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\topmatter 
\title \fourteenpoint\bf HYP
\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 HYP}
\rightheadtext{The MATHEMATICA package \tenpoint\bf HYP}
\document


This is a MATHEMATICA package for handling  hypergeometric
series. It provides quite a few tools for
\roster
\item "(A)" manipulating factorial expressions
\item "(B)" transforming binomial sums into  hypergeometric
notation
\item "(C)" summing  hypergeometric series
\item "(D)" transforming  hypergeometric series
\item "(E)" applying contiguous relations
\item "(F)" doing formal limits of  hypergeometric
expressions
\item "(G)" transforming  hypergeometric MATHEMATICA expressions
into \TeX-code.
\item "(H)" applying Gosper's \cite{\GospAB} and Zeilberger's
\cite{\ZeilAM, \ZeilAN, \ZeilAV} algorithms
\endroster
The tools for items (A), (B), (F), (G) are contained in the file \hbox{\tt
hyp.m}, 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.m}.) The file \hbox{\tt hyp.m}
 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.m}, those for (D) are the contents of the
files \hbox{\tt transfor.m} and \hbox{\tt transfor.mli},
and those for (E) are the contents of the
file \hbox{\tt contig.m}. You also have access to summation and
transformation formulas in form of equations. This is the contents of
the files \hbox{\tt summatio.mgl} and \hbox{\tt transfor.mgl}, 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}. The tools for item (H) rely on Peter Paule
and Markus Schorn's {\sl Mathematica} implementation of Gosper's and 
Zeilberger's algorithms. 
The current version~1.1 or updates can be received via
e-mail request to \hbox{\tt peter.paule\@risc.uni-linz.ac.at}.

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  hypergeometric series is
required (cf\. \cite{\BailAA, sec.~1.1, 2.1; \SlatAC, sec.~1.1; 
\GaRaAA, pp.~1--6}). This handbook provides you
with a list of the rules, functions, summations, transformations that
are available. 
The main sources for identities that are included in
this package have been the 
books \cite{\GaRaAA, \SlatAC, \BailAA}. They contain a fairly comprehensive collection
of known summation and transformation formulas for 
hypergeometric series. In particular, the (almost) complete
Appendices of \cite{\SlatAC} and 
\cite{\GaRaAA} (for $q\uparrow 1$) are 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 Hypergeometric notation\endhead
All the notation and terminology is adopted from \cite{\GaRaAA, pp.~1--6}.
The {\it (generalized) hypergeometric series} is defined by
$${}_r F_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; 
z\right]=\sum _{n=0} ^{\infty}\frac {\po{a_1}{n}\cdots\po{a_r}{n}}
{n!\,\po{b_1}{n}\cdots\po{b_s}{n}} z^n\ ,$$
where the rising factorial
$(a)_n$ is given by $(a)_n:=a(a+1)\cdots(a+n-1)$,
$n\ge1$, $(a)_0:=1$. 
A hypergeometric series $_{r+1}F_r$ is called {\it very well-poised} if
$a_i+b_i=1+a_0$ for $i=1,2,\dots,r$,
and among the parameters $a_i$ occurs $1+a_0/2$. We
use the standard abbreviation for very well-poised hypergeometric
series,
$$_{r+1}V_r(a_0;a_2,a_3,\dots,a_r;z):={}_{r+1}F_r\!\[\matrix
a_0,1+a_0/2,a_2,a_3,\dots,a_r\\
a_0/2,1+a_0-a_2,1+a_0-a_3,\dots,1+a_0-a_r\endmatrix; z\].$$
The {\it bilateral hypergeometric series} is defined by
$${}_r H_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; 
z\right]=\sum _{n=-\infty} ^{\infty}\frac {\po{a_1}{n}\cdots\po{a_r}{n}}
{\po{b_1}{n}\cdots\po{b_s}{n}} z^n\ .$$
We also use the compact
Gasper-Rahman notation
