)       (a; q)     a  (-; q)
           \MATHvStrich      -n       \MATHvStrich          n            n        a    n
Out[3]=  \MATHphi  \MATHvStrich  a, q  ; q, q \MATHvStrich  - ------- == -(-------) + ----------
        2 1\MATHvStrich               \MATHvStrich     c            c          (c; q)
           \MATHloEck    c          \MATHroEck    (-; q)       (-; q)            n
                               a    n       a    n
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, 
Hoch, GlTausche, Ers.

\Name Subst
\Description Function that substitutes \hbox{\tt RS} instead of
\hbox{\tt LS} at position
    \hbox{\tt Position} in \hbox{\tt Expr}. The parameters \hbox{\tt LS} and
\hbox{\tt RS} are optional. If they are
    omitted, the right-hand side ``\hbox{\tt RS}" of \hbox{\tt Gleichung} is substituted
    instead of the left-hand side ``\hbox{\tt LS}" of \hbox{\tt Gleichung}. 
\Usage Subst[Expr,Position,LS,RS].
\Example
\MATH
In[1]:= SUM[pq[a,k]/pq[b,k]/pq[q,k],%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck     (a; q)
         \MATHbackslash              k
Out[1]=   \MATHgroesser   ---------------
         /   (b; q)  (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck       k       k
         k=0
\goodbreakpoint%
In[2]:= Subst[\%,%
\MATHlbrace 1%
\MATHrbrace ,pq[a,k]/pq[b,k],ph[%
\MATHlbrace b/a,q\MATHhoch -k%
\MATHrbrace ,%
\MATHlbrace b%
\MATHrbrace ,q,a*q\MATHhoch k]]
\goodbreakpoint%
                %
\MATHluEck  b   -k          %
\MATHruEck 
                %
\MATHvStrich  -, q          k %
\MATHvStrich 
          \MATHinfty    \MATHphi  %
\MATHvStrich  a     ; q, a q  %
\MATHvStrich 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 2 1%
\MATHvStrich                  %
\MATHvStrich 
         \MATHbackslash       %
\MATHloEck    b             %
\MATHroEck 
Out[2]=   \MATHgroesser   ----------------------
         /          (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck              k
         k=0
\goodbreakpoint%
In[3]:= Sgl2102
Do you want to set values for the equation? [y|n]: y
a=b/a
c=b
n=k
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
           %
\MATHluEck  b   -k          %
\MATHruEck     (a; q)
           %
\MATHvStrich  -, q          k %
\MATHvStrich           k
Out[3]=  \MATHphi  %
\MATHvStrich  a     ; q, a q  %
\MATHvStrich  == -------
        2 1%
\MATHvStrich                  %
\MATHvStrich     (b; q)
           %
\MATHloEck    b             %
\MATHroEck           k
\goodbreakpoint%
In[4]:= GlTausche
\goodbreakpoint%
        (a; q)        %
\MATHluEck  b   -k          %
\MATHruEck 
              k       %
\MATHvStrich  -, q          k %
\MATHvStrich 
Out[4]= ------- ==  \MATHphi  %
\MATHvStrich  a     ; q, a q  %
\MATHvStrich 
        (b; q)     2 1%
\MATHvStrich                  %
\MATHvStrich 
              k       %
\MATHloEck    b             %
\MATHroEck 
\goodbreakpoint%
In[5]:= Gleichung
\goodbreakpoint%
        (a; q)        %
\MATHluEck  b   -k          %
\MATHruEck 
              k       %
\MATHvStrich  -, q          k %
\MATHvStrich 
Out[5]= ------- ==  \MATHphi  %
\MATHvStrich  a     ; q, a q  %
\MATHvStrich 
        (b; q)     2 1%
\MATHvStrich                  %
\MATHvStrich 
              k       %
\MATHloEck    b             %
\MATHroEck 
\goodbreakpoint%
In[6]:= Subst[\%1,%
\MATHlbrace 1%
\MATHrbrace ]
\goodbreakpoint%
                %
\MATHluEck  b   -k          %
\MATHruEck 
                %
\MATHvStrich  -, q          k %
\MATHvStrich 
          \MATHinfty    \MATHphi  %
\MATHvStrich  a     ; q, a q  %
\MATHvStrich 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 2 1%
\MATHvStrich                  %
\MATHvStrich 
         \MATHbackslash       %
\MATHloEck    b             %
\MATHroEck 
Out[6]=   \MATHgroesser   ----------------------
         /          (q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck              k
         k=0
\endMATH
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, GlTausche, Ers,
    PosListe.

\Name SUM
\Description This is HYPQ's internal object for entering sums.
It should be used instead of Mathematica's \hbox{\tt Sum}.
\Usage SUM[Summand,{summation-index,lower-bound,upper-bound}].
\Example See the examples for \hbox{\tt SUMph} and \hbox{\tt
SUMInfinity}.
\Seealso SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMSammle, 
SUMTausche.

\Name SUMErw1
\Description Rule that extends a {\tt SUM[]} at the top. 
\vskip6pt
\leavevmode\hphantom{Description: }
$\sum\limits _{k=l} ^{n} \text {Expr} \to \sum\limits _{k=l} ^{n+m} \text{Expr}-\sum\limits _{k=n+1} ^{n+m} \text{Expr}$.
\vskip6pt
\leavevmode\hskip10pt 
   The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.SUMErw1.
\Example
\MATH
In[1]:= SUM[a[k],%
\MATHlbrace k,0,N%
\MATHrbrace ]
\goodbreakpoint%
          N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash    
Out[1]=   \MATHgroesser   a[k]
         /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMErw1
top-extend by: 3
\goodbreakpoint%
                                          3 + N
                                          %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
                                           \MATHbackslash    
Out[2]= -a[1 + N] - a[2 + N] - a[3 + N] +   \MATHgroesser   a[k]
                                           /   
                                          %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
                                           k=0
\goodbreakpoint%
In[3]:= \%\%/.SUMErw1
top-extend by: M
\goodbreakpoint%
        M + N        M + N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck         %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash             \MATHbackslash    
Out[3]=   \MATHgroesser   a[k] -    \MATHgroesser    a[k]
         /            /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0        k=1 + N
\endMATH
\Seealso SUM, SUMErw2, SUMZerl, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe.

\Name SUMErw2
\Description Rule that extends a \hbox{\tt SUM[]} at the bottom. 
\vskip6pt
\leavevmode\hphantom{Description: }
$   \sum\limits _{k=l} ^{n} \text{Expr} \to \sum\limits _{k=l-m} ^{n} \text{Expr}-\sum\limits _{k=l-m} ^{l-1} \text{Expr}$.
\vskip6pt
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.SUMErw2.
\Example
\MATH
In[1]:= SUM[a[k],%
\MATHlbrace k,0,N%
\MATHrbrace ]
\goodbreakpoint%
          N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash    
Out[1]=   \MATHgroesser   a[k]
         /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMErw2
bottom-extend by: 3
\goodbreakpoint%
                                   N
                                 %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
                                  \MATHbackslash    
Out[2]= -a[-3] - a[-2] - a[-1] +   \MATHgroesser   a[k]
                                  /   
                                 %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
                                 k=-3
\goodbreakpoint%
In[3]:= \%\%/.SUMErw2
bottom-extend by: M
\goodbreakpoint%
           -1            N
          %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck         %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
           \MATHbackslash             \MATHbackslash    
Out[3]= -(  \MATHgroesser   a[k]) +   \MATHgroesser   a[k]
           /            /   
          %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
          k=-M         k=-M
\endMATH
\Seealso SUM, SUMErw1, SUMZerl, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe.

\Name SUMExpand
\Description Rule that expands \hbox{\tt SUM}s.
\Usage Expr/.SUMExpand.
\Example
\MATH
In[1]:= SUM[(x[k]-y[k])\MATHhoch 2,%
\MATHlbrace k,0,(m+n)/2%
\MATHrbrace ]
\goodbreakpoint%
        m + n
        -----
          2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck              2
Out[1]=  \MATHbackslash    (x[k] - y[k])
          \MATHgroesser   
         /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMExpand
\goodbreakpoint%
        m   n        m   n               m   n
        - + -        - + -               - + -
        2   2        2   2               2   2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck     2   %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck                %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck     2
Out[2]=  \MATHbackslash    x[k]  +  \MATHbackslash    -2 x[k] y[k] +  \MATHbackslash    y[k]
          \MATHgroesser             \MATHgroesser                    \MATHgroesser   
         /            /                   /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck                %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0          k=0                 k=0
\endMATH
\Seealso SUM, SimplifyP, Expandq, MinusOne, PSort.

\Name SUMInfinity
\Description Rule that changes the upper bound of a \hbox{\tt SUM[]} to \hbox{\tt Infinity}.
\Usage Expr/.SUMInfinity.
\Example 
\MATH 
In[1]:= SUM[pq[q\MATHhoch -n,k]/pq[q,k]*q\MATHhoch k,\MATHlbrace k,0,n\MATHrbrace ]
\goodbreakpoint%
          n   k   -n
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q  (q  ; q)
         \MATHbackslash               k
Out[1]=   \MATHgroesser   ------------
         /     (q; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck         k
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMInfinity
\goodbreakpoint%
          \MATHinfty    k   -n
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q  (q  ; q)
         \MATHbackslash               k
Out[2]=   \MATHgroesser   ------------
         /     (q; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck         k
         k=0
\endMATH 
\Seealso SUM, SUMph, SUMPh, SUMps, Ers, PosListe.

\Name SumListe
\Description List of all summation formulas.
\Usage SumListe.
\Seealso SumListe\$gl, SListe.

\Name SumListe\$gl
\Description List of all summation formulas.
\Usage SumListe\$gl.
\Seealso SumListe.

\Name SUMPh
\Description Rule that transforms a \hbox{\tt SUM[]} into multibasic 
hypergeometric notation, if possible. If the upper bound is not 
\hbox{\tt Infinity} you have to apply \hbox{\tt SUMInfinity} first (if allowed).
\Usage Expr/.SUMPh.
\Example
\MATH
In[1]:= SUM[pq[a,n]*pq[b,n,q\MATHhoch 2]/pq[c,n,q\MATHhoch 2]/pq[q,n]*z\MATHhoch n,%
\MATHlbrace n,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty    n              2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q)  (b; q )
         \MATHbackslash             n        n
Out[1]=   \MATHgroesser   -------------------
         /         2
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck  (c; q )  (q; q)
         n=0         n       n
\goodbreakpoint%
In[2]:= \%/.SUMPh
\goodbreakpoint%
         %
\MATHluEck                  %
\MATHruEck 
         %
\MATHvStrich  a: b      2    %
\MATHvStrich 
Out[2]= \MATHphi %
\MATHvStrich       ; q, q ; z %
\MATHvStrich 
         %
\MATHvStrich  -: c           %
\MATHvStrich 
         %
\MATHloEck                  %
\MATHroEck 
\endMATH
\Seealso SUM, Ph, SUMRegeln, SUMph, SUMInfinity, PhSUM, Phph, phPh, Ers, PosListe.

\Name SUMph
\Description Rule that transforms a \hbox{\tt SUM[]} into basic hypergeometric notation,
if possible. If the upper bound is not \hbox{\tt Infinity} you have to apply
\hbox{\tt SUMInfinity} first (if allowed).
\Usage Expr/.SUMph.
\Example 
\MATH 
In[1]:= SUM[pq[q\MATHhoch -n,k]/pq[q,k]*q\MATHhoch k,\MATHlbrace k,0,Infinity\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty    k   -n
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q  (q  ; q)
         \MATHbackslash               k
Out[1]=   \MATHgroesser   ------------
         /     (q; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck         k
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMph
\goodbreakpoint%
           \MATHluEck            \MATHruEck 
           \MATHvStrich   -n       \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich  q  ; q, q \MATHvStrich 
        1 0\MATHvStrich            \MATHvStrich 
           \MATHloEck   -        \MATHroEck 
\goodbreakpoint%
In[3]:= SUM[(1-q\MATHhoch (k+2))*pq[%
\MATHlbrace q\MATHhoch -n,a%
\MATHrbrace ,%
\MATHlbrace b,c,q%
\MATHrbrace ,k+1]*q\MATHhoch (k*(k+1)/2),
        %
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty                                 -n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck                             (q  , a; q)
         \MATHbackslash     (k (1 + k))/2       2 + k             1 + k
Out[3]=   \MATHgroesser   q              (1 - q     ) -----------------
         /                               (b, c, q; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck                                         1 + k
         k=0
\goodbreakpoint%
In[4]:= \%/.SUMph
\goodbreakpoint%
                           %
\MATHluEck        1 - n      3        %
\MATHruEck 
                           %
\MATHvStrich  a q, q     , q, q         %
\MATHvStrich            -n
        (1 - q) (1 + q)  \MATHphi  %
\MATHvStrich                    ; q, -q %
\MATHvStrich  (a; q)  (q  ; q)
                        4 4%
\MATHvStrich              2   2         %
\MATHvStrich        1         1
                           %
\MATHloEck   b q, c q, q , q          %
\MATHroEck 
Out[4]= ------------------------------------------------------------------
                             (b; q)  (c; q)  (q; q)
                                   1       1       1
\endMATH 
\Seealso SUM, ph, SUMRegeln, SUMps, SUMInfinity, phSUM, Ers, PosListe.

\Name SUMps
\Description Rule that transforms a bilateral \hbox{\tt SUM[]} into 
basic hypergeometric notation,
if possible. If the upper bound is not \hbox{\tt Infinity} you have to apply
\hbox{\tt SUMInfinity} first (if allowed).
If the lower bound is not \hbox{\tt Infinity} then \hbox{\tt SUMph} is applied.
\Usage Expr/.SUMps.
\Example
\MATH
In[1]:= SUM[pq[a,n]/pq[b,n]*z\MATHhoch n,%
\MATHlbrace n,-Infinity,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty    n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck z  (a; q)
         \MATHbackslash             n
Out[1]=   \MATHgroesser   ----------
         /    (b; q)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck        n
        n=-\MATHinfty 
\goodbreakpoint%
In[2]:= \%/.SUMps
\goodbreakpoint%
            %
\MATHluEck          %
\MATHruEck 
            %
\MATHvStrich  a       %
\MATHvStrich 
Out[2]=  ps %
\MATHvStrich   ; q, z %
\MATHvStrich 
        1  1%
\MATHvStrich  b       %
\MATHvStrich 
            %
\MATHloEck          %
\MATHroEck 
\endMATH
\Seealso SUM, ps, SUMRegeln, SUMph, SUMInfinity, psSUM, Ers, PosListe.

\Name SUMRegeln
\Description Rule that transforms the expressions in a \hbox{\tt
SUM[]} into a
form that could also be expressed in basic hypergeometric notation.
This is useful, if you want to convert a \hbox{\tt SUM[]} into basic
hypergeometric
notation but without using the \hbox{\tt ph[]}-notation. In particular, expressions
of the form $(-1)^{dk}$, where $d$ is an integer and $k$ is the summation
index, will simplify. 
\Usage Expr/.SUMRegeln.
\Example 
\MATH 
In[1]:= SUM[Binomialq[n,i]*Binomialq[m,k-i]*q\MATHhoch ((n-i)*(k-i)),\MATHlbrace %
i,0,Infinity\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck  %
                  \MATHluEck         \MATHruEck   \MATHluEck    \MATHruEck  
         \MATHbackslash     (-i + k) (-i + n) \MATHvStrich    m    %
\MATHvStrich   \MATHvStrich  n \MATHvStrich  
Out[1]=   \MATHgroesser   q                  \MATHvStrich         %
\MATHvStrich   \MATHvStrich    \MATHvStrich  
         /                      \MATHvStrich  -i + k \MATHvStrich   %
\MATHvStrich  i \MATHvStrich  
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck   %
                 \MATHloEck         \MATHroEck   \MATHloEck    \MATHroEck  
         i=0                              q      q
\goodbreakpoint%
In[2]:= \%/.SUMRegeln
\goodbreakpoint%
                \MATHinfty     i   -k        -n
              \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck  %
q  (q  ; q)  (q  ; q)
         k n   \MATHbackslash                i         i     1 - k + m
        q    (  \MATHgroesser   ------------------------) (q         ; q)
               /             1 - k + m                      k
              \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck %
(q; q)  (q         ; q)
               i=0       i                i
Out[2]= -----------------------------------------------------
                               (q; q)
                                     k
\endMATH 
\Seealso SUM, ph, ps, SUMph, SUMps, phSUM, psSUM, MinusOne, Ers, PosListe.

\Name SUMSammle
\Description Rule that causes all terms of an expression \hbox{\tt
Expr}, which involves  
a \hbox{\tt SUM[]} to be put into the \hbox{\tt SUM[]}.
\Usage Expr/.SUMSammle.
\Example 
\MATH 
In[1]:= pq[a,n]/pq[b,m]*(-1)\MATHhoch n*SUM[1/pq[q,k],\MATHlbrace k,0,Infinity\MATHrbrace ]
\goodbreakpoint%
                 \MATHinfty 
               \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck 
            n   \MATHbackslash       1
        (-1)  (  \MATHgroesser   -------) (a; q)
                /   (q; q)         n
               \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       k
                k=0
Out[1]= ----------------------------
                  (b; q)
                        m
\goodbreakpoint%
In[2]:= \%/.SUMSammle
\goodbreakpoint%
          \MATHinfty        n
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck  (-1)  (a; q)
         \MATHbackslash                 n
Out[2]=   \MATHgroesser   ---------------
         /   (b; q)  (q; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck       m       k
         k=0
\endMATH 
\Seealso SUM, SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, 
pqzus, pqinfzus, Ers,\linebreak PosListe.

\Name SUMShift
\Description Rule that shifts the index in a \hbox{\tt SUM[]}.
\vskip6pt
\leavevmode\hphantom{Description: }
$   \sum\limits _{k=l} ^{n} \text{Expr}(k) \to \sum\limits _{k=l-m} ^{n-m}
\text{Expr}(k+m)$.
\vskip6pt
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.SUMShift.
\Example
\MATH
In[1]:= SUM[a[k],%
\MATHlbrace k,3,N%
\MATHrbrace ]
\goodbreakpoint%
          N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash    
Out[1]=   \MATHgroesser   a[k]
         /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=3
\goodbreakpoint%
In[2]:= \%/.SUMShift
shift summation index by: 3
\goodbreakpoint%
        -3 + N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash    
Out[2]=   \MATHgroesser    a[3 + k]
         /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\endMATH
\Seealso SUM, SUMErw1, SUMErw2, SUMTausche, SUMZerl, SUMRegeln, SUMUmkehr, Ers, PosListe.

\Name SUMTausche
\Description Rule that exchanges summations. You should
apply \hbox{\tt SUMSammle} before applying \hbox{\tt SUMTausche}.
\vskip6pt
\leavevmode\hphantom{Description: }
$   \sum\limits _{k_1=l_1} ^{n_1}\sum\limits _{k_2=l_2} ^{n_2}\text {Expr}
\to \sum\limits _{k_2=l_2} ^{n_2}\sum\limits _{k_1=l_1} ^{n_1}\text {Expr}$.
\Usage Expr/.SUMTausche.
\Example
\MATH
In[1]:= SUM[SUM[Binomialq[n,k+l],%
\MATHlbrace k,0,n1%
\MATHrbrace ],%
\MATHlbrace l,0,n2%
\MATHrbrace ]
\goodbreakpoint%
         n2   n1
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck        %
\MATHruEck  
         \MATHbackslash     \MATHbackslash    %
\MATHvStrich    n   %
\MATHvStrich  
Out[1]=   \MATHgroesser     \MATHgroesser   %
\MATHvStrich        %
\MATHvStrich  
         /    /   %
\MATHvStrich  k + l %
\MATHvStrich  
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck %
\MATHloEck        %
\MATHroEck  
         l=0  k=0          q
\goodbreakpoint%
In[2]:= \%/.SUMTausche
\goodbreakpoint%
         n1   n2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck        %
\MATHruEck  
         \MATHbackslash     \MATHbackslash    %
\MATHvStrich    n   %
\MATHvStrich  
Out[2]=   \MATHgroesser     \MATHgroesser   %
\MATHvStrich        %
\MATHvStrich  
         /    /   %
\MATHvStrich  k + l %
\MATHvStrich  
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck %
\MATHloEck        %
\MATHroEck  
         k=0  l=0          q
\endMATH
\Seealso SUM, SUMErw1, SUMErw2, SUMSammle, SUMShift, SUMZerl, SUMRegeln, 
    SUMUmkehr, Ers, PosListe.

\Name SUMUmkehr
\Description Rule that reverses the order of summation. \hbox{\tt SUMUmkehr}
applies to \hbox{\tt SUM[]} as well as \hbox{\tt ph[]}.
\Usage Expr/.SUMUmkehr.
\Example 
\MATH 
In[1]:= ph[\MATHlbrace a,q\MATHhoch -n\MATHrbrace ,\MATHlbrace b\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck               \MATHruEck 
           \MATHvStrich      -n       \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich  a, q  ; q, z \MATHvStrich 
        2 1\MATHvStrich               \MATHvStrich 
           \MATHloEck    b          \MATHroEck 
\goodbreakpoint%
In[2]:= \%/.SUMUmkehr
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
                            1 - n
                       -n  q
                    \MATHluEck  q  , ------        1 + n \MATHruEck  (a; q)
            n  n    \MATHvStrich         b        b q      \MATHvStrich        n
        (-1)  z   \MATHphi  \MATHvStrich             ; q, -------- \MATHvStrich  -------
                 2 1\MATHvStrich     1 - n          a z    \MATHvStrich  (b; q)
                    \MATHloEck    q                      \MATHroEck        n
                        ------
                          a
Out[2]= ------------------------------------------------
                          (n (1 + n))/2
                         q
\goodbreakpoint%
In[3]:= ph[\MATHlbrace q\MATHhoch -m,q\MATHhoch -n\MATHrbrace ,\MATHlbrace b\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck                 \MATHruEck 
           \MATHvStrich   -m   -n       \MATHvStrich 
Out[3]=  \MATHphi  \MATHvStrich  q  , q  ; q, z \MATHvStrich 
        2 1\MATHvStrich                 \MATHvStrich 
           \MATHloEck     b           \MATHroEck 
\goodbreakpoint%
In[4]:= \%/.SUMUmkehr
Is m a nonnegative integer?
[y|n]:  n
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
                            1 - n                      -m
                    \MATHluEck   -n  q             1 + m + n \MATHruEck  (q  ; q)
            n  n    \MATHvStrich  q  , ------     b q          \MATHvStrich          n
        (-1)  z   \MATHphi  \MATHvStrich         b   ; q, ------------ \MATHvStrich  ---------
                 2 1\MATHvStrich                       z       \MATHvStrich   (b; q)
                    \MATHloEck   1 + m - n                   \MATHroEck         n
                      q
Out[4]= ------------------------------------------------------
                             (n (1 + n))/2
                            q
\goodbreakpoint%
In[5]:= SUM[pq[q\MATHhoch -n,k]/pq[q,k],\MATHlbrace k,0,Infinity\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty     -n
        \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (q  ; q)
         \MATHbackslash            k
Out[5]=   \MATHgroesser   ---------
         /    (q; q)
        \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck        k
         k=0
\goodbreakpoint%
In[6]:= \%/.SUMUmkehr
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
           n   k + k n   -n
         \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q        (q  ; q)
          \MATHbackslash                     k    -n
        (  \MATHgroesser   ------------------) (q  ; q)
          /        (q; q)                 n
         \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck            k
          k=0
Out[6]= -----------------------------------
                      (q; q)
                            n
\endMATH 
\Seealso SUM, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, SUMRegeln, Ers, PosListe.