$$ {(a_1,a_2,\dots,a_r)_n} :=(a_1)_n\,(a_2)_n\,\cdots\,(a_r)_n$$
and 
$$\Ga\bmatrix a_1,a_2,\dots,a_r\\b_1,b_2,\dots,b_s\endbmatrix:=
\frac {\Ga(a_1)\,\Ga(a_2)\,\cdots\,\Ga(a_r)} 
{\Ga(b_1)\,\Ga(b_2)\,\cdots\,\Ga(b_s)}\ .$$


\head The file \tt hyp.m\endhead
The objects which are defined in the file hyp.m 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 baszerl}, \hbox{\tt
baszus}, \hbox{\tt Binomialp}, 
 \hbox{\tt Div}, \hbox{\tt Drucke},
\hbox{\tt Ers}, \hbox{\tt erw1}, \hbox{\tt erw2}, \hbox{\tt Expandq},
\hbox{\tt F}, \hbox{\tt FH}, \hbox{\tt Factorialp}, \hbox{\tt FCancel},
\hbox{\tt FEinf}, \hbox{\tt FFormat}, \hbox{\tt FOrdne}, \hbox{\tt
FPerm}, \hbox{\tt FSUM},
\hbox{\tt FTausche}, \hbox{\tt GAMMA},
\hbox{\tt Gleichung}, \hbox{\tt GlTausche}, \hbox{\tt GOSPER},
\hbox{\tt Gzerl}, \hbox{\tt Gzus}, \hbox{\tt H}, \hbox{\tt HEinf}, 
\hbox{\tt HF}, \hbox{\tt Hoch}, \hbox{\tt HOrdne}, \hbox{\tt HPerm}, 
\hbox{\tt HShift}, \hbox{\tt HSUM}, \hbox{\tt
hypAttributes}, \hbox{\tt inv}, \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 Multinomialp}, 
\hbox{\tt neg1}, \hbox{\tt neg2}, 
\hbox{\tt p}, \hbox{\tt P}, \hbox{\tt paufl}, 
\hbox{\tt PosListe}, \hbox{\tt PSort},
\hbox{\tt pzerl}, \hbox{\tt pzus}, \hbox{\tt RS}, \hbox{\tt SchreibeZahl}, 
\hbox{\tt SimplifyP}, \hbox{\tt Sub}, \hbox{\tt Subst},
\hbox{\tt SUM},
\hbox{\tt SUMErw1}, \hbox{\tt SUMErw2}, \hbox{\tt SUMExpand}, 
\hbox{\tt SUMF}, \hbox{\tt SUMH}, 
\hbox{\tt SUMInfinity}, \hbox{\tt SUMRegeln},
\hbox{\tt SUMSammle}, \hbox{\tt SUMShift}, \hbox{\tt SUMTausche}, 
\hbox{\tt SUMUmkehr}, \hbox{\tt SUMZerl}, \hbox{\tt TeX}, \hbox{\tt TeXFV},
\hbox{\tt TeXMat}, 
\hbox{\tt trans}, \hbox{\tt V}, \hbox{\tt ZB},
\hbox{\tt zerl}, \hbox{\tt zus1},
\hbox{\tt zus2}, \hbox{\tt zus3}.  \par}
\medskip\noindent
These objects can be divided into 10 groups: There are the basic
objects,
\medskip\noindent
{\leftskip20pt\rightskip20pt\noindent
\hbox{\tt Binomialp}, \hbox{\tt F}, \hbox{\tt Factorialp},
\hbox{\tt GAMMA}, \hbox{\tt H}, \hbox{\tt Multinomialp}, 
\hbox{\tt p}, \hbox{\tt SUM}, \hbox{\tt V},\par}
\medskip\noindent
the rules for manipulating factorial expressions
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt baszerl}, \hbox{\tt baszus},
\hbox{\tt erw1}, \hbox{\tt erw2}, \hbox{\tt Expandq}, 
\hbox{\tt Gzerl}, \hbox{\tt Gzus}, \hbox{\tt inv},
\hbox{\tt lina1}, \hbox{\tt lina2}, \hbox{\tt linz}, \hbox{\tt MinusOne},
\hbox{\tt neg1}, \hbox{\tt neg2}, \hbox{\tt paufl}, 
\hbox{\tt pzerl}, \hbox{\tt pzus}, 
\hbox{\tt trans}, \hbox{\tt zerl}, \hbox{\tt zus1}, \hbox{\tt zus2},
\hbox{\tt zus3},  \par}
\medskip\noindent
the rules for manipulating sums and  hypergeometric series,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt FCancel}, \hbox{\tt FEinf}, \hbox{\tt FFormat}, \hbox{\tt FH}, 
\hbox{\tt FOrdne}, \hbox{\tt FSUM}, \hbox{\tt FPerm}, \hbox{\tt
FTausche}, \hbox{\tt HEinf}, \hbox{\tt HF}, \hbox{\tt HOrdne}, 
\hbox{\tt HPerm}, \hbox{\tt HShift}, \hbox{\tt HSUM}, 
\hbox{\tt SUMErw1}, \hbox{\tt SUMErw2}, \hbox{\tt SUMExpand}, 
\hbox{\tt SUMF}, \hbox{\tt SUMH},
\hbox{\tt SUMInfinity}, \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  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 hypergeometric expressions,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt SimplifyP},  \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 TeXFV}, \hbox{\tt TeXMat}, \par}
\medskip\noindent
two objects for on-line help,
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt hypAttributes}, \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.