\Name SUMZerl
\Description Rule that splits a \hbox{\tt SUM[]} into two parts. 
\vskip6pt
\leavevmode\hphantom{Description: }
$   \sum\limits _{k=l} ^{n} \text{Expr} \to \sum\limits _{k=l} ^{l+m-1} \text{Expr}
+\sum\limits _{k=l+m} ^{n} \text{Expr}$.
\vskip6pt
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.SUMZerl.
\Example
\MATH
In[1]:= SUM[a[k],%
\MATHlbrace k,0,N%
\MATHrbrace ]
\goodbreakpoint%
          N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash    
Out[1]=   \MATHgroesser   a[k]
         /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMZerl
bottom-split by: 3
\goodbreakpoint%
                               N
                             %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
                              \MATHbackslash    
Out[2]= a[0] + a[1] + a[2] +   \MATHgroesser   a[k]
                              /   
                             %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
                              k=3
\goodbreakpoint%
In[3]:= \%\%/.SUMZerl
bottom-split by: M
\goodbreakpoint%
        -1 + M         N
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck         %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash             \MATHbackslash    
Out[3]=   \MATHgroesser    a[k] +   \MATHgroesser   a[k]
         /            /   
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0          k=M
\endMATH
\Seealso SUM, SUMErw1, SUMErw2, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe.

\Name T2101
\Description Transformation formula (\cite{\GaRaAA}, (1.4.1); Appendix (III.1)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ b,a z;q)_\infty}}
{{( \let\over/  c,z;q)_\infty}} 
   {}_2\phi _1\!\left [ \matrix \let\over/ {c\over b},z\\ \let\over/  a z\endmatrix ;q,b\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2102
\Description Transformation formula (\cite{\GaRaAA}, (1.4.5); Appendix (III.2)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ {c\over b},b z;q)_\infty}}
{{( \let\over/  c,z;q)_\infty}} 
   {}_2\phi _1\!\left [ \matrix \let\over/ {{a b z}\over c},b\\ \let\over/  b z\endmatrix ;q,{c\over
    b}\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2103
\Description Transformation formula (\cite{\GaRaAA}, (1.4.6); Appendix (III.3)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ {{a b z}\over c};q)_\infty}}
{{( \let\over/  z;q)_\infty}} 
   {}_2\phi _1\!\left [ \matrix \let\over/ {c\over a},{c\over b}\\ \let\over/  c\endmatrix ;q,{{a b
    z}\over c}\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2104
\Description Transformation formula (\cite{\GaRaAA}, (1.5.4); Appendix (III.4)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ a z;q)_\infty}}
{{( \let\over/  z;q)_\infty}} 
   {}_2\phi _2\!\left [ \matrix \let\over/ a,{c\over b}\\ \let\over/  c,a z\endmatrix ;q,b z\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2105
\Description Transformation formula (\cite{\GaRaAA}, (3.2.4); Appendix (III.5)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ {{a b z}\over c};q)_\infty}}
{{( \let\over/  {{b z}\over c};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ a,{c\over b},0\\ \let\over/  c,{{c q}\over {b
    z}}\endmatrix ;q,q\right ]
$$
provided the $_3\phi_2$ is terminating.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2106
\Description Transformation formula (\cite{\GaRaAA}, (1.5.6); Appendix (III.6)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ b,{q^{-n}}\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  {{b^n} {z^n}\over {q^n}} {{(\let\over/ {c\over b};q)}_{n}\over{{(\let\over/
    c;q)}_{n}}}
 {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},{q\over z},{{{q^{1 -
       n}}}\over c}\\ \let\over/  {{b {q^{1 - n}}}\over c},0\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2107
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii); Appendix (III.7)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ b,{q^{-n}}\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
{{{(\let\over/ {c\over b};q)}_{n}}\over 
    {{(\let\over/ c;q)}_{n}}}
  {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},b,{{b z}\over {c {q^n}}}\\ \let\over/  {{b {q^{1
       - n}}}\over c},0\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2108
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii); Appendix (III.8)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ b,{q^{-n}}\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
{{(\let\over/ {c\over b};q)}_{n}\over 
    {{(\let\over/ c;q)}_{n}}
}  {b^n} {}_3\phi _1\!\left [ \matrix \let\over/ {q^{-n}},b,{q\over z}\\ \let\over/  {{b {q^{1 -
       n}}}\over c}\endmatrix ;q,{z\over c}\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2109
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.2, reversed) 
in form of a rule.
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a, b\\ \let \over /
   {{b q}\over a}\endmatrix ;q, {\displaystyle t} \right ] \longrightarrow  
  { {(\let \over / b, a t ;q) _\infty} \over {(\let \over / t, {b\over a} ;q)
    _\infty} } {} _{4} W _{3} ({\displaystyle {t\over q}; {1\over a}}; q,
    {\displaystyle b})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2110
\Description Transformation formula (\cite{\GaRaAA}, (3.4.7)) 
in form of a rule.
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a, b\\ \let \over /
   {{a q}\over b}\endmatrix ;q, {\displaystyle {{q x}\over {b^2}}} \right
   ] \longrightarrow  { {(\let \over / {{q x}\over b}, {{a q x^2}\over {b^2}} ;q)
    _\infty} \over {(\let \over / {{a q x}\over b}, {{q x^2}\over
    {b^2}} ;q) _\infty} }
{} _{8} W _{7} ({\displaystyle {{a x}\over b};
    x, {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a q}}, -
    {\sqrt{a q}}  }; q, {\displaystyle {{q x}\over
    {{b^2}}}})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2111
\Description Transformation formula (\cite{\GaRaAA}, (3.5.4)) 
in form of a rule.\NoBlackBoxes
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, b^2\\ \let \over /
   {{a^2 q^2}\over {b^2}}\endmatrix ;q^2, {\displaystyle
   {{q^2 x^2}\over {{b^4}}}} \right ] \longrightarrow  
  { {(\let \over / {{a q}\over {b^2}}, {{q x^2}\over {b^2}} ;q)
    _\infty} \over {(\let \over / {{a^2 q}\over {b^2}},
    {{a q x^2}\over {b^2}} ;q) _\infty} } 
   { {(\let \over / {{a^2 q x^2}\over {b^2}},
    {{a^2 q^2 x^2}\over {{b^4}}} ;q^2) _\infty} \over {(\let \over /
    {{q x^2}\over {b^2}}, {{q^2 x^2}\over {{b^4}}} ;q^2) _\infty}
    } {} _{8} W _{7} ({\displaystyle {{a x^2}\over {b^2}}; a, x, -x,
    {{{\sqrt{q}} x}\over b}, -{{{\sqrt{q}} x}\over b}}; q, {\displaystyle
    {{a q}\over {b^2}}})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2112
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(i), reversed) 
in form of a rule.\BlackBoxes
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a, a q\\ \let \over /
   b^2 q\endmatrix ;q^2, {\displaystyle z^2} \right ] \longrightarrow  
  { {(\let \over / a z ;q) _\infty} \over {(\let \over / z ;q) _\infty} } 
   {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, -b\\ \let \over /
    b^2, a z\endmatrix ;q, {\displaystyle -z} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2161
\Description Transformation formula (\cite{\GaRaAA}, (3.3.5); Appendix (III.31)) in form of a rule.
$$\multline {}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ {{a b z}\over c},{q\over c};q)_\infty}}
{{( \let\over/  {{a z}\over
     c},{q\over a};q)_\infty}} 
    {}_2\phi _1\!\left [ \matrix \let\over/ {c\over a},{{c q}\over {a b z}}\\ \let\over/  {{c q}\over
     {a z}}\endmatrix ;q,{{b q}\over c}\right ] \\- 
   \frac {{( \let\over/ b,{q\over c},{c\over a},{{a z}\over
     q},{{{q^2}}\over {a z}};q)_\infty}}
{{( \let\over/  {c\over q},{{b q}\over c},{q\over a},{{a
     z}\over c},{{c q}\over {a z}};q)_\infty}} 
    {}_2\phi _1\!\left [ \matrix \let\over/ {{a q}\over c},{{b q}\over c}\\ \let\over/  {{{q^2}}\over
     c}\endmatrix ;q,z\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2162
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.32)) in form of a rule.
$${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/  c\endmatrix ;q,z\right ] \longrightarrow 
  \frac {{( \let\over/ b,{c\over a},a z,{q\over {a z}};q)_\infty}}
{{( \let\over/  c,{b\over
     a},z,{q\over z};q)_\infty}} 
    {}_2\phi _1\!\left [ \matrix \let\over/ a,{{a q}\over c}\\ \let\over/  {{a q}\over b}\endmatrix
     ;q,{{c q}\over {a b z}}\right ] + 
   \frac {{( \let\over/ a,{c\over b},b z,{q\over {b z}};q)_\infty}}
{{( \let\over/  c,{a\over
     b},z,{q\over z};q)_\infty}} 
    {}_2\phi _1\!\left [ \matrix \let\over/ b,{{b q}\over c}\\ \let\over/  {{b q}\over a}\endmatrix
     ;q,{{c q}\over {a b z}}\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2163
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.8) 
in form of a rule.
$$
{} _{2} \phi _{1} \! \left [ \matrix \let \over / a, b\\ \let \over /
   c\endmatrix ;q, {\displaystyle x} \right ] \longrightarrow  
  { {(\let \over / b, {c\over a}, a x ;q) _\infty} \over {(\let \over /
     {b\over a}, c, x ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {c\over b}, 0\\ \let
     \over / {{a q}\over b}, a x\endmatrix ;q, {\displaystyle q} \right ] + 
   { {(\let \over / a, {c\over b}, b x ;q) _\infty} \over {(\let \over /
     {a\over b}, c, x ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / b, {c\over a}, 0\\ \let
     \over / {{b q}\over a}, b x\endmatrix ;q, {\displaystyle q} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2201
\Description Transformation formula (\cite{\GaRaAA}, (1.5.4), reversed; Appendix (III.4), reversed) in form 
of a rule.
$${}_2\phi _2\!\left [ \matrix \let\over/ a,b\\ \let\over/  c,d\endmatrix ;q,{{c d}\over {a b}}\right
   ] \longrightarrow \frac {{( \let\over/ {d\over a};q)_\infty}}
{{( \let\over/  d;q)_\infty}} 
   {}_2\phi _1\!\left [ \matrix \let\over/ a,{c\over b}\\ \let\over/  c\endmatrix ;q,{d\over a}\right
    ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2202
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii), reversed) 
in form of a rule.
$$
{} _{2} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2\\ \let \over /
   a^2 b^2 q, a^2 z^2\endmatrix ;q^2, {\displaystyle
   a^2 q z^2} \right ] \longrightarrow  
  {{({\let \over / z}; q) _{\infty} ({\let \over / - a^2 z  }; q) 
   _{\infty}  }\over 
     {({\let \over / a^2 z^2}; q^2) _{\infty} }} 
  {{      {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, a b,
      - a b  \\ \let \over / a^2 b^2, - a^2 z 
       \endmatrix ;q, {\displaystyle z} \right ]}}
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3101
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii), reversed; Appendix (III.8), reversed) in form 
  of a rule.
$$
{}_3\phi _1\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/  {{b c
z}\over {{q^n}}}\endmatrix
   ;q,z\right ] \longrightarrow
{b^n} 
{{(\let\over/ {q\over {c z}};q)}_{n}\over 
    {{(\let\over/ {q\over {b c z}};q)}_{n}}}
{}_2\phi _1\!\left [ \matrix \let\over/ {q^{-n}},b\\ \let\over/  {q\over
       {c z}}\endmatrix ;q,{{q}\over c}\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3201
\Description Transformation formula (\cite{\GaRaAA}, (3.2.4), reversed; 
Appendix (III.5), reversed) in form 
  of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ a,b,0\\ \let\over/  c,d\endmatrix ;q,q\right ] \longrightarrow 
  \frac {{( \let\over/ {q \over d};q)_\infty}}
{{( \let\over/  {{a q}
\over d};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ a,{c\over b}\\ \let\over/ 
    c\endmatrix ;q,{{b q}\over {d}}\right ]
$$
provided the $_3\phi_2$ is terminating.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3202
\Description Transformation formula (\cite{\GaRaAA}, (1.5.6), reversed; Appendix (III.6), reversed) in form 
  of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/  d,0\endmatrix ;q,q\right ] \longrightarrow 
{{b^n} {c^n}\over {d^n}}
{      {(\let\over/ {{{q^{1 - n}}}\over c};q)}_{n}\over 
    { {(\let\over/ {{{q^{1 - n}}}\over d};q)}_{n}}
}   {}_2\phi _1\!\left [ \matrix \let\over/ {q^{-n}},{d\over c}\\ \let\over/  {{{q^{1 -
       n}}}\over c}\endmatrix ;q,{q\over b}\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3203
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii),
reversed; Appendix (III.7), reversed) in form 
  of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/  d,0\endmatrix ;q,q\right ] \longrightarrow 
{{(\let\over/ {{b {q^{1 - n}}}\over d};q)}_{n}
      \over {{(\let\over/ {{{q^{1 - n}}}\over d};q)}_{n}}
}  {}_2\phi _1\!\left [ \matrix \let\over/ {q^{-n}},b\\ \let\over/  {{b {q^{1 - n}}}\over
       d}\endmatrix ;q,{{c q}\over d}\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3204
\Description Transformation formula (\cite{\GaRaAA}, (3.2.7); Appendix (III.9)) in form of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/  d,e\endmatrix ;q,{{d e}\over {a b
   c}}\right ] \longrightarrow \frac {{( \let\over/ {e\over a},{{d e}\over {b c}};q)_\infty}}
{{( \let\over/ 
    e,{{d e}\over {a b c}};q)_\infty}} 
   {}_3\phi _2\!\left [ \matrix \let\over/ a,{d\over b},{d\over c}\\ \let\over/  d,{{d e}\over {b
    c}}\endmatrix ;q,{e\over a}\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3205
\Description Transformation formula (\cite{\GaRaAA}, (3.2.10); Appendix (III.10)) in form of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/  d,e\endmatrix ;q,{{d e}\over {a b
   c}}\right ] \longrightarrow \frac {{( \let\over/ b,{{d e}\over {a b}},{{d e}\over {b
    c}};q)_\infty}}
{{( \let\over/  d,e,{{d e}\over {a b c}};q)_\infty}} 
   {}_3\phi _2\!\left [ \matrix \let\over/ {d\over b},{e\over b},{{d e}\over {a b c}}\\ \let\over/ 
    {{d e}\over {a b}},{{d e}\over {b c}}\endmatrix ;q,b\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3206
\Description Transformation formula (\cite{\GaRaAA}, (3.2.3); Appendix (III.11)) in form of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/  d,e\endmatrix ;q,q\right ] \longrightarrow 
{{b^n} {c^n} \over {d^n} }
{{(\let\over/ {{d e}\over {b c}};q)}_{n}
     \over {{(\let\over/ e;q)}_{n}}
}  {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},{d\over b},{d\over c}\\ \let\over/ 
       d,{{d e}\over {b c}}\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3207
\Description Transformation formula (\cite{\GaRaAA}, (3.2.2); Appendix (III.12)) in form of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/  d,e\endmatrix ;q,q\right ] \longrightarrow 
  {c^n}{ {(\let\over/ {e\over c};q)}_{n}
\over {{(\let\over/ e;q)}_{n}}}
 {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},c,{d\over b}\\ \let\over/  d,{{c {q^{1 -
       n}}}\over e}\endmatrix ;q,{{b q}\over e}\right ]
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3208
\Description Transformation formula (\cite{\GaRaAA}, (3.2.5); Appendix (III.13)) in form of a rule.
$${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/  d,e\endmatrix ;q,{{d e {q^n}}\over
   {b c}}\right ] \longrightarrow 
{      {(\let\over/ {e\over c};q)}_{n}\over {{(\let\over/ e;q)}_{n}}}
{}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},c,{d\over
       b}\\ \let\over/  d,{{c {q^{1 - n}}}\over e}\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3209
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.26; Appendix (III.14)) in form of a rule.
$$
{}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},b,c\\ \let\over/  {{{q^{1 - n}}}\over b},{{{q^{1 -
   n}}}\over c}\endmatrix ;q,{{{q^{1 - n}} z}\over {b c}}\right ] \longrightarrow 
  \frac {{( \let\over/ {z\over {{q^n}}};q)_\infty}}
{{( \let\over/  z;q)_\infty}} 
   {}_5\phi _4\!\left [ \matrix \let\over/ {q^{{{-n}\over 2}}},-{q^{{{-n}\over
    2}}},{q^{{1\over 2} - {n\over 2}}},-{q^{{1\over 2} - {n\over 2}}},{{{q^{1
    - n}}}\over {b c}}\\ \let\over/  {{{q^{1 - n}}}\over b},{{{q^{1 - n}}}\over c},{z\over
    {{q^n}}},{q\over z}\endmatrix ;q,q\right ]
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3210
\Description Transformation formula (\cite{\GaRaAA}, (3.2.6)) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, {q^{-n}}\\ \let
   \over / e, d\endmatrix ;q, {\displaystyle {{d e q^n}\over {a b}}} \right ] \longrightarrow  
  {\left( \frac {d e q^{n-1}} {a} \right) }^n
{{({\let \over / {{a q^{1-n}}\over d}, {{a q^{1-n}}\over e}}; q) _{n}}\over
       {({\let \over / d, e}; q) _{n}}}
  {{  {} _{3} \phi _{2} \! \left [ \matrix \let \over / a,
       {{a b q^{1-n}}\over {d e}}, {q^{-n}}\\ \let \over / {{a q^{1-n}}\over e},
       {{a q^{1-n}}\over d}\endmatrix ;q, {\displaystyle {q\over b}} \right ] 
       }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3211
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.1) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z\\ \let \over
   / a b {\sqrt{q}}, - a b {\sqrt{q}}  \endmatrix ;q,
   {\displaystyle q} \right ] \longrightarrow  
  {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z^2\\ \let
   \over / a^2 b^2 q, 0\endmatrix ;q^2, {\displaystyle q^2} \right ]
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3212
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.1, reversed) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z^2\\ \let
   \over / a^2 b^2 q, 0\endmatrix ;q^2, {\displaystyle q^2} \right ]
  \longrightarrow {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z\\ \let
   \over / a b {\sqrt{q}}, - a b {\sqrt{q}}  \endmatrix ;q,
   {\displaystyle q} \right ]
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3213
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(i)) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, -b\\ \let \over /
   b^2, a z\endmatrix ;q, {\displaystyle -z} \right ] \longrightarrow  
  { {(\let \over / z ;q) _\infty} \over {(\let \over / a z ;q) _\infty} } 
   {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, a q\\ \let \over /
    b^2 q\endmatrix ;q^2, {\displaystyle z^2} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3214
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii)) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, -b\\ \let \over /
   b^2, - a z  \endmatrix ;q, {\displaystyle z} \right ] \longrightarrow  
 {{({\let \over / a z^2}; q^2) _{\infty} }\over 
    {({\let \over / z}; q) _{\infty}  
      ({\let \over / - a z  }; q) _{\infty} }} 
 {{  {} _{2} \phi _{2} \! \left [ \matrix \let \over / a, {{b^2}\over a}\\
       \let \over / b^2 q, a z^2\endmatrix ;q^2, {\displaystyle
       a q z^2} \right ]}}
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3215
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.3) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {q\over a}, z\\ \let
   \over / c, -q\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
  { {(\let \over / -1, -{{q z}\over c} ;q) _\infty} \over {(\let \over /
    -{q\over c}, -z ;q) _\infty} } 
   {} _{3} \phi _{2} \! \left [ \matrix \let \over / {c\over a}, {{a c}\over
    q}, z^2\\ \let \over / c^2, 0\endmatrix ;q^2, {\displaystyle q^2}
    \right ]
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3216
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.3, reversed) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / c, {{a^2}\over {c q}},
   z^2\\ \let \over / a^2, 0\endmatrix ;q^2, {\displaystyle q^2}
   \right ] \longrightarrow  { {(\let \over / -{q\over a}, -z ;q) _\infty} \over {(\let \over
    / -1, -{{q z}\over a} ;q) _\infty} } 
   {} _{3} \phi _{2} \! \left [ \matrix \let \over / {a\over c}, {{c q}\over
    a}, z\\ \let \over / a, -q\endmatrix ;q, {\displaystyle q} \right ]
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3217
\Description Transformation formula (\cite{\GaRaAA}, (3.2.11)) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d,
   e\endmatrix ;q, {\displaystyle {{d e}\over {a b c}}} \right ] \longrightarrow 
{ {(\let \over / {{d e}\over {a b}}, {{d e}\over {a c}} ;q) _\infty}
    \over {(\let \over / {{d e}\over a}, {{d e}\over {a b c}} ;q) _\infty}
    }
  {} _{7} \phi _{7} \! \left [ \matrix \let \over / {{d e}\over {a q}},
    {{{\sqrt{d e q}}}\over {{\sqrt{a}}}}, -{{{\sqrt{d e q}}}\over
    {{\sqrt{a}}}}, {e\over a}, {d\over a}, b, c\\ \let \over /
    {{{\sqrt{d e}}}\over {{\sqrt{a q}}}}, -{{{\sqrt{d e}}}\over
    {{\sqrt{a q}}}}, d, e, {{d e}\over {a b}}, {{d e}\over {a c}},
    0\endmatrix ;q, {\displaystyle {{d e}\over {b c}}} \right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3261
\Description Transformation formula (\cite{\GaRaAA}, (3.3.3); Appendix (III.33)) in form of a rule.
$$\multline {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/  d,e\endmatrix ;q,{{d e}\over {a b
   c}}\right ] \longrightarrow \frac {{( \let\over/ {e\over b},{e\over c},{{c q}\over
     a},{q\over d};q)_\infty}}
{{( \let\over/  e,{{c q}\over d},{q\over a},{e\over {b c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ c,{d\over a},{{c q}\over e}\\ \let\over/  {{c
     q}\over a},{{b c q}\over e}\endmatrix ;q,{{b q}\over d}\right ]\\- 
   \frac {{( \let\over/ {q\over d},{{e q}\over d},b,c,{d\over a},{{d
     e}\over {b c q}},{{b c {q^2}}\over {d e}};q)_\infty}}
{{( \let\over/  {d\over q},e,{{b q}\over
     d},{{c q}\over d},{q\over a},{e\over {b c}},{{b c q}\over e};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {{a q}\over d},{{b q}\over d},{{c
     q}\over d}\\ \let\over/  {{{q^2}}\over d},{{e q}\over d}\endmatrix ;q,{{d e}\over {a
     b c}}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3262
\Description Transformation formula (\cite{\GaRaAA}, (3.3.1); Appendix (III.34)) in form of a rule.
$$
 {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/  d,e\endmatrix ;q,{{d e}\over {a b
   c}}\right ] \longrightarrow \frac {{( \let\over/ {e\over b},{e\over c};q)_\infty}}
{{( \let\over/  e,{e\over {b
     c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {d\over a},b,c\\ \let\over/ 
     d,{{b c q}\over e}\endmatrix ;q,q\right ] + 
   \frac {{( \let\over/ {d\over a},b,c,{{d e}\over {b c}};q)_\infty}}
{{( \let\over/ 
    d,e,{{b c}\over
     e},{{d e}\over {a b c}};q)_\infty}} 
    {}_3\phi _2\!\left [ \matrix \let\over/ {e\over b},{e\over c},{{d e}\over {a b c}}\\ \let\over/ 
     {{d e}\over {b c}},{{e q}\over {b c}}\endmatrix ;q,q\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3263
\Description Transformation formula (\cite{\GaRaAA}, (3.3.3),
reversed; Appendix (III.33)) in form of a rule.
$$\multline
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d,
   e\endmatrix ;q, {\displaystyle {{d e}\over {a b c}}} \right ] 
\longrightarrow 
  { {(\let \over / {{d q}\over {b c}}, {e\over c}, a, b, {{a b q}\over {d e}},
     {{d e}\over {a b}} ;q) _\infty} \over {(\let \over / {{a q}\over e},
     {q\over c}, d, {{a b}\over d}, {{d e}\over {a b c}}, e ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / {q\over b}, {{d e}\over
     {a b c}}, {d\over b}\\ \let \over / {{d q}\over {a b}}, {{d q}\over {b
     c}}\endmatrix ;q, {\displaystyle {{b q}\over e}} \right ] \\ + 
   { {(\let \over / {{a q}\over c}, {d\over b}, {d\over a}, {q\over e} ;q)
     _\infty} \over {(\let \over / {{a q}\over e}, {q\over c}, d, {d\over {a
     b}} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{a
     q}\over d}, {e\over c}, a\\ \let \over / {{a b q}\over d}, {{a q}\over
     c}\endmatrix ;q, {\displaystyle {{b q}\over e}} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3264
\Description Transformation formula (\cite{\GaRaAA}, (3.3.1),
reversed; Appendix (III.34)) in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d,
   e\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow 
  { {(\let \over / {{b c q}\over e}, {q\over e} ;q) _\infty} \over {(\let
     \over / {{c q}\over e}, {{b q}\over e} ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / {d\over a}, b, c\\ \let
     \over / d, {{b c q}\over e}\endmatrix ;q, {\displaystyle {{a q}\over e}}
     \right ] - { {(\let \over / {q\over e}, a, b, c, {{d q}\over e} ;q)
     _\infty} \over {(\let \over / {{c q}\over e}, {{b q}\over e}, d, {e\over
     q}, {{a q}\over e} ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{c q}\over e}, {{b
     q}\over e}, {{a q}\over e}\\ \let \over / {{d q}\over e}, {{{q^2}}\over
     e}\endmatrix ;q, {\displaystyle q} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3265
\Description Transformation formula (\cite{\GaRaAA}, (3.4.1); Appendix (III.35)) in form of a rule.
$$\multline {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/  {{a q}\over b},{{a q}\over c}\endmatrix
   ;q,{{a q x}\over {b c}}\right ] \longrightarrow 
  \frac {{( \let\over/ a x;q)_\infty}}
{{( \let\over/  x;q)_\infty}} 
    {}_5\phi _4\!\left [ \matrix \let\over/ {\sqrt{a}},-{\sqrt{a}},{\sqrt{a}}
     {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}}  ,{{a q}\over {b c}}\\ \let\over/ 
     {{a q}\over b},{{a q}\over c},a x,{q\over x}\endmatrix ;q,q\right ]\\+ 
   \frac {{( \let\over/ a,{{a q}\over {b c}},{{a q x}\over b},{{a q
     x}\over c};q)_\infty}}
{{( \let\over/  {{a q}\over b},{{a q}\over c},{{a q x}\over {b c}},{1\over
     x};q)_\infty}} {}_5\phi _4\!\left [ \matrix \let\over/ {\sqrt{a}} x,-
     {\sqrt{a}} x  ,{\sqrt{a}} {\sqrt{q}} x,- {\sqrt{a}}
     {\sqrt{q}} x  ,{{a q x}\over {b c}}\\ \let\over/  {{a q x}\over b},{{a q
     x}\over c},q x,a {x^2}\endmatrix ;q,q\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3266
\Description Transformation formula (\cite{\GaRaAA}, (3.5.2)) 
in form of a rule.
$$\multline
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, x^2, y^2\\ \let
   \over / a^2 b^2, b^2 x^2 y^2\endmatrix ;q^2, {\displaystyle
   b^2 q} \right ] \longrightarrow  { {(\let \over / b^2 ;q^2) _\infty} \over {(\let
     \over / a^2 b^2 ;q^2) _\infty} } 
    { {(\let \over / a, - a b^2   ;q) _\infty} \over {(\let
     \over / -1, b^2 ;q) _\infty} } 
    {} _{5} \phi _{4} \! \left [ \matrix \let \over / -a, - b x 
     , b x, - b y  , b y\\ \let \over / -q, - a b^2
      , - b x y  , b x y\endmatrix ;q, {\displaystyle
     q} \right ]\\
 + { {(\let \over / b^2 ;q^2) _\infty} \over {(\let \over
     / a^2 b^2 ;q^2) _\infty} } 
    { {(\let \over / -a, a b^2 ;q) _\infty} \over {(\let \over / -1, b^2
     ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / a,
     b x, - b x  , b y, - b y  \\ \let \over /
     -q, a b^2, b x y, - b x y  \endmatrix ;q,
     {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3267
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.6) 
in form of a rule.
$$\multline
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d,
   e\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
  { {(\let \over / {q\over e}, {{a b q}\over e}, {{a c q}\over e}, {d\over
     a} ;q) _\infty} \over {(\let \over / d, {{a q}\over e}, {{b q}\over e},
     {{c q}\over e} ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {{a q}\over e},
     {{a b c q}\over {d e}}\\ \let \over / {{a b q}\over e},
     {{a c q}\over e}\endmatrix ;q, {\displaystyle {d\over a}} \right ]\\ - 
   { {(\let \over / {q\over e}, a, b, c, {{d q}\over e} ;q) _\infty} \over
     {(\let \over / {e\over q}, {{a q}\over e}, {{b q}\over e}, {{c q}\over
     e}, d ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over /
     {{a q}\over e}, {{b q}\over e}, {{c q}\over e}\\ \let \over /
     {{q^2}\over e}, {{d q}\over e}\endmatrix ;q, {\displaystyle q} \right
     ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3268
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.6, reversed) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d,
   e\endmatrix ;q, {\displaystyle {{d e}\over {a b c}}} \right ] \longrightarrow  
  { {(\let \over / {{d e}\over {b c}}, b, {d\over a}, {e\over a} ;q)
     _\infty} \over {(\let \over / {b\over a}, d, e, {{d e}\over {a b c}}
     ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a,
     {d\over b}, {e\over b}\\ \let \over / {{d e}\over {b c}}, {{a q}\over
     b}\endmatrix ;q, {\displaystyle q} \right ] + 
   { {(\let \over / a, {d\over b}, {e\over b}, {{d e}\over {a c}} ;q)
     _\infty} \over {(\let \over / d, e, {{d e}\over {a b c}}, {a\over b}
     ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / b,
     {d\over a}, {e\over a}\\ \let \over / {{b q}\over a}, {{d e}\over
     {a c}}\endmatrix ;q, {\displaystyle q} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3269
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.8, reversed) 
in form of a rule.
$$
{} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, 0\\ \let \over / c,
   x\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
  { {(\let \over / {q\over c}, {{a b q}\over c}, {x\over a} ;q) _\infty}
     \over {(\let \over / {{a q}\over c}, {{b q}\over c}, x ;q) _\infty} } 
    {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, {{a q}\over c}\\
     \let \over / {{a b q}\over c}\endmatrix ;q, {\displaystyle {x\over a}}
     \right ] - { {(\let \over / a, b, {{q x}\over c}, {q\over c} ;q)
     _\infty} \over {(\let \over / {c\over q}, {{a q}\over c}, {{b q}\over
     c}, x ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over /
     {{a q}\over c}, {{b q}\over c}, 0\\ \let \over / {{q^2}\over c},
     {{q x}\over c}\endmatrix ;q, {\displaystyle q} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4201
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.4, reversed) 
in form of a rule.
$$
{} _{4} \phi _{2} \! \left [ \matrix \let \over / a, b, -b, {q^{-n}}\\ \let
   \over / {{a {q^{1 - n}}}\over d}, b^2\endmatrix ;q, {\displaystyle
   -{q\over d}} \right ] \longrightarrow  a^{-n}
  {{({\let \over / d}; q) _{n}}\over {({\let \over / {d\over a}}; q)
       _{n}}}
   {{{} _{4} \phi _{3} \! \left [ \matrix \let \over /
       a, a q, {q^{1 - n}}, {q^{-n}}\\ \let \over / d, d q,
       b^2 q\endmatrix ;q^2, {\displaystyle q^2} \right ] 
       }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4301
\Description Transformation formula (\cite{\GaRaAA}, (2.10.4); Appendix (III.15)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/  e,f,{{a b c {q^{1 - n}}}\over {e
   f}}\endmatrix ;q,q\right ] \longrightarrow 
  {a^n}
{   {{(\let\over/ {e\over a},{f\over a};q)}_{n}}\over{{(\let\over/ e,f;q)}_{n}}}
 {}_4\phi _3\!\left [ \matrix \let\over/ {q^{-n}},a,{{a c {q^{1 - n}}}\over {e
    f}},{{a b {q^{1 - n}}}\over {e f}}\\ \let\over/  {{a b c {q^{1 - n}}}\over {e f}},{{a
    {q^{1 - n}}}\over e},{{a {q^{1 - n}}}\over f}\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4302
\Description Transformation formula (\cite{\GaRaAA}, (3.2.9); Appendix (III.16)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/  e,f,{{a b c {q^{1 - n}}}\over {e
   f}}\endmatrix ;q,q\right ] \longrightarrow 
{{{(\let\over/ a,{{e f}\over {a b}},{{e f}\over {a
    c}};q)}_{n}}\over{{(\let\over/ e,f,{{e f}\over {a b c}};q)}_{n}}
}  {}_4\phi _3\!\left [ \matrix \let\over/ {q^{-n}},{e\over a},{f\over a},{{e f}\over {a b
    c}}\\ \let\over/  {{e f}\over {a b}},{{e f}\over {a c}},{{{q^{1 - n}}}\over
    a}\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4303
\Description Transformation formula (\cite{\GaRaAA}, (2.5.1), reversed; Appendix (III.19)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/  e,f,{{a b c {q^{1 - n}}}\over {e
   f}}\endmatrix ;q,q\right ] \longrightarrow 
{{{(\let\over/ {{e f}\over {a b}},{{e f}\over {a
    c}};q)}_{n}}\over{{(\let\over/ {{e f}\over a},{{e f}\over {a b c}};q)}_{n}}
}  {}_8W _7({{e f}\over {a q}};{f\over a},{e\over a},b,c,{q^{-n}};q,{{e f
    {q^n}}\over {b c}}) 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4304
\Description Transformation formula (\cite{\GaRaAA}, (2.10.7); Appendix (III.20)) in form of a rule.
$$
{}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/ d, e,{{a b c {q^{1 - n}}}\over {d
   e}}\endmatrix ;q,q\right ] \longrightarrow 
  \frac {{( \let\over/ {{d e {q^n}}\over c},{{d e {q^n}}\over b},{{d e
    {q^n}}\over a},{{d e {q^n}}\over {a b c}};q)_\infty}}
{{( \let\over/  {{d e {q^n}}\over {b c}},{{d e
    {q^n}}\over {a c}},{{d e {q^n}}\over {a b}},d e {q^n};q)_\infty}}
     {}_8W _7(d e {q^{-1 + n}};a,b,c,d {q^n},e {q^n};q,{{d e}\over {a b c}})
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4305
\Description Transformation formula (\cite{\GaRaAA}, (3.10.13); Appendix (III.21)) in form of a rule.
$${}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},c,d\\ \let\over/  a b {\sqrt{q}},- a b
   {\sqrt{q}}  ,- c d  \endmatrix ;q,q\right ] \longrightarrow 
  {}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},{c^2},{d^2}\\ \let\over/  {a^2} {b^2} q,-
   c d  ,- c d q  \endmatrix ;{q^2},{q^2}\right ]
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4306
\Description Transformation formula (\cite{\GaRaAA}, (3.10.13), reversed; Appendix (III.21),
reversed) in form of a rule.
$$  {}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},{c^2},{d^2}\\ \let\over/  {a^2} {b^2} q,-
   c d  ,- c d q  \endmatrix ;{q^2},{q^2}\right ]
\longrightarrow 
  {}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},c,d\\ \let\over/  a b {\sqrt{q}},- a b
   {\sqrt{q}}  ,- c d  \endmatrix ;q,q\right ] 
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4307
\Description Transformation formula (\cite{\GaRaAA}, (8.8.3); Appendix (III.22)) in form of a rule.
$$
{{{}_4\phi _3\!\left [ \matrix \let\over/ a,b,a b z,{{a b}\over z}\\ \let\over/  a b
      {\sqrt{q}},- a b {\sqrt{q}}  ,- a b 
      \endmatrix ;q,q\right ]}^2} \longrightarrow 
  {}_5\phi _4\!\left [ \matrix \let\over/ {a^2},{b^2},a b,a b z,{{a b}\over z}\\ \let\over/  a b
   {\sqrt{q}},- a b {\sqrt{q}}  ,- a b  ,{a^2}
   {b^2}\endmatrix ;q,q\right ]
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4308
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.2) 
in form of a rule.
$$
{} _{4} W _{3} ({\displaystyle a; b}; q, {\displaystyle t}) \longrightarrow  
  { {(\let \over / a q, b t ;q) _\infty} \over {(\let \over / t,
    {{a q}\over b} ;q) _\infty} } 
   {} _{2} \phi _{1} \! \left [ \matrix \let \over / {1\over b}, t\\ \let
    \over / b q t\endmatrix ;q, {\displaystyle a q} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4309
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(i)) 
in form of a rule.
$$\multline
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, c, d\\ \let \over /
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}\endmatrix ;q,
   {\displaystyle {{a^2 {q^3}}\over {b^2 c^2 d^2}}} \right ] \\\longrightarrow  
  { {(\let \over / {{a q^2}\over {b c d}}, {{{a^3} {q^3}}\over
    {b^2 c^2 d^2}} ;q) _\infty} \over {(\let \over /
    {{a^2 q^2}\over {b c d}}, {{a^2 {q^3}}\over
    {b^2 c^2 d^2}} ;q) _\infty} } 
   {} _{10} W _{9} ({\displaystyle {{{a^2} q}\over {b c d}}; {\sqrt{a}},
    -{\sqrt{a}}, {\sqrt{a q}}, - {\sqrt{a q}}
     , {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over
    {b c}}}; q, {\displaystyle {{a {q^2}}\over {b c d}}})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4310
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii)) 
in form of a rule.
$$
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, c, d\\ \let \over /
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}\endmatrix ;q,
   {\displaystyle -{{a q^2}\over {b c d}}} \right ] \longrightarrow  
  { {(\let \over / a q, -q, {{{a^{{3\over 2}}} q^2}\over {b c d}},
    -{{{a^{{3\over 2}}} q^2}\over {b c d}} ;q) _\infty} \over {(\let
    \over / {{a^2 q^2}\over {b c d}}, -{{a q^2}\over {b c d}},
    {\sqrt{a}} q, - {\sqrt{a}} q   ;q) _\infty} } 
   {} _{8} W _{7} ({\displaystyle {{{a^2} q}\over {b c d}}; {\sqrt{a}},
    -{\sqrt{a}}, {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over
    {b c}}}; q, {\displaystyle -q})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4311
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8)) 
in form of a rule.
$$
{} _{4} W _{3} ({\displaystyle a; b}; q, {\displaystyle {x\over
   {b^2 q}}}) \longrightarrow  { {(\let \over / {{a x^2}\over {b^2}}, {x\over {b q}}
    ;q) _\infty} \over {(\let \over / {{a q x}\over b}, {{x^2}\over
    {b^2 q}} ;q) _\infty} } 
   {} _{8} W _{7} ({\displaystyle {{a x}\over b}; {\sqrt{a q}},
    - {\sqrt{a q}}  , {\sqrt{a}} q, -
    {\sqrt{a}} q  , x}; q, {\displaystyle {x\over {{b^2} q}}})$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4312
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.4) 
in form of a rule.
$$
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, a q, {q^{1 - n}},
   {q^{-n}}\\ \let \over / d, d q, b^2 q\endmatrix ;q^2, {\displaystyle
   q^2} \right ] \longrightarrow  {a^n}
{{({\let \over / {d\over a}}; q) _{n}}\over {({\let \over / d}; q) _{n}}}
  {} _{4} \phi _{2} \! \left [ \matrix \let \over /
    a, b, -b, {q^{-n}}\\ \let \over / {{a {q^{1 - n}}}\over d},
    b^2\endmatrix ;q, {\displaystyle -{q\over d}} \right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4313
\Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15) 
in form of a rule.
$$\multline 
 {} _{4} \phi _{3} \! \left [       
       \matrix  a, b, c, d\\
{b q}/{a}, {c q}/{a}, {d q}/{a}\endmatrix ;q,
    {\displaystyle \frac{q^ 2}{a^ 2}} \right ] 
\longrightarrow
     \frac {({a}/{d},
     {b q}/{d}, {c q}/{d}, {a b c}/{d} ;q) _\infty} 
  {( {q}/{d}, {a b}/{d},
     {a c}/{d}, {b c q}/{d} ;q) _\infty}\\
\times
  {} _{12} W _{11} \left({\displaystyle \frac{b c}{d};
     \frac{{\sqrt{b c q}}}{{\sqrt{a d}}}, -
     \frac{{\sqrt{b c q}}}{{\sqrt{a d}}}  ,
     \frac{{\sqrt{b c}} q}{{\sqrt{a d}}}, -
     \frac{{\sqrt{b c}} q}{{\sqrt{a d}}}  , \frac{a b}{d},
     \frac{a c}{d}, a, b, c}; q, {\displaystyle \frac{q}{a}}\right) 
\endmultline$$
provided at least one of $a,b,c$ is of the form $q^-n$, $n=0,1,2,\dots$.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4361
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16) 
in form of a rule.
$$\multline
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, - {\sqrt{a}} q
    , b, c\\ \let \over / -{\sqrt{a}}, {{a q}\over b}, {{a q}\over
   c}\endmatrix ;q, {\displaystyle x} \right ] \longrightarrow  
  \left( 1 - {{b c x}\over {{\sqrt{a}} q}} \right)  
    { {(\let \over / b c x ;q) _\infty} \over {(\let \over / {{b c x}\over
     {a q}} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over /
     {\sqrt{a}}, - {\sqrt{a}} q  , {\sqrt{a q}},
     -{\sqrt{a q}}, {{a q}\over {b c}}\\ \let \over / {{a q}\over b},
     {{a q}\over c}, {{a q^2}\over {b c x}}, b c x\endmatrix ;q,
     {\displaystyle q} \right ]\\ + 
   \left( 1 - {\sqrt{a}} \right)  
    { {(\let \over / a q, {{a q}\over {b c}}, c x, b x ;q) _\infty} \over
     {(\let \over / {{a q}\over b}, {{a q}\over c}, x, {{a q}\over
     {b c x}} ;q) _\infty} } 
    {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{b c x}\over
     {{\sqrt{a}} q}}, -{{b c x}\over {{\sqrt{a}}}}, {{b c x}\over
     {{\sqrt{a q}}}}, -{{b c x}\over {{\sqrt{a q}}}}, x\\ \let \over /
     c x, b x, {{b c x}\over a}, {{b^2 c^2 x^2}\over
     {a q}}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4362
\Description Transformation formula (\cite{\GaRaAA}, (2.10.10),
reversed; Appendix (III.36)) in form of a rule.
$$\multline
{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, c, d\\ \let \over / e,
   f, {{a b c d q}\over {e f}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow 
   { {(\let \over / {{e f}\over {a b}}, {{e f}\over {a c}}, {{e f}\over {a
     d}}, {{e f}\over {a b c d}} ;q) _\infty} \over {(\let \over / {{e f}\over
     a}, {{e f}\over {a b c}}, {{e f}\over {a b d}}, {{e f}\over {a c d}} ;q)
     _\infty} } {} _{8} W _{7} ({\displaystyle {{e f}\over {a q}}; {f\over a},
     {e\over a}, b, c, d}; q, {\displaystyle {{e f}\over {b c d}}})
\\- { {(\let \over / {{e f}\over {a b c d}}, a, b, c, d, {{{e^2} f}\over
       {a b c d}}, {{e {f^2}}\over {a b c d}} ;q) _\infty} \over {(\let \over
       / {{e f}\over {a b c}}, {{e f}\over {a b d}}, {{e f}\over {a c d}}, e,
       f, {{e f}\over {b c d}}, {{a b c d}\over {e f}} ;q) _\infty} } 
      {} _{4} \phi _{3} \! \left [ \matrix \let \over / {{e f}\over {a b c}},
       {{e f}\over {a b d}}, {{e f}\over {a c d}}, {{e f}\over {b c d}}\\ \let
       \over / {{{e^2} f}\over {a b c d}}, {{e {f^2}}\over {a b c d}}, {{e f
       q}\over {a b c d}}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5401
\Description Transformation formula (\cite{\GaRaAA}, (2.8.3); Appendix (III.25)) in form of a rule.
$$\multline {}_5\phi _4\!\left [ \matrix \let\over/ a,b,c,d,{q^{-n}}\\ \let\over/  {{a q}\over b},{{a q}\over
   c},{{a q}\over d},{{{b^2} {c^2} {d^2} {q^{-2 - n}}}\over {{a^2}}}\endmatrix
   ;q,q\right ] \longrightarrow
{   {{(\let\over/ {{a {q^2}}\over {b c d}},{{{a^3} {q^3}}\over {{b^2} {c^2}
    {d^2}}};q)}_{n}}\over{{(\let\over/ {{{a^2} {q^2}}\over {b c d}},{{{a^2} {q^3}}\over
    {{b^2} {c^2} {d^2}}};q)}_{n}}}\\
 {}_{12}W _{11}({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a
    q}\over {b d}},{{a q}\over {b c}},{\sqrt{a}},-{\sqrt{a}},{\sqrt{a}}
    {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}}  ,{{{a^3} {q^{3 +
    n}}}\over {{b^2} {c^2} {d^2}}},{q^{-n}};q,q) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5402
\Description Transformation formula (\cite{\GaRaAA}, (2.8.4); Appendix (III.26)) in form of a rule.
$$\multline 
{}_5\phi _4\!\left [ \matrix \let\over/ {q^{-n}},b,c,d,e\\ \let\over/  {{{q^{1 - n}}}\over b},{{{q^{1
   - n}}}\over c},{{{q^{1 - n}}}\over d},{b^2} {c^2} {d^2} e {q^{-2 + 2
   n}}\endmatrix ;q,q\right ] \longrightarrow 
{   {{(\let\over/ {{{q^{2 - 2 n}}}\over {b c d e}},{{{q^{3 - 3 n}}}\over {{b^2} {c^2}
    {d^2}}};q)}_{n}}\over{{(\let\over/ {{{q^{2 - 2 n}}}\over {b c d}},{{{q^{3 - 3 n}}}\over
    {{b^2} {c^2} {d^2} e}};q)}_{n}}
}\\
  {}_{12}W _{11}({{{q^{1 - 2 n}}}\over {b c d}};{{{q^{1 - n}}}\over {c d}},{{{q^{1
    - n}}}\over {b d}},{{{q^{1 - n}}}\over {b c}},{q^{{{-n}\over
    2}}},-{q^{{{-n}\over 2}}},{q^{{1\over 2} - {n\over 2}}},-{q^{{1\over 2} -
    {n\over 2}}},e,{{{q^{3 - 3 n}}}\over {{b^2} {c^2} {d^2} e}};q,q) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5403
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4), reversed) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / x^2, y^2, a, a q,
   {q^{-2 n}}\\ \let \over / a^2, a b, a b q, {{{q^{2 -
   2 n}} x^2 y^2}\over {a^2 b^2}}\endmatrix ;q^2, {\displaystyle
   q^2} \right ] \\
\longrightarrow {{({\let \over / {{a^2 b^2}\over {x^2}}, {{a^2 b^2}\over
    {y^2}}}; q^2) _{n}}\over {({\let \over / a^2 b^2,
    {{a^2 b^2}\over {x^2 y^2}}}; q^2) _{n}}}
  {} _{10} W _{9} ({\displaystyle -{{a b}\over q}; b, x,
    -x, y, -y, -{q^{-n}}, {q^{-n}}}; q, {\displaystyle
    {{{a^3} {b^2} {q^{2 n}}}\over {{x^2} {y^2}}}})
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5404
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.26, Appendix
(III.14), reversed) 
in form of a rule.
$$
{} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{q^{1 - n}}}\over {b c}},
   -{q^{{1\over 2} - {n\over 2}}}, {q^{{1\over 2} - {n\over 2}}},
   -{q^{{{-n}\over 2}}}, {q^{{{-n}\over 2}}}\\ \let \over / {{{q^{1 -
   n}}}\over b}, {{{q^{1 - n}}}\over c}, {z\over {{q^n}}}, {q\over
   z}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
  { {(\let \over / z ;q) _\infty} \over {(\let \over / {z\over {{q^n}}} ;q)
    _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {q^{-n}}, b,
    c\\ \let \over / {{{q^{1 - n}}}\over b}, {{{q^{1 - n}}}\over c}\endmatrix
    ;q, {\displaystyle {{{q^{1 - n}} z}\over {b c}}} \right ]
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5405
\Description Transformation formula (\cite{\GaRaAA}, (8.8.3), Appendix
(III.22), reversed) 
in form of a rule.
$$
{} _{5} \phi _{4} \! \left [ \matrix \let \over / {a^2}, {b^2}, a b, z,
   {{{a^2} {b^2}}\over z}\\ \let \over / a b {\sqrt{q}}, -
   a b {\sqrt{q}}  , - a b  , {a^2} {b^2}\endmatrix
   ;q, {\displaystyle q} \right ] \longrightarrow  
  {{{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, z,
      {{{a^2} {b^2}}\over z}\\ \let \over / a b {\sqrt{q}}, -
      a b {\sqrt{q}}  , - a b  \endmatrix ;q,
      {\displaystyle q} \right ]}^2}
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5461
\Description Transformation formula (\cite{\GaRaAA}, (3.4.4)) 
in form of a rule.
$$\multline
{} _{5} W _{4} ({\displaystyle a; b, c}; q, {\displaystyle
   {{{\sqrt{a q}} x}\over {b c}}}) \longrightarrow  
  \left( 1 - x^2 \right)  { {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x
     ;q) _\infty} \over {(\let \over / {x\over {{\sqrt{a q}}}} ;q) _\infty} }
      {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a q}},
     -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q  ,
     {{a q}\over {b c}}\\ \let \over / {{a q}\over b}, {{a q}\over c},
     {\sqrt{a}} {q^{{3\over 2}}} x, {{{\sqrt{a}} {q^{{3\over 2}}}}\over
     x}\endmatrix ;q, {\displaystyle q} \right ]\\ + 
   { {(\let \over / a q, {{a q}\over {b c}}, {{{\sqrt{a q}} x}\over b},
     {{{\sqrt{a q}} x}\over c} ;q) _\infty} \over {(\let \over /
     {{a q}\over b}, {{a q}\over c}, {{{\sqrt{a q}} x}\over {b c}},
     {{{\sqrt{a q}}}\over x} ;q) _\infty} } 
    {} _{5} \phi _{4} \! \left [ \matrix \let \over / x, -x, {\sqrt{q}} x,
     - {\sqrt{q}} x  , {{{\sqrt{a q}} x}\over {b c}}\\ \let
     \over / {{{\sqrt{a q}} x}\over b}, {{{\sqrt{a q}} x}\over c},
     {{{\sqrt{q}} x}\over {{\sqrt{a}}}}, q x^2\endmatrix ;q,
     {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5462
\Description Transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, first form) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a q}},
   -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q  , {{b c}\over
   {a q}}\\ \let \over / b, c, {\sqrt{a}} {q^{{3\over 2}}} x,
   {{{\sqrt{a}} {q^{{3\over 2}}}}\over x}\endmatrix ;q, {\displaystyle q}
   \right ] \longrightarrow \frac {1} {{1 - x^2}}
 {{{ {(\let \over / {x\over {{\sqrt{a q}}}} ;q) _\infty} \over
        {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x ;q) _\infty} } 
       {} _{5} W _{4} ({\displaystyle a; {{a q}\over b}, {{a q}\over c}}; q,
        {\displaystyle {{b c x}\over {{{\left( a q \right) }^{{3\over
        2}}}}}})}} \\- 
\frac {1} {{1 - x^2}}
   {{{ {(\let \over / {x\over {{\sqrt{a q}}}}, a q, {{b c}\over {a q}},
        {{b x}\over {{\sqrt{a q}}}}, {{c x}\over {{\sqrt{a q}}}} ;q)
        _\infty} \over {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x, b, c,
        {{b c x}\over {{{\left( a q \right) }^{{3\over 2}}}}},
        {{{\sqrt{a q}}}\over x} ;q) _\infty} } 
       {} _{5} \phi _{4} \! \left [ \matrix \let \over / x, -x, {\sqrt{q}} x,
        - {\sqrt{q}} x  , {{b c x}\over {{{\left( a q \right)
        }^{{3\over 2}}}}}\\ \let \over / {{b x}\over {{\sqrt{a q}}}},
        {{c x}\over {{\sqrt{a q}}}}, {{{\sqrt{q}} x}\over {{\sqrt{a}}}},
        q x^2\endmatrix ;q, {\displaystyle q} \right ]}}
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5463
\Description Transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, second form) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / x, -x, {\sqrt{q}} x,
   - {\sqrt{q}} x  , {{b c}\over {{\sqrt{a q}} x}}\\ \let
   \over / b, c, {{{\sqrt{q}} x}\over {{\sqrt{a}}}}, q x^2\endmatrix ;q,
   {\displaystyle q} \right ] \\\longrightarrow  
  { {(\let \over / {{b {\sqrt{a q}}}\over x}, {{c {\sqrt{a q}}}\over x},
     {{b c}\over {{\sqrt{a q}} x}}, {{{\sqrt{a q}}}\over x} ;q) _\infty}
     \over {(\let \over / a q, {{b c}\over {x^2}}, b, c ;q) _\infty} } 
    {} _{5} W _{4} ({\displaystyle a; {{{\sqrt{a q}} x}\over b},
     {{{\sqrt{a q}} x}\over c}}; q, {\displaystyle {{b c}\over
     {{\sqrt{a q}} x}}}) \\
- \left( 1 - x^2 \right)  
    { {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x, {{b {\sqrt{a q}}}\over
     x}, {{c {\sqrt{a q}}}\over x}, {{b c}\over {{\sqrt{a q}} x}},
     {{{\sqrt{a q}}}\over x} ;q) _\infty} \over {(\let \over / {x\over
     {{\sqrt{a q}}}}, a q, {{b c}\over {x^2}}, b, c ;q) _\infty} } 
    {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a q}},
     -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q  ,
     {{b c}\over {x^2}}\\ \let \over / {{b {\sqrt{a q}}}\over x},
     {{c {\sqrt{a q}}}\over x}, {\sqrt{a}} {q^{{3\over 2}}} x,
     {{{\sqrt{a}} {q^{{3\over 2}}}}\over x}\endmatrix ;q, {\displaystyle q}
     \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5464
\Description Transformation formula (\cite{\GaRaAA}, (3.5.2), reversed) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / a, x, -x, y, -y\\ \let \over
   / -q, a b^2, {{x y}\over b}, -{{x y}\over b}\endmatrix ;q,
   {\displaystyle q} \right ] \\\longrightarrow  
  { {(\let \over / a^2 b^2 ;q^2) _\infty} \over {(\let \over / b^2
     ;q^2) _\infty} } { {(\let \over / -1, b^2 ;q) _\infty} \over {(\let
     \over / -a, a b^2 ;q) _\infty} } 
    {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, {{x^2}\over
     {b^2}}, {{y^2}\over {b^2}}\\ \let \over / a^2 b^2,
     {{x^2 y^2}\over {b^2}}\endmatrix ;q^2, {\displaystyle b^2 q}
     \right ] - { {(\let \over / a, - a b^2   ;q) _\infty}
     \over {(\let \over / -a, a b^2 ;q) _\infty} } 
    {} _{5} \phi _{4} \! \left [ \matrix \let \over / -a, -x, x, -y, y\\ \let
     \over / -q, - a b^2  , -{{x y}\over b}, {{x y}\over
     b}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5465
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7), reversed) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}}, -{\sqrt{a}}, c,
   d, e\\ \let \over / {\sqrt{a b}}, -{\sqrt{a b}}, a, {{c d e q}\over
   {a b}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow  
  - { {(\let \over / a b q, {{{a^3} {b^3}}\over
       {c^2 d^2 e^2}} ;q^2) _\infty} \over {(\let \over / a q,
       {{{a^3} b^2}\over {c^2 d^2 e^2}} ;q^2) _\infty} } 
      { {(\let \over / c, d, e, {{a^2 b}\over {c d e}}, {{a b}\over
       {c d e}} ;q) _\infty} \over {(\let \over / {{c d e}\over {a b}},
       a b, {{a b}\over {c d}}, {{a b}\over {c e}}, {{a b}\over {d e}}
       ;q) _\infty} } \\
  {} _{5} \phi _{4} \! \left [ \matrix \let \over /
       {{{a^{{3\over 2}}} b}\over {c d e}}, -{{{a^{{3\over 2}}} b}\over
       {c d e}}, {{a b}\over {c d}}, {{a b}\over {c e}}, {{a b}\over
       {d e}}\\ \let \over / {{{{\left( a b \right) }^{{3\over 2}}}}\over
       {c d e}}, -{{{{\left( a b \right) }^{{3\over 2}}}}\over {c d e}},
       {{a^2 b}\over {c d e}}, {{a b q}\over {c d e}}\endmatrix ;q,
       {\displaystyle q} \right ] \\  + 
   { {(\let \over / {{a b}\over c}, {{a b}\over d}, {{a b}\over e},
     {{a b}\over {c d e}} ;q) _\infty} \over {(\let \over / a b,
     {{a b}\over {c d}}, {{a b}\over {c e}}, {{a b}\over {d e}} ;q)
     _\infty} }
{} _{10} W _{9} ({\displaystyle {{a b}\over q}; b, c, c q,
     d, d q, e, e q}; {q^2}, {\displaystyle {{{a^3} {b^2}}\over
     {{c^2} {d^2} {e^2}}}})\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5466
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, first form) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / a, - a q  ,
   a {\sqrt{q}}, - a {\sqrt{q}}  , {{b c}\over {a^2 q}}\\
   \let \over / b, c, x, {{a^2 q^2}\over x}\endmatrix ;q, {\displaystyle
   q} \right ] 
\longrightarrow  \frac {1} {({1 - {{a q}\over x}})}{{{ {(\let \over / {q\over x} ;q) _\infty} \over {(\let \over
        / {{a^2 q^2}\over x} ;q) _\infty} } 
       {} _{4} \phi _{3} \! \left [ \matrix \let \over / a^2, - a q
         , {{a^2 q}\over b}, {{a^2 q}\over c}\\ \let \over / -a,
        b, c\endmatrix ;q, {\displaystyle {{b c}\over {a^2 x}}} \right ]}
      } \\- \frac { (1 - a)} {(1 - {{a q}\over x})}
   {{  { {(\let \over / {q\over x}, a^2 q,
        {{b c}\over {a^2 q}}, {{b q}\over x}, {{c q}\over x} ;q)
        _\infty} \over {(\let \over / {{a^2 q^2}\over x}, b, c,
        {{b c}\over {a^2 x}}, {x\over q} ;q) _\infty} } 
       {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{a q}\over x},
        -{{a q^2}\over x}, {{a {q^{{3\over 2}}}}\over x},
        -{{a {q^{{3\over 2}}}}\over x}, {{b c}\over {a^2 x}}\\ \let \over
        / {{b q}\over x}, {{c q}\over x}, {{q^2}\over x},
        {{a^2 {q^3}}\over {x^2}}\endmatrix ;q, {\displaystyle q} \right ]
       }}
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5467
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, second form) 
in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / a, - a q  ,
   a {\sqrt{q}}, - a {\sqrt{q}}  , {{b c x}\over
   {a^2 q^2}}\\ \let \over / b, c, x, a^2 q\endmatrix ;q,
   {\displaystyle q} \right ] \\\longrightarrow  \frac {1} {({1 - {{a q}\over x}})}
  {{{ {(\let \over / {{b q}\over x}, {{c q}\over x}, {{b c x}\over
        {a^2 q^2}}, {q\over x} ;q) _\infty} \over {(\let \over /
        {{a^2 {q^3}}\over {x^2}}, {{b c}\over {a^2 q}}, b, c ;q)
        _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over /
        {{a^2 q^2}\over {x^2}}, -{{a q^2}\over x},
        {{a^2 q^2}\over {b x}}, {{a^2 q^2}\over {c x}}\\ \let
        \over / -{{a q}\over x}, {{b q}\over x}, {{c q}\over x}\endmatrix
        ;q, {\displaystyle {{b c x}\over {a^2 q^2}}} \right ]} }\\
 - \frac {( 1 - a )} {(1 - {{a q}\over x})}
       { {(\let \over / {{a^2 q^2}\over x}, {{b q}\over x}, {{c q}\over
        x}, {{b c x}\over {a^2 q^2}}, {q\over x} ;q) _\infty} \over
        {(\let \over / {x\over q}, {{a^2 {q^3}}\over {x^2}}, {{b c}\over
        {a^2 q}}, b, c ;q) _\infty} } 
       {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{a q}\over x},
        -{{a q^2}\over x}, {{a {q^{{3\over 2}}}}\over x},
        -{{a {q^{{3\over 2}}}}\over x}, {{b c}\over {a^2 q}}\\ \let \over
        / {{b q}\over x}, {{c q}\over x}, {{q^2}\over x},
        {{a^2 q^2}\over x}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5468
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.25) 
in form of a rule.
$$\align
{} _{5} \phi &_{4} \! \left [ \matrix \let \over / a, b, c, d, e\\ \let \over /
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{b^2 c^2 d^2 e}\over
   {a^2 q^2}}\endmatrix ;q, {\displaystyle q} \right ] \\&\longrightarrow  
  - { {(\let \over / a, b, c, d, e, {{a^2 q^3}\over {b^2 c^2
       d^2 e}}, {{a^3 {q^4}}\over {b^3 c^2 d^2 e}}, {{a^3
       {q^4}}\over {b^2 c^3 d^2 e}}, {{a^3 {q^4}}\over {b^2 c^2
       d^3 e}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over
       c}, {{a q}\over d}, {{b^2 c^2 d^2 e}\over {a^2 q^3}}, {{a^3
       q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b c^2 d^2
       e}}, {{a^2 q^3}\over {b^2 c d^2 e}}, {{a^2 q^3}\over {b^2
       c^2 d e}}, {{a^2 q^3}\over {b^2 c^2 d^2}} ;q) _\infty}
}\\&\hskip3cm
      {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{a^2 q^3}\over
       {b^2 c^2 d^2}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}},
       {{a^2 q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c
       d^2 e}}, {{a^2 q^3}\over {b^2 c^2 d e}}\\ \let \over /
       {{a^2 {q^4}}\over {b^2 c^2 d^2 e}}, {{a^3 {q^4}}\over {b^3
       c^2 d^2 e}}, {{a^3 {q^4}}\over {b^2 c^3 d^2 e}}, {{a^3
       {q^4}}\over {b^2 c^2 d^3 e}}\endmatrix ;q, {\displaystyle q}
       \right ] \\&\hskip.3cm  + { {(\let \over / {{a q^2}\over {b c d}}, {{a^2
     q^2}\over {b c d e}}, {{a^3 q^3}\over {b^2 c^2 d^2}}, {{a^2
     q^3}\over {b^2 c^2 d^2 e}} ;q) _\infty} \over {(\let \over /
     {{a^2 q^2}\over {b c d}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}},
     {{a^2 q^3}\over {b^2 c^2 d^2}}, {{a q^2}\over {b c d e}} ;q)
     _\infty} }\\&\hskip3cm
 {} _{12} W _{11} ({\displaystyle {{a^2 q}\over {b c d}};
     {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a}} {\sqrt{q}}, - {\sqrt{a}}
     {\sqrt{q}}  , {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over
     {b c}}, e, {{a^3 q^3}\over {b^2 c^2 d^2 e}}}; q, {\displaystyle
     q})\\&\hskip.3cm
 + { {(\let \over / a, e, {{a q}\over {c d}}, {{a q}\over {b d}}, {{a
     q}\over {b c}}, {{a^2 q^3}\over {b^2 c^2 d^2 e}}, {{a^2
     q^3}\over {b^2 c d e}}, {{a^2 q^3}\over {b c^2 d e}}, {{a^2
     q^3}\over {b c d^2 e}}
     ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c},
     {{a q}\over d}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2
     q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d^2 e}},
     {{a^2 q^3}\over {b^2 c^2 d e}}, {{a^2 q^3}\over {b^2 c^2
     d^2}}, {{b c d e}\over {a q^2}} ;q) _\infty} } \\&\hskip.5cm
  {{(\let \over / {{{a^5} {q^7}}\over {{b^4} {c^4} {d^4}
     e^2}}; q)_{\infty}}\over 
  {(\let \over / {{{a^4} {q^6}}\over {b^3 c^3 d^3 e^2 }};q)_{\infty} }} 
   {} _{12} W _{11} ({\displaystyle {{{a^4} {q^5}}\over {b^3 c^3 d^3
     e^2}}; {{{a^{{3\over 2}}} q^2}\over {b c d e}}, -{{{a^{{3\over 2}}}
     q^2}\over {b c d e}}, {{{a^{{3\over 2}}} {q^{{5\over 2}}}}\over {b c d
     e}}, -{{{a^{{3\over 2}}} {q^{{5\over 2}}}}\over {b c d e}}, {{a
     q^2}\over {b c d}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2
     q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d^2 e}},
     {{a^2 q^3}\over {b^2 c^2 d e}}}; q, {\displaystyle q})
\endalign$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5469
\Description Transformation formula (\cite{\GaRaAA}, (3.4.1),
reversed; Appendix (III.35)) in form of a rule.
$$\multline
{} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}}, -{\sqrt{a}},
   {\sqrt{a}} {\sqrt{q}}, - {\sqrt{a}} {\sqrt{q}}  , {{b c}\over
   {a q}}\\ \let \over / b, c, a x, {q\over x}\endmatrix ;q, {\displaystyle q}
   \right ] \longrightarrow { {(\let \over / x ;q) _\infty} \over {(\let \over / a x ;q)
     _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {{a
     q}\over b}, {{a q}\over c}\\ \let \over / b, c\endmatrix ;q,
     {\displaystyle {{b c x}\over {a q}}} \right ] \\ - 
   { {(\let \over / x, a, {{b c}\over {a q}}, b x, c x ;q) _\infty} \over
     {(\let \over / a x, b, c, {{b c x}\over {a q}}, {1\over x} ;q) _\infty} }
      {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}} x, -
     {\sqrt{a}} x  , {\sqrt{a}} {\sqrt{q}} x, - {\sqrt{a}}
     {\sqrt{q}} x  , {{b c x}\over {a q}}\\ \let \over / b x, c x, q x,
     a {x^2}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T6501
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii)) 
in form of a rule.
$$\multline
{} _{6} \phi _{5} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, b, c, d,
   {q^{-n}}\\ \let \over / {\sqrt{a}}, {{a q}\over b}, {{a q}\over c},
   {{a q}\over d}, {{b^2 c^2 d^2 {q^{-1 - n}}}\over
   {a^2}}\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow  
{{({\let \over / {{a q}\over {b c d}}, {{{a^3} q^2}\over
    {b^2 c^2 d^2}}, -{{{a^{{3\over 2}}} q^2}\over {b c d}}}; q)
    _{n}}\over {({\let \over / {{a^2 q^2}\over {b c d}},
    {{a^2 q^2}\over {b^2 c^2 d^2}}, -{{{a^{{3\over 2}}} q}\over
    {b c d}}}; q) _{n}}}
  {} _{12} W _{11} ({\displaystyle {{{a^2} q}\over {b c d}}; {{a q}\over
    {c d}}, {{a q}\over {b d}}, {{a q}\over {b c}}, {\sqrt{a}} q,
    -{\sqrt{a}}, {\sqrt{a q}}, - {\sqrt{a q}}
     , {{{a^3} {q^{2 + n}}}\over {{b^2} {c^2} {d^2}}}, {q^{-n}}}; q,
    {\displaystyle q})
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7601
\Description Transformation formula (\cite{\GaRaAA}, (2.8.5); Appendix (III.27)) in form of a rule.
$$\multline {}_7\phi _6\!\left [ \matrix \let\over/ a,{\sqrt{a}} q,- {\sqrt{a}} q 
   ,b,c,d,{q^{-n}}\\ \let\over/  {\sqrt{a}},-{\sqrt{a}},{{a q}\over b},{{a q}\over c},{{a
   q}\over d},{{{b^2} {c^2} {d^2}}\over {{a^2} {q^n}}}\endmatrix ;q,q\right ] 
   \longrightarrow 
{{{\left(\let\over/  1 - {{{a^3} {q^{1 + 2 n}}}\over {{b^2} {c^2} {d^2}}}
\right)}\over {\let\over/ (1 - {{{a^3} q}\over {{b^2} {c^2} {d^2}}})}}  
{      {{(\let\over/ {a\over {b c d}},{{{a^3} q}\over {{b^2} {c^2}
       {d^2}}};q)}_{n}}\over{{(\let\over/ {{{a^2} {q^2}}\over {b c d}},{{{a^2} q}\over
       {{b^2} {c^2} {d^2}}};q)}_{n}}}}\\
{}_{12}W _{11}({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b
       d}},{{a q}\over {b c}},{\sqrt{a}} {\sqrt{q}},- {\sqrt{a}}
       {\sqrt{q}}  ,{\sqrt{a}} q,- {\sqrt{a}} q  ,{{{a^3}
       {q^{1 + n}}}\over {{b^2} {c^2} {d^2}}},{q^{-n}};q,q) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7701
\Description Transformation formula (\cite{\GaRaAA}, (3.2.11),
reversed) in form of a rule.
$$
{} _{7} \phi _{7} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, -
   {\sqrt{a}} q  , b, c, d, e\\ \let \over / {\sqrt{a}}, -{\sqrt{a}},
   {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e},
   0\endmatrix ;q, {\displaystyle {{{a^2} {q^2}}\over {b c d e}}} \right ]
    \longrightarrow { {(\let \over / a q, {{a q}\over {d e}} ;q) _\infty} \over {(\let \over
    / {{a q}\over d}, {{a q}\over e} ;q) _\infty} }
{} _{3} \phi _{2} \! \left [ \matrix \let \over / {{a q}\over {b c}},
    d, e\\ \let \over / {{a q}\over b}, {{a q}\over c}\endmatrix ;q,
    {\displaystyle {{a q}\over {d e}}} \right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8701
\Description Transformation formula (\cite{\GaRaAA}, (2.10.10) terminated; Appendix (III.17)) in form of a rule.
$$
{}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow 
  \frac {{( \let\over/ a q,{{a q}\over {d e}},{{a q}\over {d f}},{{a
    q}\over {e f}};q)_\infty}}
{{( \let\over/  {{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over
    {d e f}};q)_\infty}} 
   {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {b c}},d,e,f\\ \let\over/  {{a q}\over b},{{a
    q}\over c},{{d e f}\over a}\endmatrix ;q,q\right ],
$$
provided the $_8\phi_7$ series converges and the $_4\phi_3$ series
terminates.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8702
\Description Transformation formula (\cite{\GaRaAA}, (2.5.1); Appendix (III.18)) in form of a rule.
$$
{}_8W _7(a;b,c,d,e,{q^{-n}};q,{{{a^2} {q^{2 + n}}}\over {b c d e}})\longrightarrow 
{   {{(\let\over/ a q,{{a q}\over {d e}};q)}_{n}}\over{{(\let\over/ {{a q}\over d},{{a q}\over
    e};q)}_{n}}
}  {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {b c}},d,e,{q^{-n}}\\ \let\over/  {{a q}\over
    b},{{a q}\over c},{{d e}\over {a {q^n}}}\endmatrix ;q,q\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8703
\Description Transformation formula (\cite{\GaRaAA}, (2.10.7), reversed; Appendix (III.20), reversed) in form 
  of a rule.
$$
{}_8W _7(a;b,c,d,e,{{a {q^{1 + n}}}\over e};q,{{a {q^{1 - n}}}\over {b c
   d}}) \longrightarrow \frac {{( \let\over/ {{a q}\over {c d}},{{a q}\over {b d}},{{a
    q}\over {b c}},a q;q)_\infty}}
{{( \let\over/  {{a q}\over d},{{a q}\over c},{{a q}\over b},{{a
    q}\over {b c d}};q)_\infty}} 
   {}_4\phi _3\!\left [ \matrix \let\over/ b,c,d,{q^{-n}}\\ \let\over/  {{a q}\over e},{{b c d}\over
    a},{e\over {{q^n}}}\endmatrix ;q,q\right ]
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8704
\Description Transformation formula (\cite{\GaRaAA}, (2.10.1); Appendix (III.23)) in form of a rule.
$$
{}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow 
  \frac {{( \let\over/ a q,{{a q}\over {e f}},{{{a^2} {q^2}}\over {b c d
    e}},{{{a^2} {q^2}}\over {b c d f}};q)_\infty}}
{{( \let\over/  {{a q}\over e},{{a q}\over f},{{{a^2}
    {q^2}}\over {b c d}},{{{a^2} {q^2}}\over {b c d e f}};q)_\infty}}
     {}_8W _7({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b
    d}},{{a q}\over {b c}},e,f;q,{{a q}\over {e f}})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8705
\Description Transformation formula (\cite{\GaRaAA}, (2.10.1), iterated; Appendix (III.24)) in form of a rule.
$$\multline {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow 
  \frac {{( \let\over/ a q,b,{{{a^2} {q^2}}\over {b d e f}},{{{a^2}
    {q^2}}\over {b c e f}},{{{a^2} {q^2}}\over {b c d f}},{{{a^2} {q^2}}\over
    {b c d e}};q)_\infty}}
{{( \let\over/  {{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over
    f},{{{a^3} {q^3}}\over {{b^2} c d e f}},{{{a^2} {q^2}}\over {b c d e
    f}};q)_\infty}} \\{}_8W _7({{{a^3} {q^2}}\over {{b^2} c d e f}};{{a
    q}\over {b c}},{{a q}\over {b d}},{{a q}\over {b e}},{{a q}\over {b
    f}},{{{a^2} {q^2}}\over {b c d e f}};q,b)
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8706
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii), reversed) 
in form of a rule.
$$
{} _{8} W _{7} ({\displaystyle a; b, -b, c, d, {{a^2 q}\over
   {b^2 c d}}}; q, {\displaystyle -q}) \longrightarrow  
  { {(\let \over / a q, -{{a q}\over {b^2}}, b q, - b q  
    ;q) _\infty} \over {(\let \over / b^2 q, -q, {{a q}\over b},
    -{{a q}\over b} ;q) _\infty} } 
   {} _{4} \phi _{3} \! \left [ \matrix \let \over / b^2, {{b^2 c}\over
    a}, {{b^2 d}\over a}, {{a q}\over {c d}}\\ \let \over / {{a q}\over
    c}, {{a q}\over d}, {{b^2 c d}\over a}\endmatrix ;q, {\displaystyle
    -{{a q}\over {b^2}}} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8707
\Description Transformation formula (\cite{\GaRaAA}, (3.4.7), reversed) 
in form of a rule.
$$
{} _{8} W _{7} ({\displaystyle b; {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a q}},
   -{\sqrt{a q}}}, x; q, {\displaystyle {{b^2 q}\over {a^2 x}}}) \longrightarrow  
  { {(\let \over / b q, {{b^2 q}\over {a^2}} ;q) _\infty} \over {(\let
    \over / {{b q}\over a}, {{b^2 q}\over a} ;q) _\infty} } 
   {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, {{a x}\over b}\\ \let
    \over / {{b q}\over x}\endmatrix ;q, {\displaystyle {{b^2 q}\over
    {a^2 x}}} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8708
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8), reversed) 
in form of a rule.
$$
{} _{8} W _{7} ({\displaystyle b; {\sqrt{a q}}, -{\sqrt{a q}},
   {\sqrt{a}} q, - {\sqrt{a}} q  , x}; q, {\displaystyle
   {{b^2}\over {a^2 q x}}}) \longrightarrow  
  { {(\let \over / b q, {{b^2}\over {a^2 q}} ;q) _\infty} \over {(\let
    \over / {{b^2}\over a}, {b\over {a q}} ;q) _\infty} } 
   {} _{4} W _{3} ({\displaystyle a; {{a x}\over b}}; q, {\displaystyle
    {{b^2}\over {a^2 q x}}})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8709
\Description Transformation formula (\cite{\GaRaAA}, (3.5.4), reversed) 
in form of a rule.
$$
{} _{8} W _{7} ({\displaystyle b;  x, -x, {{{\sqrt{b q}}}\over
   {{\sqrt{a}}}}, -{{{\sqrt{b q}}}\over {{\sqrt{a}}}}}, a; q,
   {\displaystyle {{b q}\over {{x^2}}}}) \longrightarrow  
  { {(\let \over / {{b q}\over a}, {{b^2 q^2}\over {a^2 x^2}}
    ;q^2) _\infty} \over {(\let \over / a b q, {{b^2 q^2}\over
    {x^2}} ;q^2) _\infty} } 
   { {(\let \over / {{a b q}\over {x^2}}, b q ;q) _\infty} \over {(\let
    \over / {{b q}\over {x^2}}, {{b q}\over a} ;q) _\infty} } 
   {} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, {{a x^2}\over
    b}\\ \let \over / {{a b q^2}\over {x^2}}\endmatrix ;q^2,
    {\displaystyle {{b^2 q^2}\over {a^2 x^2}}} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8710
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10)) 
in form of a rule.
$$\multline
{} _{8} W _{7} ({\displaystyle a; b, c, c q, d, d q}; q^2, {\displaystyle
   {{a^2 q^2}\over {b c^2 d^2}}}) \\\longrightarrow  
  { {(\let \over / a q, {{a q}\over {b c}}, {{a q}\over {c d}},
    -{{a q}\over {c d}}, {{a q}\over {{\sqrt{b}} d}}, -{{a q}\over
    {{\sqrt{b}} d}} ;q) _\infty} \over {(\let \over / {{a q}\over b},
    {{a q}\over c}, {{a q}\over d}, -{{a q}\over d}, {{a q}\over
    {{\sqrt{b}} c d}}, -{{a q}\over {{\sqrt{b}} c d}} ;q) _\infty} } 
   {} _{8} W _{7} ({\displaystyle -{a\over d}; c, {\sqrt{b}}, -{\sqrt{b}},
    {{{\sqrt{a q}}}\over d}, -{{{\sqrt{a q}}}\over d}}; q, {\displaystyle
    {{a q}\over {b c}}})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8711
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10), reversed) 
in form of a rule.
$$\multline
{} _{8} W _{7} ({\displaystyle a; b, -b, d, -d, c}; q, {\displaystyle
   {{a^2 q^2}\over {b^2 c d^2}}}) \\\longrightarrow  
  { {(\let \over / {{a^2 q^2}\over {b^2 d^2}}, {{a^2 q^2}\over
    {c d^2}}, - a q  , a q, -{{a q}\over {b c}},
    {{a q}\over {b c}} ;q) _\infty} \over {(\let \over /
    {{a^2 q^2}\over {d^2}}, {{a^2 q^2}\over {b^2 c d^2}},
    -{{a q}\over c}, {{a q}\over c}, -{{a q}\over b}, {{a q}\over b} ;q)
    _\infty} } {} _{8} W _{7} ({\displaystyle {{a^2 q}\over {d^2}};
    b^2, c, c q, -{{a q}\over {d^2}}, -{{a q^2}\over {d^2}}};
    q^2, {\displaystyle {{a^2 q^2}\over {b^2 c^2}}})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8761
\Description Transformation formula (\cite{\GaRaAA}, (2.10.10); Appendix (III.36)) in form of a rule.
$$\multline {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow 
  \frac {{( \let\over/ a q,{{a q}\over {d e}},{{a q}\over {d
f}},{{a q}\over {e f}};q)_\infty}}
{{( \let\over/ 
     {{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over {d e
     f}};q)_\infty}} {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {b
     c}},d,e,f\\ \let\over/  {{a q}\over b},{{a q}\over c},{{d e f}\over a}\endmatrix
     ;q,q\right ] \\+ \frac {{( \let\over/ a q,{{a q}\over {b
     c}},d,e,f,{{{a^2} {q^2}}\over {b d e f}},{{{a^2} {q^2}}\over {c d e f}};q)_\infty}}
{{( \let\over/ 
     {{a q}\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over
     f},{{{a^2} {q^2}}\over {b c d e f}},{{d e f}\over {a q}};q)_\infty}}  {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {d e}},{{a q}\over {d
     f}},{{a q}\over {e f}},{{{a^2} {q^2}}\over {b c d e f}}\\ \let\over/  {{{a^2}
     {q^2}}\over {b d e f}},{{{a^2} {q^2}}\over {c d e f}},{{a {q^2}}\over {d
     e f}}\endmatrix ;q,q\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8762
\Description Transformation formula (\cite{\GaRaAA}, (2.11.1); Appendix (III.37)) in form of a rule.
$$\multline {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \\\longrightarrow 
  - \frac {{( \let\over/ a q,{b\over a},{{b q}\over c},{{b q}\over
       d},{{b q}\over e},{{b q}\over f},d,e,f,{{a q}\over {b c}},
{{b d e f}\over {a^2}}, {{a^2 q}\over {b d e f}};q)_\infty}}
{{( \let\over/  {a\over
       b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over f},{{b
       d}\over a},{{b e}\over a}, {{b f}\over {a}}, {{d e f}\over
{a}}, {{a q}\over {d e f}}, {{q}\over {c}}, {{b^2 q}\over {a}};q)_\infty}} 
   \\   {}_8W _7({{{b^2}}\over a};b,{{b c}\over a},{{b d}\over a},{{b e}\over
       a},{{b f}\over a};q,{{{a^2} {q^2}}\over {b c d e f}})  \\+ 
   \frac {{( \let\over/ a q,{{a q}\over {d e}},{{a q}\over {d f}},{{a
     q}\over {e f}},{{e q}\over c},{{f q}\over c}, {{b}\over {a}}, 
     {{b e f}\over a};q)_\infty}}
{{( \let\over/  {{a
     q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over {d e f}},{q\over
     c},{{e f q}\over c},{{b e}\over a},{{b f}\over a};q)_\infty}} 
  \\  {}_8W _7({{e f}\over c};{{a q}\over {b c}},{{a q}\over {c d}},{{e f}\over
     a},e,f;q,{{b d}\over a})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8763
\Description Transformation formula (\cite{\GaRaAA}, (2.11.1),
reversed; Appendix (III.37)) in form of a rule.
$$\multline
{} _{8} W _{7} ({\displaystyle a; b, c, d, e, f}; q, {\displaystyle {{{a^2}
   {q^2}}\over {b c d e f}}}) \\ \longrightarrow
  { {(\let \over / {{c e f}\over a}, {{f q}\over d}, {{e q}\over d}, {c\over
     a}, {{a q}\over {e f}}, a q, {{a q}\over {b f}}, {{a q}\over {b e}} ;q)
     _\infty} \over {(\let \over / {{e f q}\over d}, {{c f}\over a}, {{c
     e}\over a}, {q\over d}, {{a q}\over f}, {{a q}\over e}, {{a q}\over {b e
     f}}, {{a q}\over b} ;q) _\infty} } 
    {} _{8} W _{7} ({\displaystyle {{e f}\over d}; {{a q}\over {b d}}, {{e
     f}\over a}, {{a q}\over {c d}}, e, f}; q, {\displaystyle {{b c}\over a}})
\\     + { {(\let \over / a q, {{{a^2} {q^2}}\over {b d e f}}, {{c q}\over b},
     {{a {q^2}}\over {b d e}}, {{a {q^2}}\over {b d f}}, {{a q}\over {c d}},
     e, f, b, {{{a^2} {q^2}}\over {b c e f}}, {{b c e f}\over {{a^2} q}} ;q)
     _\infty} \over {(\let \over / {{c f}\over a}, {{c e}\over a}, {q\over d},
     {{a q}\over f}, {{a q}\over e}, {{a q}\over b}, {{b e f}\over {a q}}, {{a
     q}\over d}, {{{a^2} {q^2}}\over {b c d e f}}, {{a q}\over c}, {{{a^2}
     {q^3}}\over {{b^2} d e f}} ;q) _\infty} } \\
    {} _{8} W _{7} ({\displaystyle {{{a^2} {q^2}}\over {{b^2} d e f}}; {{a
     q}\over {b d}}, {q\over b}, {{{a^2} {q^2}}\over {b c d e f}}, {{a q}\over
     {b f}}, {{a q}\over {b e}}}; q, {\displaystyle {{b c}\over a}})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8764
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.15) 
in form of a rule.
$$\multline
{} _{8} W _{7} ({\displaystyle a^2; a b, a c, a d, a e, a f}; q,
   {\displaystyle {{q^2}\over {a b c d e f}}}) \\\longrightarrow  
  -\frac {a} {b}{{ { {(\let \over / a^2 q, a c, {b\over a}, {c\over a}, {{b q}\over
         d}, {{b q}\over e}, {{b q}\over f}, {q\over {b d}}, {q\over
         {b e}}, {q\over {b f}} ;q) _\infty} \over {(\let \over /
         {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {q\over {a d}},
         {q\over {a e}}, {q\over {a f}}, b^2 q, b c, {a\over b}, {c\over
         b} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle b^2; a b, b c,
         b d, b e, b f}; q, {\displaystyle {{q^2}\over
         {a b c d e f}}})}}\\ - 
   \frac {a} {c}{{ { {(\let \over / a^2 q, a b, {b\over a}, {c\over a}, {{c q}\over
        d}, {{c q}\over e}, {{c q}\over f}, {q\over {c d}}, {q\over
        {c e}}, {q\over {c f}} ;q) _\infty} \over {(\let \over /
        {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {q\over {a d}},
        {q\over {a e}}, {q\over {a f}}, c^2 q, b c, {a\over c}, {b\over
        c} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle c^2; a c, b c,
        c d, c e, c f}; q, {\displaystyle {{q^2}\over
        {a b c d e f}}})}}
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8810
\Description Transformation formula (\cite{\GaRaAA}, (5.6.1); Appendix
(III.38)) in form of a rule.
\NoBlackBoxes
$$\multline
{} _{8} \psi _{8} \! \left [ \matrix \let \over / {\sqrt{a}} q, -
   {\sqrt{a}} q  , b, c, d, e, f, g\\ \let \over / {\sqrt{a}},
   -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d},
   {{a q}\over e}, {{a q}\over f}, {{a q}\over g}\endmatrix ;q,
   {\displaystyle {{{a^3} {q^2}}\over {b c d e f g}}} \right ]
\\
\hskip-7pt\longrightarrow 
{ {(\let \over / f, {f\over a}, a q, {q\over a}, q, {{a q}\over {b g}},
     {{a q}\over {c g}}, {{a q}\over {d g}}, {{a q}\over {e g}},
     {{g q}\over b}, {{g q}\over c}, {{g q}\over d}, {{g q}\over e} ;q)
     _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c},
     {{a q}\over d}, {{a q}\over e}, {q\over b}, {q\over c}, {q\over d},
     {q\over e}, {q\over g}, {{a q}\over g}, {f\over g}, {{f g}\over a},
     {{{g^2} q}\over a} ;q) _\infty} }
  {} _{8} W _{7} ({\displaystyle {{{g^2}}\over a}; {{b g}\over a},
     {{c g}\over a}, {{d g}\over a}, {{e g}\over a}, {{f g}\over a}}; q,
     {\displaystyle {{{a^3} {q^2}}\over {b c d e f g}}})
\\     + 
    { {(\let \over / g, {g\over a}, a q, {q\over a}, q, {{a q}\over {b f}},
     {{a q}\over {c f}}, {{a q}\over {d f}}, {{a q}\over {e f}},
     {{f q}\over b}, {{f q}\over c}, {{f q}\over d}, {{f q}\over e} ;q)
     _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c},
     {{a q}\over d}, {{a q}\over e}, {q\over b}, {q\over c}, {q\over d},
     {q\over e}, {q\over f}, {{a q}\over f}, {g\over f}, {{f g}\over a},
     {{{f^2} q}\over a} ;q) _\infty} }
   {} _{8} W _{7} ({\displaystyle {{{f^2}}\over a}; {{b f}\over a},
     {{c f}\over a}, {{d f}\over a}, {{e f}\over a}, {{f g}\over a}}; q,
     {\displaystyle {{{a^3} {q^2}}\over {b c d e f g}}}) 
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10901
\Description Transformation formula (\cite{\GaRaAA}, (2.9.1); Appendix (III.28)) in form of a rule.
\BlackBoxes
$$\multline {}_{10}W _9(a;b,c,d,e,f,{{{a^3} {q^{2 + n}}}\over {b c d e f}},{q^{-n}};q,q)\\\longrightarrow 
{   {{(\let\over/ a q,{{a q}\over {e f}},{{{a^2} {q^2}}\over {b c d e}},{{{a^2} {q^2}}\over
    {b c d f}};q)}_{n}}\over{{(\let\over/ {{a q}\over e},{{a q}\over f},{{{a^2} {q^2}}\over
    {b c d e f}},{{{a^2} {q^2}}\over {b c d}};q)}_{n}}
}  {}_{10}W _9({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b d}},{{a
    q}\over {b c}},e,f,{{{a^3} {q^{2 + n}}}\over {b c d e f}},{q^{-n}};q,q) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10902
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(i), reversed) 
in form of a rule.
$$
{} _{10} W _{9} ({\displaystyle a; b, -b, b {\sqrt{q}}, - b {\sqrt{q}}
    , c, d, {{a^2 q}\over {b^2 c d}}}; q, {\displaystyle
   {{a q}\over {b^2}}}) \longrightarrow  { {(\let \over / a q, {{a^2 q}\over {{b^4}}}
    ;q) _\infty} \over {(\let \over / {{a q}\over {b^2}}, {{a^2 q}\over
    {b^2}} ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over /
    b^2, {{b^2 c}\over a}, {{b^2 d}\over a}, {{a q}\over {c d}}\\
    \let \over / {{a q}\over c}, {{a q}\over d}, {{b^2 c d}\over
    a}\endmatrix ;q, {\displaystyle {{a^2 q}\over {{b^4}}}} \right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10903
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.19) 
in form of a rule.
$$\multline
{} _{10} W _{9} ({\displaystyle a; b, c, d, e, f, g, {q^{-n}}}; q,
   {\displaystyle q}) \\ 
\longrightarrow  {e^n}
{{({\let \over / a q, {{a q}\over {c e}},
    {{a q}\over {d e}}, {{a q}\over {e f}}, {{a q}\over {e g}}, b}; q)
    _{n}}\over {({\let \over / {{a q}\over c}, {{a q}\over d}, {{a q}\over
    e}, {{a q}\over f}, {{a q}\over g}, {b\over e}}; q) _{n}}}
 {} _{10} W _{9} ({\displaystyle {e\over
    {b {q^n}}}; e, {{a q}\over {b c}}, {{a q}\over {b d}}, {{a q}\over
    {b f}}, {{a q}\over {b g}}, {e\over {a {q^n}}}, {q^{-n}}}; q,
    {\displaystyle q})  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10904
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4)) 
in form of a rule.
$$\multline
{} _{10} W _{9} ({\displaystyle a; b, x, -x, y, -y, -{q^{-n}}, {q^{-n}}}; q,
   {\displaystyle -{{{a^3} {q^{3 + 2 n}}}\over {b x^2 y^2}}})\\ \longrightarrow  
{{({\let \over / a^2 q^2, {{a^2 q^2}\over {x^2 y^2}}};
    q^2) _{n}}\over {({\let \over / {{a^2 q^2}\over {x^2}},
    {{a^2 q^2}\over {y^2}}}; q^2) _{n}}}
  {} _{5} \phi _{4} \! \left [ \matrix \let \over / {q^{-2 n}}, x^2, y^2,
    -{{a q}\over b}, -{{a q^2}\over b}\\ \let \over / {{x^2 y^2}\over
    {a^2 {q^{2 n}}}}, {{a^2 q^2}\over {b^2}}, - a q 
    , - a q^2  \endmatrix ;q^2, {\displaystyle q^2} \right
    ]  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10905
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.21(iii)) 
in form of a rule.\NoBlackBoxes
$$\multline
{} _{10} \phi _{9} \! \left [ \matrix \let \over / a, {\sqrt{a}} q,
   - {\sqrt{a}} q  , b, c, {a\over {b c}}, {C\over {A {q^n}}}, {1\over
   {B C {q^n}}}, {B\over {A {q^n}}}, {q^{-n}}\\ \let \over / {\sqrt{a}},
   -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, b c q, {1\over
   {C {q^n}}}, {{B C}\over {A {q^n}}}, {1\over {B {q^n}}}, {1\over
   {A {q^n}}}\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow  
{{({\let \over / a q, b q, c q, {{a q}\over {b c}}, {{A q}\over B},
    {{A q}\over C}, B C q}; q) _{n}}\over {({\let \over / A q, B q, C q,
    {{A q}\over {B C}}, {{a q}\over b}, {{a q}\over c}, b c q}; q)
    _{n}}}
  {} _{10} \phi _{9} \! \left [ \matrix \let \over / A, {\sqrt{A}} q,
   - {\sqrt{A}} q  , B, C, {A\over {B C}}, {c\over {a {q^n}}},
    {1\over {b c {q^n}}}, {b\over {a {q^n}}}, {q^{-n}}\\ \let \over /
    {\sqrt{A}}, -{\sqrt{A}}, {{A q}\over B}, {{A q}\over C}, B C q,
    {1\over {c {q^n}}}, {{b c}\over {a {q^n}}}, {1\over {b {q^n}}},
    {1\over {a {q^n}}}\endmatrix ;q, {\displaystyle q} \right ] 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10906
\Description Transformation formula (\cite{\RaVeAA}, (7.7), reversed) 
in form of a rule.\BlackBoxes
$$\multline
{} _{10} W _{9} ({\displaystyle a^2 {q^n}; c, d, e, a {q^{{1\over 2} +
   n}}, - a {q^{{1\over 2} + n}}  , {{{a^4} {q^{1 + n}}}\over
   {c d e}}, {q^{-n}}}; q, {\displaystyle -{q^{1 + n}}}) \\\longrightarrow  
{{({\let \over / {{a^2 q}\over c}, {{a^2 q}\over d}, {{a^2 q}\over
    e}, {{a^2 q}\over {c d e}}}; q) _{n}}\over {({\let \over / a^2 q,
    {{a^2 q}\over {c d}}, {{a^2 q}\over {c e}}, {{a^2 q}\over
    {d e}}}; q) _{n}}}
  {} _{12} W _{11} ({\displaystyle a^2; {q^{-2 n}}, c, c q, d, d q, e,
    e q, {{{a^4} {q^{1 + n}}}\over {c d e}}, {{{a^4} {q^{2 + n}}}\over
    {c d e}}}; q^2, {\displaystyle q^2}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10907
\Description Transformation formula (\cite{\RaVeAA}, (7.8), reversed) 
in form of a rule.
$$\multline
{} _{10} W _{9} ({\displaystyle {{a^2}\over e}; {{a {\sqrt{q}}}\over e},
   -{{a {\sqrt{q}}}\over e}, c, d, e, {{{a^4} {q^{1 + n}}}\over {c d e}},
   {q^{-n}}}; q, {\displaystyle -{q\over e}}) \\\longrightarrow  
{{({\let \over / {{a^2 q}\over c}, {{a^2 q}\over d}, {{a^2 q}\over
    e}, {{a^2 q}\over {c d e}}}; q) _{n}}\over {({\let \over / a^2 q,
    {{a^2 q}\over {c d}}, {{a^2 q}\over {c e}}, {{a^2 q}\over
    {d e}}}; q) _{n}}}
  {} _{12} W _{11} ({\displaystyle a^2; e^2, c, c q, d, d q,
    {{{a^4} {q^{1 + n}}}\over {c d e}}, {{{a^4} {q^{2 + n}}}\over
    {c d e}}, {q^{1 - n}}, {q^{-n}}}; q^2, {\displaystyle q^2}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10961
\Description Transformation formula (\cite{\GaRaAA}, (2.12.9); Appendix (III.39)) in form of a rule.
$$\multline {}_{10}W_9(a; b,c,d,e,f,g,{{{a^3} {q^2}}\over {b c d e f g}}; q,q)\\ 
\longrightarrow
    \frac {{( \let\over/ a q,{b\over a},f,g,{{{a^3} {q^2}}\over {b c d e f
     g}},{{b q}\over f},{{b q}\over g},{{{b^2} c d e f g}\over {{a^3} q}},{{a
     q}\over {d e}},{{a q}\over {c e}},{{a q}\over {c d}}
     ;q)_\infty}}
{{( \let\over/  {{{b^2} c d e}\over {{a^2}}},{{{a^2}
     q}\over {b c d e}},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a
     q}\over f},{{a q}\over g},{{b c d e f g}\over {{a^2} q}},{{b c}\over
     a},{{b d}\over a},{{b e}\over a};q)_\infty}} \\
 \frac {{( \let\over/ {{b d e}\over a},{{b
     c e}\over a},{{b c d}\over a} ;q)_\infty}}
{{(\let\over/ 
   {{b f}\over a},{{b g}\over a},{{{a^2}
     {q^2}}\over {c d e f g}};q)_\infty}} 
    {}_{10}W_9({{{b^2} c d e}\over {{a^2} q}};
      b,{{b c}\over a},{{b d}\over a},{{b e}\over a},
      {{b c d e f}\over {{a^2} q}},{{b c d e g}\over {{a^2} q}},
      {{a q}\over {f g}}; q,q) \\+ 
   \frac {{( \let\over/ a q,{b\over a},{{{a^2} {q^2}}\over {c d e
     f}},{{{a^2} {q^2}}\over {c d e g}},{{b f g}\over a},{{b c d e f}\over
     {{a^2} q}},{{b c d e g}\over {{a^2} q}},{{a q}\over {f g}};q)_\infty}}
{{( \let\over/  {{{a^2}
     {q^2}}\over {c d e}},{{b c d e}\over {{a^2} q}},{{a q}\over f},{{a
     q}\over g},{{b c d e f g}\over {{a^2} q}},{{b f}\over a},{{b g}\over
     a},{{{a^2} {q^2}}\over {c d e f g}};q)_\infty}} \\
    {}_{10}W_9({{{a^2} q}\over {c d e}}; b,{{a q}\over {d e}},{{a q}\over {c e}},
      {{a q}\over {c d}},f,g,{{{a^3} {q^2}}\over {b c d e f g}}; q,q)\\ -
   \frac {{( \let\over/ a q,{b\over a},c,d,e,f,g,{{{a^3} {q^2}}\over {b c
     d e f g}},{{b q}\over c},{{b q}\over d},{{b q}\over e},{{b q}\over f}
  ;q)_\infty}}
{{( \let\over/  {{{b^2} q}\over
     a},{a\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over
     f},{{a q}\over g},{{b c d e f g}\over {{a^2} q}},{{b c}\over a},{{b
     d}\over a},{{b e}\over a},{{b f}\over a};q)_\infty}} \\
 \frac {{( \let\over/ 
        {{b q}\over g},{{{b^2} c d e f g}\over {{a^3} q}}
    ;q)_\infty}}
{{( \let\over/ {{b g}\over a},{{{a^2}
     {q^2}}\over {c d e f g}};q)_\infty}} 
    {}_{10}W_9({{{b^2}}\over a}; b,{{b c}\over a},{{b d}\over a},{{b e}\over a},
      {{b f}\over a},{{b g}\over a},{{{a^2} {q^2}}\over {c d e f g}}; q,q)
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10962
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7)) 
in form of a rule.
$$\multline
{} _{10} W _{9} ({\displaystyle a; b, c, c q, d, d q, e, e q}; q^2,
   {\displaystyle {{{a^3} {q^3}}\over {b c^2 d^2 e^2}}}) \longrightarrow  
  { {(\let \over / a q, {{a q}\over {c d}}, {{a q}\over {c e}},
     {{a q}\over {d e}} ;q) _\infty} \over {(\let \over / {{a q}\over c},
     {{a q}\over d}, {{a q}\over e}, {{a q}\over {c d e}} ;q) _\infty} }
     {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{\sqrt{a q}}}\over
     {{\sqrt{b}}}}, -{{{\sqrt{a q}}}\over {{\sqrt{b}}}}, c, d, e\\ \let \over
     / {\sqrt{a q}}, -{\sqrt{a q}}, {{a q}\over b}, {{c d e}\over
     a}\endmatrix ;q, {\displaystyle q} \right ]\\+ 
   { {(\let \over / a q^2, {{{a^3} {q^3}}\over {c^2 d^2 e^2}}
     ;q^2) _\infty} \over {(\let \over / {{a q^2}\over b},
     {{{a^3} {q^3}}\over {b c^2 d^2 e^2}} ;q^2) _\infty} } 
    { {(\let \over / c, d, e, {{a^2 q^2}\over {b c d e}} ;q) _\infty}
     \over {(\let \over / {{a q}\over c}, {{a q}\over d}, {{a q}\over e},
     {{c d e}\over {a q}} ;q) _\infty} } \\
    {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{{\left( a q \right)
     }^{{3\over 2}}}}\over {{\sqrt{b}} c d e}}, -{{{{\left( a q \right)
     }^{{3\over 2}}}}\over {{\sqrt{b}} c d e}}, {{a q}\over {c d}},
     {{a q}\over {c e}}, {{a q}\over {d e}}\\ \let \over / {{{{\left( a q
     \right) }^{{3\over 2}}}}\over {c d e}}, -{{{{\left( a q \right)
     }^{{3\over 2}}}}\over {c d e}}, {{a^2 q^2}\over {b c d e}},
     {{a q^2}\over {c d e}}\endmatrix ;q, {\displaystyle q} \right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T10963
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.30) 
in form of a rule.
$$\align
{} _{10} W& _{9} ({\displaystyle a; b, c, d, e, f, g, {{a^3 q^2}\over
    {b c d e f g}}}; q, {\displaystyle q}) \\&\longrightarrow  -
   { {(\let \over / {{b q}\over f}, {{b q}\over g},
     {{b^2 c d e f g}\over {a^3 q}}, a q, c, d, e, f, g,
     {{a^3 q^2}\over {b c d e f g}}, {b\over a}, {{b q}\over c},
     {{b q}\over d}, {{b q}\over e} ;q) _\infty} \over {(\let \over /
     {{a q}\over f}, {{a q}\over g}, {{b c d e f g}\over {a^2 q}},
     {{b^2 q}\over a}, {{b c}\over a}, {{b d}\over a}, {{b e}\over a},
     {{b f}\over a}, {{b g}\over a}, {{a^2 q^2}\over {c d e f g}},
     {a\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e} ;q)
     _\infty} }\\&\hskip3cm
 {} _{10} W _{9} ({\displaystyle {b^2\over a}; b,
     {{b c}\over a}, {{b d}\over a}, {{b e}\over a}, {{b f}\over a},
     {{b g}\over a}, {{a^2 q^2}\over {c d e f g}}}; q,
     {\displaystyle q})\\&\hskip1cm
 + { {(\let \over / {{b e g}\over a}, {{b f g}\over
     a}, a q, {b\over a}, {{a^3 q^2}\over {b c d e f g}},
     {{b^2 c d e f g}\over {a^3 q}}, {{a q}\over {c g}},
     {{a q}\over {d g}}, {{a q}\over {e g}}, {{a q}\over {f g}},
     {{b c g}\over a}, {{b d g}\over a} ;q) _\infty} \over {(\let \over /
     {{b e}\over a}, {{b f}\over a}, {{b^2 c d e f g^2}\over
     {a^3 q}}, {{b g}\over a}, {{a^3 q^2}\over
     {b c d e f g^2}}, {{a q}\over g}, {{a q}\over c}, {{a q}\over
     d}, {{a q}\over e}, {{a q}\over f}, {{b c}\over a}, {{b d}\over a}
     ;q) _\infty} }\\&\hskip3cm
 {} _{10} W _{9} ({\displaystyle
     {{b^2 c d e f g^2}\over {a^3 q^2}}; b,
     {{b d e f g}\over {a^2 q}}, {{b c e f g}\over {a^2 q}},
     {{b c d f g}\over {a^2 q}}, {{b c d e g}\over {a^2 q}},
     {{b g}\over a}, g}; q, {\displaystyle q}) \\&\hskip1cm+ 
   { {(\let \over / {{a^2 q^2}\over {c d f g}}, {{a^2 q^2}\over
     {c d e g}}, a q, {b\over a}, g, {{b q}\over g},
     {{b d e f g}\over {a^2 q}}, {{b c e f g}\over {a^2 q}},
     {{b c d f g}\over {a^2 q}}, {{b c d e g}\over {a^2 q}}
     ;q) _\infty} \over {(\let \over / {{b e}\over a}, {{b f}\over a},
     {{a^3 q^3}\over {c d e f g^2}}, {{a^2 q^2}\over
     {c d e f g}}, {{b c d e f g^2}\over {a^3 q^2}},
     {{b c d e f g}\over {a^2 q}}, {{a q}\over c}, {{a q}\over d},
     {{a q}\over e}, {{a q}\over f} ;q)
     _\infty} }\\&\hskip3cm
  {{(\let \over /{{a^2 q^2}\over {d e f g}}, 
   {{a^2 q^2}\over {c e f g}};q)_{\infty}}\over
   {(\let \over / {{b c}\over a}, {{b d}\over a}; q)_{\infty}}}
 {} _{10} W _{9} ({\displaystyle {{a^3 q^2}\over
     {c d e f g^2}}; b, {{a q}\over {c g}}, {{a q}\over {d g}},
     {{a q}\over {e g}}, {{a q}\over {f g}}, {{a^2 q^2}\over
     {c d e f g}}, {{a^3 q^2}\over {b c d e f g}}}; q,
     {\displaystyle q})
\endalign$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T101010
\Description Transformation formula (\cite{\GaRaAA}, (5.6.3); Appendix
(III.40)) in form of a rule.
$$\multline
{} _{10} \psi _{10} \! \left [ \matrix \let \over / {\sqrt{a}} q, -
   {\sqrt{a}} q  , b, c, d, e, f, g, h, k\\ \let \over / {\sqrt{a}},
   -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d},
   {{a q}\over e}, {{a q}\over f}, {{a q}\over g}, {{a q}\over h},
   {{a q}\over k}\endmatrix ;q, {\displaystyle {{{a^4} {q^3}}\over
   {b c d e f g h k}}} \right ] \\
\longrightarrow
    { {(\let \over / g, h, {g\over a}, {h\over a}, a q, {q\over a}, q,
     {{a q}\over {b k}}, {{a q}\over {c k}}, {{a q}\over {d k}},
     {{a q}\over {e k}}, {{a q}\over {f k}}, {{k q}\over b}, {{k q}\over
     c}, {{k q}\over d}, {{k q}\over e}, {{k q}\over f} ;q) _\infty} \over
     {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d},
     {{a q}\over e}, {{a q}\over f}, {q\over b}, {q\over c}, {q\over d},
     {q\over e}, {q\over f}, {{h k}\over a}, {{g k}\over a}, {h\over k},
     {g\over k}, {q\over k}, {{a q}\over k}, {{{k^2} q}\over a} ;q) _\infty}
     }\\
\times
  {} _{10} W _{9} ({\displaystyle {{{k^2}}\over a}; {{b k}\over a},
     {{c k}\over a}, {{d k}\over a}, {{e k}\over a}, {{f k}\over a},
     {{h k}\over a}, {{g k}\over a}}; q, {\displaystyle {{{a^4} {q^3}}\over
     {b c d e f g h k}}}) \\
 +     { {(\let \over / g, k, {g\over a}, {k\over a}, a q, {q\over a}, q,
     {{a q}\over {b h}}, {{a q}\over {c h}}, {{a q}\over {d h}},
     {{a q}\over {e h}}, {{a q}\over {f h}}, {{h q}\over b}, {{h q}\over
     c}, {{h q}\over d}, {{h q}\over e}, {{h q}\over f} ;q) _\infty} \over
     {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d},
     {{a q}\over e}, {{a q}\over f}, {q\over b}, {q\over c}, {q\over d},
     {q\over e}, {q\over f}, {{g h}\over a}, {{h k}\over a}, {g\over h},
     {k\over h}, {q\over h}, {{a q}\over h}, {{{h^2} q}\over a} ;q) _\infty}
     }\\
\times
{} _{10} W _{9} ({\displaystyle {{{h^2}}\over a}; {{b h}\over a},
     {{c h}\over a}, {{d h}\over a}, {{e h}\over a}, {{f h}\over a},
     {{g h}\over a}, {{h k}\over a}}; q, {\displaystyle {{{a^4} {q^3}}\over
     {b c d e f g h k}}}) \\
 +     { {(\let \over / h, k, {h\over a}, {k\over a}, a q, {q\over a}, q,
     {{a q}\over {b g}}, {{a q}\over {c g}}, {{a q}\over {d g}},
     {{a q}\over {e g}}, {{a q}\over {f g}}, {{g q}\over b}, {{g q}\over
     c}, {{g q}\over d}, {{g q}\over e}, {{g q}\over f} ;q) _\infty} \over
     {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d},
     {{a q}\over e}, {{a q}\over f}, {q\over b}, {q\over c}, {q\over d},
     {q\over e}, {q\over f}, {{g h}\over a}, {{g k}\over a}, {h\over g},
     {k\over g}, {q\over g}, {{a q}\over g}, {{{g^2} q}\over a} ;q) _\infty}
     }\\
\times
{} _{10} W _{9} ({\displaystyle {{{g^2}}\over a}; {{b g}\over a},
     {{c g}\over a}, {{d g}\over a}, {{e g}\over a}, {{f g}\over a},
     {{g h}\over a}, {{g k}\over a}}; q, {\displaystyle {{{a^4} {q^3}}\over
     {b c d e f g h k}}}) 
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121101
\Description Transformation formula (\cite{\GaRaAA}, (2.8.3), reversed; Appendix (III.25), reversed) in form 
  of a rule.
$$\multline {}_{12}W _{11}(a;b,c,d,{{a {\sqrt{q}}}\over {{\sqrt{b}} {\sqrt{c}}
   {\sqrt{d}}}},-{{a {\sqrt{q}}}\over {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},{{a
   q}\over {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},-{{a q}\over {{\sqrt{b}}
   {\sqrt{c}} {\sqrt{d}}}},\\
  b c d {q^n},{q^{-n}};q,q) \longrightarrow 
{{{(\let\over/ a q,{{{b^2} {c^2} {d^2}}\over {{a^2} q}};q)}_{n}}\over{{(\let\over/ {{b
    c d}\over a},b c d;q)}_{n}}
}  {}_5\phi _4\!\left [ \matrix \let\over/ {{{a^2} q}\over {b c d}},{{a q}\over {c d}},{{a
    q}\over {b d}},{{a q}\over {b c}},{q^{-n}}\\ \let\over/  {{a q}\over b},{{a q}\over
    c},{{a q}\over d},{{{a^2} {q^{2 - n}}}\over {{b^2} {c^2} {d^2}}}\endmatrix
    ;q,q\right ] 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121102
\Description Transformation formula (\cite{\GaRaAA}, (2.8.4), reversed; Appendix (III.26), reversed) in form 
  of a rule.
$$\multline {}_{12}W _{11}({\sqrt{b}} {\sqrt{c}} {\sqrt{d}} {q^{-{1\over 2} - {n\over
   2}}};b,c,d,{q^{{{-n}\over 2}}},-{q^{{{-n}\over 2}}},{q^{{1\over 2} -
   {n\over 2}}},-{q^{{1\over 2} - {n\over 2}}},e,{{b c d}\over e};q,q)\\\longrightarrow 
{   {{(\let\over/ {\sqrt{b}} {\sqrt{c}} {\sqrt{d}} {q^{{1\over 2} - {n\over 2}}},{{b c
    d}\over e};q)}_{n}}\over{{(\let\over/ {{{\sqrt{b}} {\sqrt{c}} {\sqrt{d}} {q^{{1\over 2}
    - {n\over 2}}}}\over e},b c d;q)}_{n}}
} \\ {}_5\phi _4\!\left [ \matrix \let\over/ {q^{-n}},{{{\sqrt{b}} {q^{{1\over 2} - {n\over
    2}}}}\over {{\sqrt{c}} {\sqrt{d}}}},{{{\sqrt{c}} {q^{{1\over 2} - {n\over
    2}}}}\over {{\sqrt{b}} {\sqrt{d}}}},{{{\sqrt{d}} {q^{{1\over 2} - {n\over
    2}}}}\over {{\sqrt{b}} {\sqrt{c}}}},e\\ \let\over/  {{{\sqrt{c}} {\sqrt{d}}
    {q^{{1\over 2} - {n\over 2}}}}\over {{\sqrt{b}}}},{{{\sqrt{b}} {\sqrt{d}}
    {q^{{1\over 2} - {n\over 2}}}}\over {{\sqrt{c}}}},{{{\sqrt{b}} {\sqrt{c}}
    {q^{{1\over 2} - {n\over 2}}}}\over {{\sqrt{d}}}},{{e {q^{1 - n}}}\over {b
    c d}}\endmatrix ;q,q\right ] 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121103
\Description Transformation formula (\cite{\GaRaAA}, (2.8.5), reversed; Appendix (III.27), reversed) in form 
  of a rule.
$$\multline {}_{12}W _{11}(a;b,c,d,{{a q}\over {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},-{{a
   q}\over {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},{{a {q^{{3\over 2}}}}\over
   {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},-{{a {q^{{3\over 2}}}}\over {{\sqrt{b}}
   {\sqrt{c}} {\sqrt{d}}}},\\
  b c d {q^{-2 + n}},{q^{-n}};q,q) \longrightarrow 
{{{\left(\let\over/  1 - {{b c d}\over {{q^2}}} \right)}  \over 
{(1 - b c d {q^{-2 + 2 n}})}}
{{{(\let\over/ a q,{{{b^2} {c^2} {d^2}}\over {{a^2} {q^3}}};q)}_{n}}\over{{(\let\over/ {{b c
       d}\over {a {q^2}}},{{b c d}\over {{q^2}}};q)}_{n}}}}\\
    {}_7\phi _6\!\left [ \matrix \let\over/ {{{a^2} q}\over {b c d}},{{a {q^{{3\over
       2}}}}\over {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},-{{a {q^{{3\over
       2}}}}\over {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},{{a q}\over {c d}},{{a
       q}\over {b d}},{{a q}\over {b c}},{q^{-n}}\\ \let\over/  {{a {\sqrt{q}}}\over
       {{\sqrt{b}} {\sqrt{c}} {\sqrt{d}}}},-{{a {\sqrt{q}}}\over {{\sqrt{b}}
       {\sqrt{c}} {\sqrt{d}}}},{{a q}\over b},{{a q}\over c},{{a q}\over
       d},{{{a^2} {q^{4 - n}}}\over {{b^2} {c^2} {d^2}}}\endmatrix ;q,q\right
       ] \endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121104
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii), reversed) 
in form of a rule.
$$\multline
{} _{12} W _{11} ({\displaystyle a; c, d, {{a^2 q}\over {b c d}},
   {\sqrt{b}} q, -{\sqrt{b}}, {\sqrt{b q}}, -{\sqrt{b q}},
   {{a^2 {q^n}}\over b}, {q^{-n}}}; q, {\displaystyle q}) \\\longrightarrow  
{{({\let \over / a q, {{a^2}\over {b^2}}, -{a\over {{\sqrt{b}}}}};
    q) _{n}}\over {({\let \over / {a\over b}, {{a^2}\over b}, -{{a q}\over
    {{\sqrt{b}}}}}; q) _{n}}}
  {} _{6} \phi _{5} \! \left [ \matrix \let \over / b, {\sqrt{b}} q,
    {{b c}\over a}, {{b d}\over a}, {{a q}\over {c d}}, {q^{-n}}\\ \let
    \over / {\sqrt{b}}, {{a q}\over c}, {{a q}\over d}, {{b c d}\over a},
    {{b^2 {q^{1 - n}}}\over {a^2}}\endmatrix ;q, {\displaystyle q} \right
    ]  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121105
\Description Transformation formula (\cite{\RaVeAA}, (7.7)) 
in form of a rule.
$$\multline
{} _{12} W _{11} ({\displaystyle a^2; c, c q, d, d q, e, e q,
   {{{a^4} {q^{1 + n}}}\over {c d e}}, {{{a^4} {q^{2 + n}}}\over
   {c d e}}, {q^{-2 n}}}; q^2, {\displaystyle q^2})\\ \longrightarrow  
{{({\let \over / a^2 q, {{a^2 q}\over {c d}}, {{a^2 q}\over
    {c e}}, {{a^2 q}\over {d e}}}; q) _{n}}\over {({\let \over /
    {{a^2 q}\over c}, {{a^2 q}\over d}, {{a^2 q}\over e},
    {{a^2 q}\over {c d e}}}; q) _{n}}}
  {} _{10} W _{9} ({\displaystyle a^2 {q^n}; c, d, e, a {q^{{1\over 2} +
    n}}, - a {q^{{1\over 2} + n}}  , {{{a^4} {q^{1 + n}}}\over
    {c d e}}, {q^{-n}}}; q, {\displaystyle -{q^{1 + n}}}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121106
\Description Transformation formula (\cite{\RaVeAA}, (7.8)) 
in form of a rule.
$$\multline
{} _{12} W _{11} ({\displaystyle a^2; e^2, c, c q, d, d q,
   {{{a^4} {q^{1 + n}}}\over {c d e}}, {{{a^4} {q^{2 + n}}}\over
   {c d e}}, {q^{1 - n}}, {q^{-n}}}; q^2, {\displaystyle q^2}) \\\longrightarrow  
{{({\let \over / a^2 q, {{a^2 q}\over {c d}}, {{a^2 q}\over
    {c e}}, {{a^2 q}\over {d e}}}; q) _{n}}\over {({\let \over /
    {{a^2 q}\over c}, {{a^2 q}\over d}, {{a^2 q}\over e},
    {{a^2 q}\over {c d e}}}; q) _{n}}}
  {} _{10} W _{9} ({\displaystyle {{a^2}\over e}; {{a {\sqrt{q}}}\over e},
    -{{a {\sqrt{q}}}\over e}, c, d, e, {{{a^4} {q^{1 + n}}}\over {c d e}},
    {q^{-n}}}; q, {\displaystyle -{q\over e}}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121107
\Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15, reversed) 
in form of a rule.
$$\multline  
{} _{12} W _{11} \left({\displaystyle d; \frac{{\sqrt{d q}}}{{\sqrt{a}}},
    - \frac{{\sqrt{d q}}}{{\sqrt{a}}}  ,
    \frac{{\sqrt{d}} q}{{\sqrt{a}}}, - \frac{{\sqrt{d}} q}{{\sqrt{a}}}
     , \frac{a d}{c}, \frac{a d}{b}, a, b, c}; q, {\displaystyle
    \frac{q}{a}}\right) \\
\longrightarrow 
     \frac {( {d q}/{b c},
     {a d}/{c}, {a d}/{b}, d q ;q) _\infty} {(
      {a d}/{b c}, {d q}/{c},
     {d q}/{b}, a d ;q) _\infty}
{} _{4} \phi _{3} \! \left [          
    \matrix  a, b, c, {b c}/{d}\\  {b q}/{a}, {c q}/{a},
     {b c q}/{a d}\endmatrix ;q, {\displaystyle \frac{q^
     2}{a^ 2}} \right ] 
\endmultline$$
provided at least one of $a,b,c$ is of the form $q^-n$, $n=0,1,2,\dots$.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T121161
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.25, reversed) 
in form of a rule.
$$\align
{} _{12} W& _{11} ({\displaystyle a; {\sqrt{b}}, -{\sqrt{b}}, {\sqrt{b}}
   {\sqrt{q}}, - {\sqrt{b}} {\sqrt{q}}  , c, d, {{a^2 q}\over
   {b c d}}, e, {{a^2 q}\over {b e}}}; q, {\displaystyle q}) \\&\longrightarrow  
  { {(\let \over / a q, {{a^2 q}\over {b e}}, {{a^2 q}\over b^2}, {{a
     q}\over {b e}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a
     q}\over e}, {{a^2 q}\over b}, {{a^2 q}\over {b^2 e}} ;q) _\infty} }
      {} _{5} \phi _{4} \! \left [ \matrix \let \over / b, {{b c}\over a}, {{b
     d}\over a}, {{a q}\over {c d}}, e\\ \let \over / {{a q}\over c}, {{a
     q}\over d}, {{b c d}\over a}, {{b^2 e}\over a^2}\endmatrix ;q,
     {\displaystyle q} \right ] \\&\hskip2cm+ 
   { {(\let \over / a q, {{a q}\over {b e}}, b, {{b c}\over a}, {{b d}\over
     a}, {{a q}\over {c d}}, e, {{a^3 q^2}\over {b^2 c e}}, {{a^3
     q^2}\over {b^2 d e}}, {{a c d q}\over {b e}} ;q) _\infty} \over
     {(\let \over / {{a q}\over b}, {{a q}\over e}, {{a^2 q}\over b}, {{a
     q}\over c}, {{a q}\over d}, {{b c d}\over a}, {{b^2 e}\over {a^2 q}},
     {{a c q}\over {b e}}, {{a d q}\over {b e}}, {{a^3 q^2}\over {b^2 c
     d e}} ;q) _\infty} }\\&\hskip4cm
 {} _{5} \phi _{4} \! \left [ \matrix \let \over /
     {{a^2 q}\over b^2}, {{a^2 q}\over {b e}}, {{a c q}\over {b e}},
     {{a d q}\over {b e}}, {{a^3 q^2}\over {b^2 c d e}}\\ \let \over /
     {{a^2 q^2}\over {b^2 e}}, {{a^3 q^2}\over {b^2 c e}}, {{a^3
     q^2}\over {b^2 d e}}, {{a c d q}\over {b e}}\endmatrix ;q,
     {\displaystyle q} \right ] \\&\hskip2cm- 
   { {(\let \over / a q, {{a q}\over {b e}}, b, e, c, d, {{a^2 q}\over {b c
     d}}, {{a^2 q^2}\over {b c e}}, {{a^2 q^2}\over {b d e}}, {{c d
     q}\over e}, {{{a^4} q^3}\over {b^3 e^2}} ;q) _\infty} \over {(\let
     \over / {{a q}\over b}, {{a q}\over e}, {{a^2 q}\over b}, {{a q}\over
     c}, {{a q}\over d}, {{b c d}\over a}, {{a c q}\over {b e}}, {{a d q}\over
     {b e}}, {{a^3 q^2}\over {b^2 c d e}}, {{b e}\over {a q}}, {{a^3
     q^3}\over {b^2 e^2}} ;q) _\infty} } \\&\hskip4cm
    {} _{12} W _{11} ({\displaystyle {{a^3 q^2}\over {b^2 e^2}}; {{a
     q}\over {{\sqrt{b}} e}}, -{{a q}\over {{\sqrt{b}} e}}, {{a {q^{{3\over
     2}}}}\over {{\sqrt{b}} e}}, -{{a {q^{{3\over 2}}}}\over {{\sqrt{b}} e}},
     {{a q}\over b}, {{a^2 q}\over {b e}}, {{a c q}\over {b e}}, {{a d
     q}\over {b e}}, {{a^3 q^2}\over {b^2 c d e}}}; q, {\displaystyle
     q})
\endalign$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name TeX
\Description Switch that changes the output of TeXForm to be usable with
   Plain-{\TeX} and \LaTeX. By default the output of TeXForm is usable with 
   \AmSTeX.
\Usage TeX.
\Example 
\MATH
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AMSTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= TeXForm[ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[2]//TeXForm= 
\MATHlbrace \MATHrbrace  \MATHtief \MATHlbrace 2\MATHrbrace  %
\MATHbackslash phi \MATHtief \MATHlbrace 1\MATHrbrace  \MATHbackslash %
! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let %
\MATHbackslash over / a, b\MATHbackslash \MATHbackslash  \MATHbackslash let %
\MATHbackslash over /
 c\MATHbackslash endmatrix ;q, \MATHlbrace \MATHbackslash displaystyle %
z\MATHrbrace  \MATHbackslash right ]
\goodbreakpoint%
In[3]:= TeX
\goodbreakpoint%
In[4]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[5]:= TeXForm[ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]]
\goodbreakpoint%
Out[5]//TeXForm= 
\MATHlbrace \MATHrbrace  \MATHtief \MATHlbrace 2\MATHrbrace  %
\MATHbackslash phi \MATHtief \MATHlbrace 1\MATHrbrace  \MATHbackslash %
! \MATHbackslash left [ \MATHbackslash matrix \MATHlbrace  %
\MATHbackslash let \MATHbackslash over / a, b\MATHbackslash cr %
\MATHbackslash let \MATHbackslash over / c\MATHrbrace 
 ;q, \MATHlbrace \MATHbackslash displaystyle z\MATHrbrace  %
\MATHbackslash right ]
\endMATH 
\Seealso AmSTeX, AmSLaTeX, LaTeX, TeXMat, TeXphW.