Also 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 PSort}, \hbox{\tt RS}, \hbox{\tt Sub},\par}
\medskip\noindent
for manipulating equations and writing expressions in a ``normalized"
form (\hbox{\tt PSort}) in order to be able to quickly check if two
expressions agree. These objects are particularly important when
using objects from \hbox{\tt summatio.mgl} and\linebreak \hbox{\tt
transfor.mgl}. Finally there are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt GOSPER}, \hbox{\tt ZB}, \par}
\medskip\noindent
for applying Gosper's and Zeilberger's algorithms. Peter Paule's and
Markus Schorn's {\sl Mathematica} implementation of these algorithms
which is needed for using {\tt GOSPER} and {\tt ZB} also provides
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Zb}, \hbox{\tt Gosper}, \hbox{\tt RunMode}, \hbox{\tt FileName}, 
\hbox{\tt SolAmount},
\hbox{\tt Fnk}, \hbox{\tt GoRat}, \hbox{\tt GoSol}, \hbox{\tt Cert}, 
  \hbox{\tt DegBound}, \hbox{\tt System}, \hbox{\tt SystemDimension}.
\par}
\medskip\noindent
Regarding these objects 
the user is referred to the documentation and the description
\cite{\PaScAA} of this implementation.


\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.m\endhead
This file basically contains the Appendix~II of \cite{\GaRaAA} for
$q\uparrow1$ 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.m} are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt S1001}, \hbox{\tt S2101},
\hbox{\tt S2103}, \hbox{\tt S2104}, \hbox{\tt S2105}, \hbox{\tt S2106}, 
\hbox{\tt S2131}, \hbox{\tt S2132}, \hbox{\tt S2210}, \hbox{\tt S2240},
\hbox{\tt S3201}, \hbox{\tt S3202}, \hbox{\tt S3204}, 
\hbox{\tt S3231}, \hbox{\tt S3232},
\hbox{\tt S3233}, \hbox{\tt S3234}, \hbox{\tt S3235}, 
\hbox{\tt S3261}, \hbox{\tt S3291}, \hbox{\tt S3340},
\hbox{\tt S4306}, \hbox{\tt S4307}, \hbox{\tt S4331}, \hbox{\tt S4332}, 
\hbox{\tt S5431}, \hbox{\tt S5432}, \hbox{\tt S5540},
\hbox{\tt S6531}, \hbox{\tt S6532}, \hbox{\tt S7631}, \hbox{\tt S7632}, 
\hbox{\tt S7691}, \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 hypergeometric
series to which the rule applies. The number $\langle$n1$\rangle$ allows to
distinguish the rules applying to 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 and its
$q$-analogue (in the package HYPQ) has the same numbering, it is within
the range 31--60 if the summation is a one-term summation and its
$q$-analogue (in the package HYPQ) has a different numbering, it is within
the range 61--90 if the summation is a two- or more-term summation 
and its
$q$-analogue (in the package HYPQ) has the same numbering, it
is within the range 90--120 if the summation is a two- or more-term 
summation and its
$q$-analogue (in the package HYPQ) has a different numbering.

For terminating series there is a check if one of the parameters is
of the form $(-n)$ 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 FSUM}, or in case
that automatic evaluating is active (cf\. \hbox{\tt P}). The rule
\hbox{\tt SListe} enables you to quickly check if one of the
summation rules \hbox{\tt S1001}--\hbox{\tt S7691} can be directly applied.