\Name TeXMat
\Description Function that writes (to be precise: appends) an expression 
  \hbox{\tt Expr} in \hbox{\tt InputForm} to a file \hbox{\tt [name].m} and the
\hbox{\tt TeXForm} of \hbox{\tt Expr} to the 
  file \hbox{\tt [name].tex}.
 The expressions are numbered automatically. The 
   number can be reset by \hbox{\tt SchreibeZahl}. The string
\hbox{\tt comment} is 
   optional. It allows to place the comment \hbox{\tt comment} above the
expression and the number
   in each of the two files.
\Usage: TeXMat[Expr,name,comment].
\Example
\MATH
In[1]:= TeXMat[pq[a,2*n,q\MATHhoch 3],filename]
\goodbreakpoint%
In[2]:= TeXMat[ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z],filename,"A basic hypergeometric \MATHtief 2\MATHbackslash phi\MATHtief 1 
        series"]
\goodbreakpoint%
In[3]:= !type filename.m
A[1]:=
pq[a, 2*n, q\MATHhoch 3]
"A basic hypergeometric \MATHtief 2\MATHbackslash \MATHbackslash phi\MATHtief 1  series"
A[2]:=
ph[%
\MATHlbrace a, b%
\MATHrbrace , %
\MATHlbrace c%
\MATHrbrace , q, z]
\goodbreakpoint%
In[3]:= !type filename.tex
A[1]:=
(%
\MATHlbrace \MATHbackslash let \MATHbackslash over / a%
\MATHrbrace ; %
\MATHlbrace q\MATHhoch 3%
\MATHrbrace ) \MATHtief %
\MATHlbrace 2\MATHbackslash ,n%
\MATHrbrace  
"A basic hypergeometric \MATHtief 2\MATHbackslash \MATHbackslash phi\MATHtief 1  series"
A[2]:=
%
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  \MATHbackslash phi \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash  \MATHbackslash let \MATHbackslash over /
  c\MATHbackslash endmatrix ;q, %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ]
\endMATH
\Seealso AmSTeX, AmSLaTeX, LaTeX, TeX, TeXphW, SchreibeZahl.