\head The file \tt transfor.m\endhead
This file basically contains the Appendix~III of \cite{\GaRaAA} for
$q\uparrow1$ 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.m} are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt T2103}, \hbox{\tt T2104},
\hbox{\tt T2106}, \hbox{\tt T2107}, \hbox{\tt T2110}, 
\hbox{\tt T2112}, \hbox{\tt T2131}, \hbox{\tt T2132}, 
\hbox{\tt T2133}, \hbox{\tt T2134}, 
\hbox{\tt T2135}, \hbox{\tt T2136}, \hbox{\tt T2137}, 
\hbox{\tt T2138}, \hbox{\tt T2139}, \hbox{\tt T2139}, 
\hbox{\tt T2140}, \hbox{\tt T2141}, \hbox{\tt T2163}, 
\hbox{\tt T2191}, \hbox{\tt T2192}, 
\hbox{\tt T3204}, \hbox{\tt T3205},
\hbox{\tt T3206}, \hbox{\tt T3207}, \hbox{\tt T3217}, 
\hbox{\tt T3231}, \hbox{\tt T3232}, \hbox{\tt T3233}, 
\hbox{\tt T3234}, \hbox{\tt T3235}, \hbox{\tt T3236}, 
\hbox{\tt T3237}, \hbox{\tt T3238}, 
\hbox{\tt T3239}, \hbox{\tt T3240}, 
\hbox{\tt T3261}, \hbox{\tt T3262}, \hbox{\tt T3263}, 
\hbox{\tt T3264}, 
\hbox{\tt T4301}, \hbox{\tt T4302}, \hbox{\tt T4303}, \hbox{\tt T4304},
\hbox{\tt T4306}, \hbox{\tt T4309}, \hbox{\tt T4310}, 
\hbox{\tt T4312}, \hbox{\tt T4313}, \hbox{\tt T4331}, \hbox{\tt T4332}, 
\hbox{\tt T4362}, \hbox{\tt T4391}, 
\hbox{\tt T5401}, \hbox{\tt T5402}, \hbox{\tt T5403}, 
\hbox{\tt T5468}, \hbox{\tt T6501}, 
\hbox{\tt T6531}, \hbox{\tt T6532}, \hbox{\tt T6533}, \hbox{\tt T6534}, 
\hbox{\tt T7631}, \hbox{\tt T7632},
\hbox{\tt T7633}, \hbox{\tt T7634}, \hbox{\tt T7635}, 
\hbox{\tt T7636}, \hbox{\tt T7637}, 
\hbox{\tt T7691}, \hbox{\tt T7692}, \hbox{\tt T7693}, 
\hbox{\tt T7694}, \hbox{\tt T7740}, \hbox{\tt T8731}, \hbox{\tt T8732}, 
\hbox{\tt T9831},
\hbox{\tt T9832}, \hbox{\tt T9833}, \hbox{\tt T9834},
\hbox{\tt T9835}, \hbox{\tt T9836}, \hbox{\tt T9837}, \hbox{\tt T9838}, 
\hbox{\tt T9891}, \hbox{\tt T9892}, \hbox{\tt T9893}, \hbox{\tt T9894}, 
\hbox{\tt T9940}, \hbox{\tt T111031}, \hbox{\tt T111032}, 
\hbox{\tt TListe}, \hbox{\tt TransListe}.  \par}
\medskip\noindent
The comments for the file \hbox{\tt summatio.m} 
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 T2103}--\hbox{\tt T9891} can be directly applied.


\head The file \tt transfor.mli\endhead
Each of the objects of this file corresponds to a transformation rule
of the file \hbox{\tt transfor.m}. 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.mli} are
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Tli2103}, \hbox{\tt Tli2104},
\hbox{\tt Tli2106}, \hbox{\tt Tli2107}, \hbox{\tt Tli2110}, 
\hbox{\tt Tli2112}, \hbox{\tt Tli2131}, \hbox{\tt Tli2132}, 
\hbox{\tt Tli2133}, \hbox{\tt Tli2134}, 
\hbox{\tt Tli2135}, \hbox{\tt Tli2136}, \hbox{\tt Tli2137}, 
\hbox{\tt Tli2138}, \hbox{\tt Tli2139}, \hbox{\tt Tli2139}, 
\hbox{\tt Tli2140}, \hbox{\tt Tli2141}, \hbox{\tt Tli2163}, 
\hbox{\tt Tli2191}, \hbox{\tt Tli2192}, 
\hbox{\tt Tli3204}, \hbox{\tt Tli3205},
\hbox{\tt Tli3206}, \hbox{\tt Tli3207}, \hbox{\tt Tli3217}, 
\hbox{\tt Tli3231}, \hbox{\tt Tli3232}, \hbox{\tt Tli3233}, 
\hbox{\tt Tli3234}, \hbox{\tt Tli3235}, \hbox{\tt Tli3236}, 
\hbox{\tt Tli3237}, \hbox{\tt Tli3238}, 
\hbox{\tt Tli3239}, \hbox{\tt Tli3240}, 
\hbox{\tt Tli3261}, \hbox{\tt Tli3262}, \hbox{\tt Tli3263}, 
\hbox{\tt Tli3264}, 
\hbox{\tt Tli4301}, \hbox{\tt Tli4302}, \hbox{\tt Tli4303}, \hbox{\tt Tli4304},
\hbox{\tt Tli4306}, \hbox{\tt Tli4309}, \hbox{\tt Tli4310}, 
\hbox{\tt Tli4312}, \hbox{\tt Tli4313}, 
\hbox{\tt Tli4331}, \hbox{\tt Tli4332}, 
\hbox{\tt Tli4362}, \hbox{\tt Tli4391}, 
\hbox{\tt Tli5401}, \hbox{\tt Tli5402}, \hbox{\tt Tli5403}, 
\hbox{\tt Tli5468}, \hbox{\tt Tli6501}, 
\hbox{\tt Tli6531}, \hbox{\tt Tli6532}, \hbox{\tt Tli6533}, 
\hbox{\tt Tli6534}, \hbox{\tt Tli7631}, \hbox{\tt Tli7632},
\hbox{\tt Tli7633}, \hbox{\tt Tli7634}, \hbox{\tt Tli7635}, 
\hbox{\tt Tli7636}, \hbox{\tt Tli7637}, 
\hbox{\tt Tli7691}, \hbox{\tt Tli7692}, \hbox{\tt Tli7693}, 
\hbox{\tt Tli7694}, \hbox{\tt Tli7740}, \hbox{\tt Tli8731}, 
\hbox{\tt Tli8732}, \hbox{\tt Tli9831},
\hbox{\tt Tli9832}, \hbox{\tt Tli9833}, \hbox{\tt Tli9834},
\hbox{\tt Tli9835}, \hbox{\tt Tli9836}, \hbox{\tt Tli9837}, 
\hbox{\tt Tli9838}, 
\hbox{\tt Tli9891}, \hbox{\tt Tli9892}, \hbox{\tt Tli9893}, 
\hbox{\tt Tli9894}, \hbox{\tt Tli9940},
\hbox{\tt Tli111031}, \hbox{\tt Tli111032}.  \par}

\head The files \tt summatio.mgl \rm and \tt transfor.mgl\endhead
These files contain the same summations, respectively transformations,
as \hbox{\tt summatio.m}, respectively \hbox{\tt transfor.m}, 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 Sgl1001}, \hbox{\tt Sgl2101},
\hbox{\tt Sgl2103}, \hbox{\tt Sgl2104}, \hbox{\tt Sgl2105}, \hbox{\tt Sgl2106}, 
\hbox{\tt Sgl2131}, \hbox{\tt Sgl2132}, \hbox{\tt Sgl2210}, \hbox{\tt Sgl2240},
\hbox{\tt Sgl3201}, \hbox{\tt Sgl3202}, \hbox{\tt Sgl3204}, 
\hbox{\tt Sgl3231}, \hbox{\tt Sgl3232},
\hbox{\tt Sgl3233}, \hbox{\tt Sgl3234}, \hbox{\tt Sgl3235}, 
\hbox{\tt Sgl3261}, \hbox{\tt Sgl3291}, \hbox{\tt Sgl3340}, 
\hbox{\tt Sgl4306}, \hbox{\tt Sgl4307}, \hbox{\tt Sgl4331}, \hbox{\tt Sgl4332}, 
\hbox{\tt Sgl5431}, \hbox{\tt Sgl5432}, \hbox{\tt Sgl5540},
\hbox{\tt Sgl6531}, \hbox{\tt Sgl6532}, \hbox{\tt Sgl7631}, \hbox{\tt Sgl7632}, 
\hbox{\tt Sgl7691}, 
\hbox{\tt SumListe\$gl},  \par}
\medskip\noindent
and
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Tgl2103}, \hbox{\tt Tgl2104},
\hbox{\tt Tgl2106}, \hbox{\tt Tgl2107}, \hbox{\tt Tgl2110}, 
\hbox{\tt Tgl2112}, \hbox{\tt Tgl2131}, \hbox{\tt Tgl2132}, 
\hbox{\tt Tgl2133}, \hbox{\tt Tgl2134}, 
\hbox{\tt Tgl2135}, \hbox{\tt Tgl2136}, \hbox{\tt Tgl2137}, 
\hbox{\tt Tgl2138}, \hbox{\tt Tgl2139}, \hbox{\tt Tgl2139}, 
\hbox{\tt Tgl2140}, \hbox{\tt Tgl2141}, \hbox{\tt Tgl2163}, 