\Name TeXphW
\Description Switch that toggles between writing very well-poised basic
  hypergeometric series in terms of \hbox{\tt W} and in terms of \hbox{\tt ph}, respectively,
  when written in TeXForm. By default very well-poised hypergeometric series 
  are written in terms of \hbox{\tt W}.
\Usage TeXphW.
\Example 
\MATH 
In[1]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= ph[\MATHlbrace a,q*Sqrt[a],-q*Sqrt[a],b,c\MATHrbrace ,\MATHlbrace %
Sqrt[a],-Sqrt[a],a*q/b,a*q/c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck  a, Sqrt[a] q, -(Sqrt[a] q), b, c       \MATHruEck 
           \MATHvStrich                                         \MATHvStrich 
Out[2]=  \MATHphi  \MATHvStrich                       a q  a q   ; q, z \MATHvStrich 
        5 4\MATHvStrich    Sqrt[a], -Sqrt[a], ---, ---          \MATHvStrich 
           \MATHloEck                        b    c           \MATHroEck 
\goodbreakpoint%
In[3]:= TeXForm[\%]
\goodbreakpoint%
Out[3]//TeXForm= 
\MATHlbrace \MATHrbrace  \MATHtief \MATHlbrace 5\MATHrbrace  W \MATHtief %
\MATHlbrace 4\MATHrbrace  (\MATHlbrace \MATHbackslash displaystyle a; b, %
c\MATHrbrace ; q, \MATHlbrace \MATHbackslash displaystyle z\MATHrbrace )
\goodbreakpoint%
In[4]:= TeXphW
\goodbreakpoint%
In[5]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses ph[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[6]:= TeXForm[\%2]
\goodbreakpoint%
Out[6]//TeXForm= 
\MATHlbrace \MATHrbrace  \MATHtief \MATHlbrace 5\MATHrbrace  \MATHbackslash %
phi \MATHtief \MATHlbrace 4\MATHrbrace  \MATHbackslash ! \MATHbackslash %
left [ \MATHbackslash matrix \MATHbackslash let \MATHbackslash over / a, %
\MATHlbrace \MATHbackslash sqrt\MATHlbrace a\MATHrbrace \MATHrbrace  q, %
-\MATHbackslash left(
 \MATHlbrace \MATHbackslash sqrt\MATHlbrace a\MATHrbrace \MATHrbrace  %
q \MATHbackslash right) , b, c\MATHbackslash \MATHbackslash  \MATHbackslash %
let \MATHbackslash over / \MATHlbrace \MATHbackslash sqrt\MATHlbrace a\MATHrbrace %
\MATHrbrace , -\MATHlbrace \MATHbackslash sqrt\MATHlbrace a\MATHrbrace %
\MATHrbrace , \MATHlbrace \MATHlbrace a
 q\MATHrbrace \MATHbackslash over b\MATHrbrace , \MATHlbrace %
\MATHlbrace a q\MATHrbrace \MATHbackslash over c\MATHrbrace \MATHbackslash %
endmatrix ;q, \MATHlbrace \MATHbackslash displaystyle z\MATHrbrace  %
\MATHbackslash right ]
\goodbreakpoint%
In[7]:= TeXphW
\goodbreakpoint%
In[8]:= hypqAttributes
\goodbreakpoint%
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\endMATH 
\Seealso ph, W, hypqAttributes.