\hbox{\tt Tgl2191}, \hbox{\tt Tgl2192}, 
\hbox{\tt Tgl3204}, \hbox{\tt Tgl3205},
\hbox{\tt Tgl3206}, \hbox{\tt Tgl3207}, \hbox{\tt Tgl3217}, 
\hbox{\tt Tgl3231}, \hbox{\tt Tgl3232}, \hbox{\tt Tgl3233}, 
\hbox{\tt Tgl3234}, \hbox{\tt Tgl3235}, \hbox{\tt Tgl3236}, 
\hbox{\tt Tgl3237}, \hbox{\tt Tgl3238}, 
\hbox{\tt Tgl3239}, \hbox{\tt Tgl3240}, 
\hbox{\tt Tgl3261}, \hbox{\tt Tgl3262}, \hbox{\tt Tgl3263}, 
\hbox{\tt Tgl3264}, 
\hbox{\tt Tgl4301}, \hbox{\tt Tgl4302}, \hbox{\tt Tgl4303}, \hbox{\tt Tgl4304},
\hbox{\tt Tgl4306}, \hbox{\tt Tgl4309}, \hbox{\tt Tgl4310}, 
\hbox{\tt Tgl4312}, \hbox{\tt Tgl4313}, 
\hbox{\tt Tgl4331}, \hbox{\tt Tgl4332}, 
\hbox{\tt Tgl4362}, \hbox{\tt Tgl4391}, 
\hbox{\tt Tgl5401}, \hbox{\tt Tgl5402}, \hbox{\tt Tgl5403}, 
\hbox{\tt Tgl5468}, \hbox{\tt Tgl6501}, 
\hbox{\tt Tgl6531}, \hbox{\tt Tgl6532}, \hbox{\tt Tgl6533}, 
\hbox{\tt Tgl6534}, \hbox{\tt Tgl7631}, \hbox{\tt Tgl7632},
\hbox{\tt Tgl7633}, \hbox{\tt Tgl7634}, \hbox{\tt Tgl7635}, 
\hbox{\tt Tgl7636}, \hbox{\tt Tgl7637}, 
\hbox{\tt Tgl7691}, \hbox{\tt Tgl7692}, \hbox{\tt Tgl7693}, 
\hbox{\tt Tgl7694}, \hbox{\tt Tgl7740}, \hbox{\tt Tgl8731}, \hbox{\tt Tgl8732}, 
\hbox{\tt Tgl9831},
\hbox{\tt Tgl9832}, \hbox{\tt Tgl9833}, \hbox{\tt Tgl9834},
\hbox{\tt Tgl9835}, \hbox{\tt Tgl9836}, \hbox{\tt Tgl9837}, 
\hbox{\tt Tgl9838}, 
\hbox{\tt Tgl9891}, \hbox{\tt Tgl9892}, \hbox{\tt Tgl9893}, 
\hbox{\tt Tgl9894}, \hbox{\tt Tgl9940}, \hbox{\tt Tgl111031}, \hbox{\tt
Tgl111032},
\hbox{\tt TransListe\$gl}.  \par}
\medskip\noindent
When calling one of these objects you will be put a question. If
the variables of the called summation or transformation are
undefined, the 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 the 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}.
(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.m\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.m} are 
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt C01},
\hbox{\tt C02},
\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 C30},
\hbox{\tt C31},
\hbox{\tt C32},
\hbox{\tt C33},
\hbox{\tt C34},
\hbox{\tt C35},
\hbox{\tt C36},
\hbox{\tt C40},
\hbox{\tt C41},
\hbox{\tt C42},
\hbox{\tt C43},
\hbox{\tt C44},
\hbox{\tt C45},
\hbox{\tt C46},
\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 those
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.m} first and then \hbox{\tt
hyp.q} and want to use \hbox{\tt Limes} with an ordinary (i.e\.
non-basic) hypergeometric expression, then you have to type \hbox{\tt
Hyp`m`Limes} instead of \hbox{\tt Limes}. (Calling \hbox{\tt Limes}
would invoke the {\it basic} 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 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.   
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
GAMMA} then would read as follows.
\MATH

In[1]:= GAMMA[2*a+1]
\goodbreakpoint%
Out[1]= Ga(1 + 2 a)
\goodbreakpoint%
In[2]:= GAMMA[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ]
\goodbreakpoint%
          [      ]
          | a, b |
Out[2]= Ga|      |
          | c, d |
          [      ]
\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
$$\smatrix \format\sa\l\s\l\s\l\se\\
\hlinefor7\\
&\text {\eightpoint German}&&\text {\eightpoint English}&&
\text {\eightpoint {\sl Mathematica} objects in HYP  }&\\
&&&&&\text {\eightpoint containing the word}&\\
\hlinefor7\\
&\text {abspalten}&&       \text {split}&&   \text {\tt lina1, lina2}&\\
\hlinefor7\\
&\text {aufl\"osen}&&       \text {dissolve}&&   \text {\tt paufl}&\\
\hlinefor7\\
&\text {drucken}&& \text {print}&&   \text {\tt Drucke}&\\
\hlinefor7\\
&\text {einf\"ugen}&&      \text {insert}&&  \text {\tt FEinf}&\\
\hlinefor7\\
&\text {ersetzen}&&\text {replace}&& \text {\tt Ers}&\\
\hlinefor7\\
&\text {erweitern}&&       \text {extend}&&  \text {\tt erw1, erw2, SUMErw1, SUMErw2}&\\
\hlinefor7\\
&\text {Gleichung}&&       \text {equation}&&\text {\tt Gleichung}&\\
\hlinefor7\\
&\text {"2 hoch 3"}&& \text {"2 to the 3"}&& \text {\tt Hoch}&\\
\hlinefor7\\
&\text {"2 mal 3"}&& \text {"2 times 3"}&& \text {\tt Mal}&\\
\hlinefor7\\
&\text {ordnen}&&  \text {order}&&   \text {\tt FOrdne}&\\
\hlinefor7\\
&\text {Regel}&&   \text {rule}&&    \text {\tt SUMRegeln}&\\
\hlinefor7\\
&\text {sammeln}&& \text {collect}&& \text {\tt SUMSammle}&\\
\hlinefor7\\
&\text {schreiben}&&       \text {write}&&   \text {\tt SchreibeZahl}&\\
\hlinefor7\\
&\text {tauschen}&&\text {exchange, interchange}&& \text {\tt  SUMTausche, GlTausche}&\\
\hlinefor7\\
&\text {umkehren}&&\text {reverse}&& \text {\tt SUMUmkehr}&\\
\hlinefor7\\
&\text {Zahl}&&    \text {number}&&  \text {\tt SchreibeZahl}&\\
\hlinefor7\\
&\text {zerlegen}&&\text {split}&&   \text {\tt zerl1, pzerl, Gzerl, SUMZerl}&\\
\hlinefor7\\
&\text {zusammen}&&\text {together}&&\text {\tt zus1, zus2, zus3}&\\
\hlinefor7\\
\endsmatrix
$$
}

\newpage
\vbox{
\head \tenpoint \bf An English--German vocabulary \endhead
$$\smatrix \format\sa\l\s\l\s\l\se\\
\hlinefor7\\
&\text {\eightpoint English}&&\text {\eightpoint German}&&
\text {\eightpoint {\sl Mathematica} objects in HYP }&\\
&&&&&\text {\eightpoint containing the word}&\\
\hlinefor7\\
&\text {collect}&& \text {sammeln}&& \text {\tt SUMSammle}&\\
\hlinefor7\\
&\text {dissolve}&&   \text {aufl\"osen}&& \text {\tt paufl}&\\
\hlinefor7\\
&\text {equation}&&\text {Gleichung}&&       \text {\tt Gleichung}&\\
\hlinefor7\\
&\text {exchange}&&\text {tauschen}&&\text {\tt SUMTausche, GlTausche}&\\
\hlinefor7\\
&\text {extend}&&  \text {erweitern}&&       \text {\tt erw1, erw2, SUMErw1, SUMErw2}&\\
\hlinefor7\\
&\text {insert}&&  \text {einf\"ugen}&&      \text {\tt FEinf}&\\
\hlinefor7\\
&\text {interchange}&&     \text {tauschen}&&\text {\tt SUMTausche, GlTausche}&\\
\hlinefor7\\
&\text {number}&&  \text {Zahl}&&    \text {\tt SchreibeZahl}&\\
\hlinefor7\\
&\text {order}&&   \text {ordnen}&&  \text {\tt FOrdne}&\\
\hlinefor7\\
&\text {print}&&   \text {drucken}&& \text {\tt Drucke}&\\
\hlinefor7\\
&\text {replace}&& \text {ersetzen}&&\text {\tt Ers}&\\
\hlinefor7\\
&\text {reverse}&& \text {umkehren}&&\text {\tt SUMUmkehr     }&\\
\hlinefor7\\
&\text {rule}&&    \text {Regel}&&   \text {\tt SUMRegeln}&\\
\hlinefor7\\
&\text {split}&&   \text {abspalten}&&       \text {\tt lina1, lina2}&\\
\hlinefor7\\
&\text {split}&&   \text {zerlegen}&&\text {\tt zerl1, pzerl, Gzerl, SUMZerl}&\\
\hlinefor7\\
&\text {"2 times 3"}&& \text {"2 mal 3"}&& \text {\tt Mal}&\\
\hlinefor7\\
&\text {"2 to the 3"}&& \text {"2 hoch 3"}&& \text {\tt Hoch}&\\
\hlinefor7\\
&\text {together}&&\text {zusammen}&&\text {\tt zus1, zus2, zus3}&\\
\hlinefor7\\
&\text {write}&&   \text {schreiben}&&       \text {\tt SchreibeZahl}&\\
\hlinefor7\\
\endsmatrix
$$
}





\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}.