\Name Tgl2101
\Description Transformation formula (\cite{\GaRaAA}, (1.4.1); Appendix (III.1)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2101}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2102
\Description Transformation formula (\cite{\GaRaAA}, (1.4.5); Appendix (III.2)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2102}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2103
\Description Transformation formula (\cite{\GaRaAA}, (1.4.6); Appendix (III.3)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2103}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2104
\Description Transformation formula (\cite{\GaRaAA}, (1.5.4); Appendix (III.4)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2104}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2105
\Description Transformation formula (\cite{\GaRaAA}, (3.2.4),
nonterminating; Appendix (III.5)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2105}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2106
\Description Transformation formula (\cite{\GaRaAA}, (1.5.6); Appendix (III.6)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2106}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2107
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii); Appendix (III.7)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2107}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2108
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii); Appendix (III.8)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2108}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2109
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.2, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T2109}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2110
\Description Transformation formula (\cite{\GaRaAA}, (3.4.7)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T2110}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2111
\Description Transformation formula (\cite{\GaRaAA}, (3.5.4)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T2111}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2112
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(i), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T2112}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2161
\Description Transformation formula (\cite{\GaRaAA}, (3.3.5); Appendix (III.31)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2161}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2162
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.32)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2162}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2163
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.8) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T2163}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2201
\Description Transformation formula (\cite{\GaRaAA}, (1.5.4), reversed; Appendix (III.4), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T2201}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2202
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T2202}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3101
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii), reversed; Appendix (III.8), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T3101}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3201
\Description Transformation formula (\cite{\GaRaAA}, (3.2.4), reversed; Appendix (III.5), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T3201}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3202
\Description Transformation formula (\cite{\GaRaAA}, (1.5.6), reversed; Appendix (III.6), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T3202}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3203
\Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii),
reversed; Appendix (III.7), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T3203}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3204
\Description Transformation formula (\cite{\GaRaAA}, (3.2.7); Appendix (III.9)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3204}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3205
\Description Transformation formula (\cite{\GaRaAA}, (3.2.10); Appendix (III.10)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3205}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3206
\Description Transformation formula (\cite{\GaRaAA}, (3.2.3); Appendix (III.11)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3206}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3207
\Description Transformation formula (\cite{\GaRaAA}, (3.2.2); Appendix (III.12)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3207}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3208
\Description Transformation formula (\cite{\GaRaAA}, (3.2.5); Appendix (III.13)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3208}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3209
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.26; Appendix (III.14)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3209}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3210
\Description Transformation formula (\cite{\GaRaAA}, (3.2.6)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3210}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3211
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.1) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3211}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3212
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.1, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3212}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3213
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(i)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3213}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3214
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3214}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3215
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.3) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3215}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3216
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.3, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3216}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3217
\Description Transformation formula (\cite{\GaRaAA}, (3.2.11)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3217}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3261
\Description Transformation formula (\cite{\GaRaAA}, (3.3.3); Appendix (III.33)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3261}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3262
\Description Transformation formula (\cite{\GaRaAA}, (3.3.1); Appendix (III.34)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3262}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3265
\Description Transformation formula (\cite{\GaRaAA}, (3.4.1); Appendix (III.35)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3265}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3266
\Description Transformation formula (\cite{\GaRaAA}, (3.5.2)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3266}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3267
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.6) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3267}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3268
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.6, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3268}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3269
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.8, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3269}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4201
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.4, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4201}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4301
\Description Transformation formula (\cite{\GaRaAA}, (2.10.4); Appendix (III.15)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4301}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4302
\Description Transformation formula (\cite{\GaRaAA}, (3.2.9); Appendix (III.16)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4302}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4303
\Description Transformation formula (\cite{\GaRaAA}, (2.5.1), reversed; Appendix (III.19)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4303}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4304
\Description Transformation formula (\cite{\GaRaAA}, (2.10.7); Appendix (III.20)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4304}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4305
\Description Transformation formula (\cite{\GaRaAA}, (3.10.13); Appendix (III.21)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4305}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4306
\Description Transformation formula (\cite{\GaRaAA}, (3.10.13), reversed; Appendix (III.21),
reversed) in form of an equation.
It is the same transformation as that in \hbox{\tt T4306}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4307
\Description Transformation formula (\cite{\GaRaAA}, (8.8.3); Appendix (III.22)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4307}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4308
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.2) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4308}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4309
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(i)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4309}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4310
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4310}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4311
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4311}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4312
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.4) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4312}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4313
\Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4313}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4361
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T4361}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5401
\Description Transformation formula (\cite{\GaRaAA}, (2.8.3); Appendix (III.25)) in form of an equation.
It is the same transformation as that in \hbox{\tt T5401}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5402
\Description Transformation formula (\cite{\GaRaAA}, (2.8.4); Appendix (III.26)) in form of an equation.
It is the same transformation as that in \hbox{\tt T5402}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5403
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5403}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5404
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.26, Appendix
(III.14), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5404}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5405
\Description Transformation formula (\cite{\GaRaAA}, (8.8.3), Appendix
(III.22), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5405}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5461
\Description Transformation formula (\cite{\GaRaAA}, (3.4.4)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5461}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5462
\Description Transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, first form) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5462}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5463
\Description Transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, second form) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5463}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5464
\Description Transformation formula (\cite{\GaRaAA}, (3.5.2), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5464}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5465
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5465}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5466
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, first form) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5466}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5467
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, second form) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5467}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5468
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.25) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T5468}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl6501
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T6501}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7601
\Description Transformation formula (\cite{\GaRaAA}, (2.8.5); Appendix (III.27)) in form of an equation.
It is the same transformation as that in \hbox{\tt T7601}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7701
\Description Transformation formula (\cite{\GaRaAA}, (3.2.11),
reversed) in form of an equation.
It is the same transformation as that in \hbox{\tt T7701}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8701
\Description Transformation formula (\cite{\GaRaAA}, (2.10.10), terminated; Appendix (III.17)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8701}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8702
\Description Transformation formula (\cite{\GaRaAA}, (2.5.1); Appendix (III.18)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8702}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8703
\Description Transformation formula (\cite{\GaRaAA}, (2.10.7), reversed; Appendix (III.20), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T8703}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8704
\Description Transformation formula (\cite{\GaRaAA}, (2.10.1); Appendix (III.23)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8704}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8705
\Description Transformation formula (\cite{\GaRaAA}, (2.10.1), iterated; Appendix (III.24)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8705}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8706
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8706}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8707
\Description Transformation formula (\cite{\GaRaAA}, (3.4.7), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8707}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8708
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8708}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8709
\Description Transformation formula (\cite{\GaRaAA}, (3.5.4), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8709}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8710
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8710}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8711
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8711}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8761
\Description Transformation formula (\cite{\GaRaAA}, (2.10.10); Appendix (III.36)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8761}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8762
\Description Transformation formula (\cite{\GaRaAA}, (2.11.1); Appendix (III.37)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8762}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8763
\Description Transformation formula (\cite{\GaRaAA}, (2.11.1),
reversed; Appendix (III.37)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8763}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8764
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.15) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T8764}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8810
\Description Transformation formula (\cite{\GaRaAA}, (5.6.1); Appendix (III.38)) in form of an equation.
It is the same transformation as that in \hbox{\tt T8810}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10901
\Description Transformation formula (\cite{\GaRaAA}, (2.9.1); Appendix (III.28)) in form of an equation.
It is the same transformation as that in \hbox{\tt T10901}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10902
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(i), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10902}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10903
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.19) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10903}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10904
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10904}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10905
\Description Transformation formula (\cite{\GaRaAA}, Ex.~3.21(iii)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10905}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10906
\Description Transformation formula (\cite{\RaVeAA}, (7.7), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10906}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10907
\Description Transformation formula (\cite{\RaVeAA}, (7.8), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10907}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10961
\Description Transformation formula (\cite{\GaRaAA}, (2.12.9); Appendix (III.39)) in form of an equation.
It is the same transformation as that in \hbox{\tt T10961}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10962
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10962}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl10963
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.30) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T10963}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl101010
\Description Transformation formula (\cite{\GaRaAA}, (5.6.3); Appendix (III.40)) in form of an equation.
It is the same transformation as that in \hbox{\tt T101010}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121101
\Description Transformation formula (\cite{\GaRaAA}, (2.8.3), reversed; Appendix (III.25), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T121101}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121102
\Description Transformation formula (\cite{\GaRaAA}, (2.8.4), reversed; Appendix (III.26), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T121102}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121103
\Description Transformation formula (\cite{\GaRaAA}, (2.8.5), reversed; Appendix (III.27), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T121103}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121104
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T121104}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121105
\Description Transformation formula (\cite{\RaVeAA}, (7.7)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T121105}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121106
\Description Transformation formula (\cite{\RaVeAA}, (7.8)) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T121106}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121107
\Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T121107}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl121161
\Description Transformation formula (\cite{\GaRaAA}, Ex.~2.25, reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T121161}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tli2101
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.4.1); Appendix (III.1))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2101}.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2 1%
\MATHvStrich   c         %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.Tli2101
\goodbreakpoint%
               %
\MATHluEck             %
\MATHruEck     (a, b z; q)     %
\MATHluEck  c          %
\MATHruEck 
               %
\MATHvStrich  a, b       %
\MATHvStrich                \MATHinfty     %
\MATHvStrich  -, z       %
\MATHvStrich 
Out[2]= %
\MATHlbrace 1,  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich , %
\MATHlbrace %
\MATHlbrace ------------  \MATHphi  %
\MATHvStrich  a   ; q, a %
\MATHvStrich %
\MATHrbrace , 
            2 1%
\MATHvStrich   c         %
\MATHvStrich      (c, z; q)   2 1%
\MATHvStrich             %
\MATHvStrich 
               %
\MATHloEck             %
\MATHroEck               \MATHinfty      %
\MATHloEck  b z        %
\MATHroEck 
 
                                   (b, a z; q)     %
\MATHluEck  c          %
\MATHruEck 
                                              \MATHinfty     %
\MATHvStrich  -, z       %
\MATHvStrich 
\MATHgroesser      %
\MATHlbrace phPerm[2, 1, u], T2101%
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace %
\MATHlbrace ------------  \MATHphi  %
\MATHvStrich  b   ; q, b %
\MATHvStrich %
\MATHrbrace , T2101%
\MATHrbrace %
\MATHrbrace 
                                    (c, z; q)   2 1%
\MATHvStrich             %
\MATHvStrich 
                                             \MATHinfty      %
\MATHloEck  a z        %
\MATHroEck 

\endMATH
The first entry in this list counts the number of
iterations of {\tt Tli}-rules (cf\. {\tt Tli2104}), 
the second entry displays the
original expression to which the {\tt Tli}-rules are applied. The subsequent
entries of the list always display an expression 
together with the sequence of rules that have to be applied to obtain
this expression from the original expression. For instance, the above
list says that the number of iterations is 1, the original expression
is ${}_2\phi_1\!\[\matrix a,b\\c\endmatrix; q, z\]$, and (if the
parameters of the original series are permuted) by the
application of {\tt T2101} two different expressions can be
obtained from the original series. The first of them is obtained by
first permuting the upper parameters by {\tt phPerm[2,1,u]} and then
applying {\tt T2101}.
\MATH
\goodbreakpoint%
In[3]:= \%1/.phPerm[2,1,u]/.T2101
\goodbreakpoint%
        (a, b z; q)     %
\MATHluEck  c          %
\MATHruEck 
                   \MATHinfty     %
\MATHvStrich  -, z       %
\MATHvStrich 
Out[3]= ------------  \MATHphi  %
\MATHvStrich  a   ; q, a %
\MATHvStrich 
         (c, z; q)   2 1%
\MATHvStrich             %
\MATHvStrich 
                  \MATHinfty      %
\MATHloEck  b z        %
\MATHroEck 

\endMATH
For an example of how to iterate {\tt Tli}-rules see {\tt Tli2104}.
\Seealso Tli2104, TransListe, Gleichung.

\Name Tli2102
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.4.5); Appendix (III.2))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2102}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2103
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.4.6); Appendix (III.3))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2103}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2104
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.5.4); Appendix (III.4))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2104}.
\Example
Here we demonstrate the iterated application of {\tt Tli}-rules.
We continue the {\tt Tli2101}-example. For information of how to
read the resulting listing confer {\tt Tli2101}.
\MATH
\goodbreakpoint%
In[4]:= \%2/.Tli2104
\goodbreakpoint%
               %
\MATHluEck             %
\MATHruEck      (a z; q)     %
\MATHluEck   c             %
\MATHruEck 
               %
\MATHvStrich  a, b       %
\MATHvStrich              \MATHinfty     %
\MATHvStrich   -, a          %
\MATHvStrich 
Out[4]= %
\MATHlbrace 2,  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich , %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace ---------  \MATHphi  %
\MATHvStrich   b    ; q, b z %
\MATHvStrich %
\MATHrbrace , T2101%
\MATHrbrace , 
            2 1%
\MATHvStrich   c         %
\MATHvStrich       (z; q)   2 2%
\MATHvStrich                 %
\MATHvStrich 
               %
\MATHloEck             %
\MATHroEck             \MATHinfty      %
\MATHloEck  a z, c         %
\MATHroEck 
\goodbreakpoint%
                 (b z; q)     %
\MATHluEck   c             %
\MATHruEck 
                         \MATHinfty     %
\MATHvStrich   -, b          %
\MATHvStrich 
\MATHgroesser      T2104%
\MATHrbrace , %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace ---------  \MATHphi  %
\MATHvStrich   a    ; q, a z %
\MATHvStrich %
\MATHrbrace , %
\MATHlbrace phPerm[2, 1, u], T2101%
\MATHrbrace %
\MATHrbrace , 
                  (z; q)   2 2%
\MATHvStrich                 %
\MATHvStrich 
                        \MATHinfty      %
\MATHloEck  b z, c         %
\MATHroEck 
\goodbreakpoint%
                 (a z, b z; q)     %
\MATHluEck     a b z       %
\MATHruEck 
                              \MATHinfty     %
\MATHvStrich  z, -----       %
\MATHvStrich 
\MATHgroesser      T2104%
\MATHrbrace , %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace --------------  \MATHphi  %
\MATHvStrich       c  ; q, c %
\MATHvStrich %
\MATHrbrace , 
                   (c, z; q)    2 2%
\MATHvStrich                 %
\MATHvStrich 
                            \MATHinfty       %
\MATHloEck  b z, a z       %
\MATHroEck 
\goodbreakpoint%
\MATHgroesser       %
\MATHlbrace phPerm[2, 1, u], T2101%
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace phPerm[2, 1, u], T2104%
\MATHrbrace %
\MATHrbrace , 
\goodbreakpoint%
        (b z, a z; q)     %
\MATHluEck     a b z       %
\MATHruEck 
                     \MATHinfty     %
\MATHvStrich  z, -----       %
\MATHvStrich 
\MATHgroesser     %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace --------------  \MATHphi  %
\MATHvStrich       c  ; q, c %
\MATHvStrich %
\MATHrbrace , T2101%
\MATHrbrace , 
          (c, z; q)    2 2%
\MATHvStrich                 %
\MATHvStrich 
                   \MATHinfty       %
\MATHloEck  a z, b z       %
\MATHroEck 
 
\MATHgroesser      %
\MATHlbrace phPerm[2, 1, u], T2104%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[5]:= \%1/.phPerm[2,1,u]/.T2101/.phPerm[2,1,u]/.T2104
\goodbreakpoint%
        (a z, b z; q)     %
\MATHluEck     a b z       %
\MATHruEck 
                     \MATHinfty     %
\MATHvStrich  z, -----       %
\MATHvStrich 
Out[5]= --------------  \MATHphi  %
\MATHvStrich       c  ; q, c %
\MATHvStrich 
          (c, z; q)    2 2%
\MATHvStrich                 %
\MATHvStrich 
                   \MATHinfty       %
\MATHloEck  b z, a z       %
\MATHroEck 
\endMATH
\Seealso Tli2101, TransListe, Gleichung.

\Name Tli2105
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.4),
nonterminating; Appendix (III.5))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2105}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2106
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.5.6); Appendix (III.6))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2106}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2107
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii); Appendix (III.7))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2107}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2108
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii); Appendix (III.8))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2108}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2109
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.2, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2109}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2110
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.7))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2110}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2111
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.4))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2111}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2112
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.2(i), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2112}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2161
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.3.5); Appendix (III.31))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2161}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2162
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Appendix (III.32))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2162}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2163
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.8)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2163}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2201
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.5.4), reversed; Appendix (III.4), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2201}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli2202
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T2202}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3101
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii), reversed; Appendix (III.8), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3101}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3201
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.4), reversed; Appendix (III.5), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3201}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3202
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (1.5.6), reversed; Appendix (III.6), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3202}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3203
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii),
reversed; Appendix (III.7), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3203}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3204
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.7); Appendix (III.9))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3204}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3205
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.10); Appendix (III.10))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3205}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3206
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.3); Appendix (III.11))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3206}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3207
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.2); Appendix (III.12))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3207}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3208
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.5); Appendix (III.13))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3208}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3209
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.26; Appendix (III.14))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3209}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3210
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.6))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3210}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3211
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.1)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3211}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3212
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.1, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3212}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3213
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.2(i))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3213}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3214
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3214}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3215
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.3)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3215}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3216
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.3, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3216}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3217
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.11))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3217}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3261
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.3.3); Appendix (III.33))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3261}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3262
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.3.1); Appendix (III.34))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3262}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3265
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.1); Appendix (III.35))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3265}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3266
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.2))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3266}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3267
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.6)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3267}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3268
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.6, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3268}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli3269
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.8, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T3269}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4201
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.4, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4201}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4301
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.4); Appendix (III.15))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4301}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4302
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.9); Appendix (III.16))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4302}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4303
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.5.1), reversed; Appendix (III.19))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4303}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4304
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.7); Appendix (III.20))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4304}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4305
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.10.13); Appendix (III.21))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4305}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4306
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.10.13), reversed; Appendix (III.21),
reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4305}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4307
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (8.8.3); Appendix (III.22))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4306}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4308
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.2)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4308}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4309
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.13(i))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4309}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4310
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4310}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4311
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.8))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4311}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4312
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.4)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4312}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4313
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~8.15)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4313}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli4361
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.16)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T4361}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5401
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.8.3); Appendix (III.25))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5401}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5402
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.8.4); Appendix (III.26))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5402}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5403
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.10.4), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5403}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5404
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.26, Appendix (III.14),
reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5404}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5405
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (8.8.3), Appendix (III.22),
reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5405}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5461
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.4))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5461}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5462
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, first form)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5462}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5463
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, second form)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5463}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5464
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.2), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5464}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5465
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.7), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5465}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5466
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, first form)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5466}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5467
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, second form)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5467}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli5468
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.25)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T5468}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli6501
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T6501}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli7601
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.8.5); Appendix (III.27))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T7601}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli7701
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.2.11), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T7701}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8701
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.10), terminated; Appendix (III.17))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8701}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8702
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.5.1); Appendix (III.18))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8702}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8703
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.7), reversed; Appendix (III.20), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8703}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8704
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.1); Appendix (III.23))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8704}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8705
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.1), iterated; Appendix (III.24))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8705}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8706
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8706}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8707
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.7), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8707}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8708
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.4.8), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8708}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8709
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.4), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8709}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8710
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.10))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8710}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8711
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.10), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8711}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8761
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.10.10); Appendix (III.36))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8761}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8762
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.11.1); Appendix (III.37))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8762}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8764
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.15)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8764}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli8810
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (5.6.1); Appendix (III.38))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T8810}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10901
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.9.1); Appendix (III.28))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10901}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10902
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.13(i), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10902}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10903
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.19)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10903}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10904
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.10.4))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10904}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10905
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~3.21(iii))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10905}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10906
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\RaVeAA}, (7.7), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10906}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10907
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\RaVeAA}, (7.8), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10907}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10961
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.12.9); Appendix (III.39))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10961}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10962
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (3.5.7))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10962}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli10963
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.30)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T10963}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli101010
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (5.6.3); Appendix (III.40))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T101010}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121101
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.8.3), reversed; Appendix (III.25), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121101}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121102
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.8.4), reversed; Appendix (III.26), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121102}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121103
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, (2.8.5), reversed; Appendix (III.27), reversed).
  of an equation
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121103}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121104
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii), reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121104}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121105
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\RaVeAA}, (7.7))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121105}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121106
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\RaVeAA}, (7.8))
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121106}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121107
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~8.15, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121107}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name Tli121161
\Description Rule that gives a list of all possible outcomes under
application of the 
transformation formula (\cite{\GaRaAA}, Ex.~2.25, reversed)
together with information of how to get them.
Also iterated use is possible.
The transformation is the same as that in \hbox{\tt T121161}.
\Seealso Tli2101, Tli2104, TransListe, Gleichung.

\Name TListe
\Description Rule that gives for a basic hypergeometric series a list of 
   applicable transformation formulas. Each entry of this list has the 
   format
\hbox{\tt $\{$ArgumentPermutations,T$\langle$number$\rangle$$\}$}, 
where \hbox{\tt ArgumentPermutations} 
   is a sequence of reorderings of the parameters of the basic hypergeometric
   series (given in terms of \hbox{\tt phPerm} and \hbox{\tt
phTausche}) and \hbox{\tt T$\langle$number$\rangle$} 
   is the name of the transformation in form of a rule which can be applied 
   subsequently. You should be aware that \hbox{\tt TListe} automatically applies
   \hbox{\tt phOrdne} before checking which transformation could be applied.
\vskip6pt
\hangafter1 \hangindent10pt\rm
\underbar{Important Note}: If the value returned by \hbox{\tt TListe}
is the empty set this does {\it not} mean that no transformation can be
applied.
You always must remember that the list of transformations included in
this package is a list of {\it basic} transformations. There are numerous
special cases of these transformations which are not contained in
this list as a separate transformation. The examples below should
illustrate these remarks.
\Usage Expr/.TListe.
\Example
\MATH
In[1]:= ph[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,q,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b       %
\MATHvStrich 
Out[1]=  \MATHphi  %
\MATHvStrich      ; q, z %
\MATHvStrich 
        2 1%
\MATHvStrich   c         %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.TListe
\goodbreakpoint%
Be sure to apply "phOrdne" before using the following information!
\goodbreakpoint%
Out[2]= %
\MATHlbrace %
\MATHlbrace T2101%
\MATHrbrace , %
\MATHlbrace T2102%
\MATHrbrace , %
\MATHlbrace T2103%
\MATHrbrace , %
\MATHlbrace T2104%
\MATHrbrace , %
\MATHlbrace T2161%
\MATHrbrace , %
\MATHlbrace T2162%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[3]:= ph[%
\MATHlbrace d,b\MATHhoch 2,q\MATHhoch -n,a\MATHhoch 2%
\MATHrbrace ,%
\MATHlbrace -a*b*Sqrt[q],a*b*Sqrt[q],-q\MATHhoch -n*d%
\MATHrbrace ,q,q]
\goodbreakpoint%
                           2   -n   2
           %
\MATHluEck            d, b , q  , a                  %
\MATHruEck 
           %
\MATHvStrich                                           %
\MATHvStrich 
Out[3]=  \MATHphi  %
\MATHvStrich                                 d  ; q, q %
\MATHvStrich 
        4 3%
\MATHvStrich  -(a b Sqrt[q]), a b Sqrt[q], -(--)       %
\MATHvStrich 
           %
\MATHloEck                                  n        %
\MATHroEck 
                                            q
\goodbreakpoint%
In[4]:= \%/.TListe
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
Be sure to apply "phOrdne" before using the following information!
\goodbreakpoint%
Out[4]= %
\MATHlbrace %
\MATHlbrace T4301%
\MATHrbrace , %
\MATHlbrace T4302%
\MATHrbrace , %
\MATHlbrace T4303%
\MATHrbrace , %
\MATHlbrace T4304%
\MATHrbrace , 
 
\MATHgroesser     %
\MATHlbrace phPerm[2,3,1,4,u], phPerm[3,1,2,l], T4305%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[5]:= \%3/.phOrdne/.phPerm[2,3,1,4,u]/.phPerm[3,1,2,l]/.T4305
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
                   2   2   2   -2 n
           %
\MATHluEck       b , a , d , q                  %
\MATHruEck 
           %
\MATHvStrich                                2   2 %
\MATHvStrich 
Out[5]=  \MATHphi  %
\MATHvStrich   2  2      d         1 - n ; q , q  %
\MATHvStrich 
        4 3%
\MATHvStrich  a  b  q, -(--), -(d q     )         %
\MATHvStrich 
           %
\MATHloEck              n                       %
\MATHroEck 
                        q
\endMATH
\vskip10pt\noindent
Now we consider two examples illustrating the note above. Though none of the 
implemented transformations can be applied, both series can be transformed,
the first by a limiting case of Watson's transformation, the second by a number
of specialized very well-poised $_8\phi_7$ transformations.
These facts are also observed by using this package.
\vskip10pt
\MATH
In[6]:= ph[%
\MATHlbrace a,q*Sqrt[a],-q*Sqrt[a],b,c,d,e%
\MATHrbrace ,%
\MATHlbrace Sqrt[a],-Sqrt[a],
        a*q/b,a*q/c,a*q/d,a*q/e,0%
\MATHrbrace ,q,a\MATHhoch 2*q\MATHhoch 2/(b*c*d*e)]
\goodbreakpoint%
           %
\MATHluEck   a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e        2  2  %
\MATHruEck 
           %
\MATHvStrich                                                a  q   %
\MATHvStrich 
Out[6]=  \MATHphi  %
\MATHvStrich                     a q  a q  a q  a q   ; q, ------- %
\MATHvStrich 
        7 7%
\MATHvStrich  Sqrt[a], -Sqrt[a], ---, ---, ---, ---, 0     b c d e %
\MATHvStrich 
           %
\MATHloEck                      b    c    d    e                 %
\MATHroEck 
\goodbreakpoint%
In[7]:= \%/.TListe
\goodbreakpoint%
Out[7]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[8]:= Tgl8702
Do you want to set values for the equation? [y|n]: n
Do you want to set a value for q in the equation? [y|n]: n
\goodbreakpoint%
           %
\MATHluEck                                             -n        2  2 + n
           %
\MATHvStrich    a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e, q         a  q
Out[8]=  \MATHphi  %
\MATHvStrich                                                 ; q, ---------
        8 7%
\MATHvStrich                     a q  a q  a q  a q     1 + n      b c d e
           %
\MATHloEck  Sqrt[a], -Sqrt[a], ---, ---, ---, ---, a q
                                 b    c    d    e
\goodbreakpoint% 
                 a q         -n               a q
       %
\MATHruEck        %
\MATHluEck  ---, d, e, q         %
\MATHruEck  (a q, ---; q)
       %
\MATHvStrich        %
\MATHvStrich  b c                  %
\MATHvStrich        d e    n
\MATHgroesser       %
\MATHvStrich  ==  \MATHphi  %
\MATHvStrich                ; q, q %
\MATHvStrich  --------------
       %
\MATHvStrich     4 3%
\MATHvStrich  a q  a q  d e        %
\MATHvStrich   a q  a q
       %
\MATHroEck        %
\MATHloEck  ---, ---, ----       %
\MATHroEck  (---, ---; q)
                  b    c      n           d    e     n
                           a q
\goodbreakpoint%
In[9]:= Limes[\%,n-\MATHgroesser Infinity]
\goodbreakpoint%
           %
\MATHluEck   a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e        2  2  %
\MATHruEck 
           %
\MATHvStrich                                                a  q   %
\MATHvStrich 
Out[9]=  \MATHphi  %
\MATHvStrich                     a q  a q  a q  a q   ; q, ------- %
\MATHvStrich  == 
        7 7%
\MATHvStrich  Sqrt[a], -Sqrt[a], ---, ---, ---, ---, 0     b c d e %
\MATHvStrich 
           %
\MATHloEck                      b    c    d    e                 %
\MATHroEck 
\goodbreakpoint% 
                            a q
                          %
\MATHluEck  ---, d, e         %
\MATHruEck 
               a q        %
\MATHvStrich  b c           a q %
\MATHvStrich 
     (a q;q)  (---;q)   \MATHphi  %
\MATHvStrich           ; q, --- %
\MATHvStrich 
            \MATHinfty   d e   \MATHinfty  3 2%
\MATHvStrich  a q  a q      d e %
\MATHvStrich 
                          %
\MATHloEck  ---, ---          %
\MATHroEck 
                             b    c
\MATHgroesser     ------------------------------------------
                  a q      a q
                 (---;q)  (---;q)
                   d    \MATHinfty    e    \MATHinfty 
\goodbreakpoint%
In[10]:= ph[%
\MATHlbrace q*Sqrt[a],-q*Sqrt[a],q,c,d,e,f%
\MATHrbrace ,%
\MATHlbrace Sqrt[a],-Sqrt[a],a*q/c,
         a*q/d,a*q/e,a*q/f%
\MATHrbrace ,q,a\MATHhoch 2*q\MATHhoch 2/(q*c*d*e*f)]
\goodbreakpoint%
            %
\MATHluEck  Sqrt[a] q, -(Sqrt[a] q), q, c, d, e, f       2     %
\MATHruEck 
            %
\MATHvStrich                                              a  q   %
\MATHvStrich 
Out[10]=  \MATHphi  %
\MATHvStrich                     a q  a q  a q  a q ; q, ------- %
\MATHvStrich 
         7 6%
\MATHvStrich  Sqrt[a], -Sqrt[a], ---, ---, ---, ---      c d e f %
\MATHvStrich 
            %
\MATHloEck                      c    d    e    f               %
\MATHroEck 
\goodbreakpoint%
In[11]:= \%/.TListe
\goodbreakpoint%
Out[11]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[12]:= \%\%/.phEinf
Add the parameter: a
\goodbreakpoint%
            %
\MATHluEck  a, Sqrt[a] q, -(Sqrt[a] q), q, c, d, e, f       2     %
\MATHruEck 
            %
\MATHvStrich                                                 a  q   %
\MATHvStrich 
Out[12]=  \MATHphi  %
\MATHvStrich                        a q  a q  a q  a q ; q, ------- %
\MATHvStrich 
         8 7%
\MATHvStrich  a, Sqrt[a], -Sqrt[a], ---, ---, ---, ---      c d e f %
\MATHvStrich 
            %
\MATHloEck                         c    d    e    f               %
\MATHroEck 
\goodbreakpoint%
In[13]:= \%/.TListe
\goodbreakpoint%
Be sure to apply "phOrdne" before using the following information!
\goodbreakpoint%
Out[13]= %
\MATHlbrace %
\MATHlbrace T8701%
\MATHrbrace , %
\MATHlbrace T8704%
\MATHrbrace , %
\MATHlbrace T8705%
\MATHrbrace , %
\MATHlbrace T8761%
\MATHrbrace , %
\MATHlbrace T8762%
\MATHrbrace %
\MATHrbrace 
\endMATH
\Seealso SListe, phPerm, phTausche, TransListe.

\Name trans
\Description $(a;q)_n \to (q^{1-n}/a;q)_n(-a)^nq^{\binom n2}$.
\Usage Expr/.trans.
\Example 
\MATH 
In[1]:= pq[a,n,q\MATHhoch 2]
\goodbreakpoint%
             2
Out[1]= (a; q )
               n
\goodbreakpoint%
In[2]:= \%/.trans
\goodbreakpoint%
                        2   2 - 2 n
            n  n  -n + n   q          2
Out[2]= (-1)  a  q        (--------; q )
                              a         n
\endMATH 
\Seealso Ers, PosListe, ManipulationsListe.

\Name TransListe
\Description List of all transformation formulas.
\Usage TransListe.
\Seealso TransListe\$gl, Liste.

\Name TransListe\$gl
\Description List of all transformation formulas.
\Usage TransListe\$gl.
\Seealso TransListe.


\Name W
\Description \hbox{\tt W[a,List,q,z]} is the very well-poised basic hypergeometric
series.
\Usage W[a,List,q,z].
\Example 
\MATH 
In[1]:= W[a,\MATHlbrace b,c\MATHrbrace ,q,z]
\goodbreakpoint%
           \MATHluEck  a, Sqrt[a] q, -(Sqrt[a] q), b, c       \MATHruEck 
           \MATHvStrich                                         \MATHvStrich 
Out[1]=  \MATHphi  \MATHvStrich                       a q  a q   ; q, z \MATHvStrich 
        5 4\MATHvStrich    Sqrt[a], -Sqrt[a], ---, ---          \MATHvStrich 
           \MATHloEck                        b    c           \MATHroEck
\endMATH 
\Seealso ph, TeXphW, PQ, phFormat.

\Name zerl
\Description \vtab $(a;q)_n \to (a;q)_m(aq^m;q)_{n-m}$,\\
           $(a;q)_\infty \to (a;q)_m(aq^m;q)_\infty$.
\endvtab
\vskip6pt
\leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.zerl.
\Example 
\MATH 
In[1]:= pq[a,2*n]
\goodbreakpoint%
Out[1]= (a; q)
              2 n
\goodbreakpoint%
In[2]:= \%/.zerl
bottom-split by: m+n
\goodbreakpoint%
                        m + n
Out[2]= (a; q)      (a q     ; q)
              m + n              -m + n
\endMATH 
\Seealso Ers, PosListe, ManipulationsListe.

\Name zus1
\Description \vtab $(a;q)_n(aq^n;q)_m \to (a;q)_{n+m}$,\\
           $(a;q)_n(aq^n;q)_\infty \to (a;q)_\infty$.
\endvtab
\Usage Expr/.zus1.
\Example 
\MATH 
In[1]:= pq[a,2*n]*pq[a*q\MATHhoch (2*n),m-n]
\goodbreakpoint%
                      2 n
Out[1]= (a; q)    (a q   ; q)
              2 n            m - n
\goodbreakpoint%
In[2]:= \%/.zus1
\goodbreakpoint%
Out[2]= (a; q)
              m + n
\goodbreakpoint%
In[3]:= pq[a,2*n]*pqinf[a*q\MATHhoch (2*n)]
\goodbreakpoint%
            2 n
Out[3]= (a q   ;q)  (a; q)
                  \MATHinfty        2 n
\goodbreakpoint%
In[4]:= \%/.zus1
\goodbreakpoint%
Out[4]= (a;q)
             \MATHinfty 
\endMATH 
\Seealso zus2, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

\Name zus2
\Description \vtab $(a;q)_n/(a;q)_m \to (aq^m;q)_{n-m}$,\\
           $(a;q)_\infty/(a;q)_m \to (aq^m;q)_\infty$.
\endvtab
\Usage Expr/.zus2.
\Example 
\MATH 
In[1]:= pq[a,m]/pq[a,n]*pq[b,m+n]
\goodbreakpoint%
        (a; q)  (b; q)
              m       m + n
Out[1]= -------------------
              (a; q)
                    n
\goodbreakpoint%
In[2]:= \%/.zus2
\goodbreakpoint%
                        n
Out[2]= (b; q)      (a q ; q)
              m + n          m - n
\goodbreakpoint%
In[3]:= pq[a,m]/pqinf[a]*pq[b,m+n]
\goodbreakpoint%
        (a; q)  (b; q)
              m       m + n
Out[3]= -------------------
              (a;q)
                   \MATHinfty 
\goodbreakpoint%
In[4]:= \%/.zus2
\goodbreakpoint%
        (b; q)
              m + n
Out[4]= -----------
             m
         (a q ;q)
                 \MATHinfty 
\endMATH 
\Seealso zus1, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

\Name zus3
\Description $(a;q)_n/(b;q)_m \to (a;q)_{n-m}$,
\vskip6pt
\leavevmode\hskip10pt  provided $aq^n=bq^m$, and
\vskip6pt
\leavevmode\hphantom{Description: }$(a;q)_\infty/(aq^n;q)_\infty \to
(a;q)_n$.
\Usage Expr/.zus3.
\Example 
\MATH 
In[1]:= pq[a*q\MATHhoch m,n]/pq[a*q\MATHhoch n,m]
\goodbreakpoint%
            m
        (a q ; q)
                 n
Out[1]= ----------
            n
        (a q ; q)
                 m
\goodbreakpoint%
In[2]:= \%/.zus3
\goodbreakpoint%
            m
Out[2]= (a q ; q)
                 -m + n
In[3]:= pqinf[a*q\MATHhoch n]/pqinf[a*q]
\goodbreakpoint%
            n
        (a q ;q)
                \MATHinfty 
Out[3]= ---------
        (a q;q)
               \MATHinfty 
\goodbreakpoint%
In[4]:= \%/.zus3
\goodbreakpoint%
              1
Out[4]= --------------
        (a q; q)
                -1 + n
\endMATH 
\Seealso zus1, zus2, erw1, erw2, Ers, PosListe, ManipulationsListe.

\Refs
\ref\no\GaRaAA \by G.~Gasper, M.~Rahman\book Basic hypergeometric series\publ
Encyclopedia of Mathematics And Its Applications~35, Cambridge
University Press\publaddr Cambridge\yr 1990\endref

\ref\no \KratAT\by C.    Krattenthaler \yr \book HYP and HYPQ --- 
{\sl Mathematica\/} packages for the manipulation of binomial sums and 
hypergeometric series, respectively $q$-binomial sums and basic 
hypergeometric series\endref

\ref\no \RaVeAA\by M.    Rahman and A. Verma \yr 1993 \paper Quadratic 
transformation formulas for basic hypergeometric series 
\jour Trans\. Amer\. Math\. Soc\. \vol 335 \pages 277--302\endref

\ref\no \WolfAA\by S. Wolfram \yr 1991
\book MATHEMATICA --- A system for doing
mathematics by computer\bookinfo second edition
\publ Addison--Wesley\publaddr New York\endref


\endRefs
\enddocument