\Usage AbsGreater[Expr].
%\Example
%\MATH
%\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}.
\Usage AbsSmaller[Expr].
%\Example
%\MATH
%\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}.
\Usage AbsUndetermined[Expr].
%\Example
%\MATH
%\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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= Add[p[c-a,n]]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck                             (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich                                     n
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  + (-a + c)  == (-a + c)  + ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich            n            n     (c)
           %
\MATHloEck           %
\MATHroEck                                  n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck                             (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich                                     n
Out[3]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  + (-a + c)  == (-a + c)  + ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich            n            n     (c)
           %
\MATHloEck           %
\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]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\goodbreakpoint%
In[2]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[2]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  c%
\MATHrbrace \MATHbackslash endmatrix ; %
\MATHlbrace \MATHbackslash displaystyle
   z%
\MATHrbrace \MATHbackslash right ] 
\goodbreakpoint%
In[3]:= AmSLaTeX
\goodbreakpoint%
In[4]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-LaTeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\goodbreakpoint%
In[5]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[5]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash begin%
\MATHlbrace matrix%
\MATHrbrace  %
\MATHlbrace  a, b%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  c%
\MATHrbrace \MATHbackslash end%
\MATHlbrace matrix%
\MATHrbrace  ;
   %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace \MATHbackslash right ] 
\endMATH 
\Seealso AmSTeX, LaTeX, TeX, TeXMat, TeXFV.

\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]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\goodbreakpoint%
In[3]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[3]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b\MATHbackslash cr c%
\MATHrbrace  ; %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ] 
\goodbreakpoint%
In[4]:= AmSTeX
\goodbreakpoint%
In[5]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\goodbreakpoint%
In[6]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[6]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  c%
\MATHrbrace \MATHbackslash endmatrix ; %
\MATHlbrace \MATHbackslash displaystyle
   z%
\MATHrbrace \MATHbackslash right ] 
\endMATH
\Seealso AmSLaTeX, LaTeX, TeX, TeXMat, TeXFV.

\Name baszerl
\Description  
\vtab $(a)_n \to m^n \prod _{k=0} ^{m-1} ((a+k)/m)_{n/m}$,\\
             $\Gamma(a) \to m^{a-1/2}(2\pi)^{(1-m)/2}
              \prod _{k=0} ^{m-1} \Gamma((a+k)/m)$.
\endvtab
\vskip6pt
\hskip10pt  The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.baszerl.
\Example 
\MATH
In[1]:= p[a,n]
\goodbreakpoint%
Out[1]= (a)
           n
\goodbreakpoint%
In[2]:= \%/.baszerl
split into ? terms: 2
\goodbreakpoint%
         n  a      1 + a
Out[2]= 2  (-)    (-----)
            2 n/2    2   n/2
\goodbreakpoint%
In[3]:= p[b,4*m]
\goodbreakpoint%
Out[3]= (b)
           4 m
\goodbreakpoint%
In[4]:= \%/.baszerl
split into ? terms: 4
\goodbreakpoint%
         4 m  b    1 + b    2 + b    3 + b
Out[4]= 4    (-)  (-----)  (-----)  (-----)
              4 m    4   m    4   m    4   m
\goodbreakpoint%
In[5]:= GAMMA[2*c]
\goodbreakpoint%
Out[5]= \MATHGamma (2 c)
\goodbreakpoint%
In[6]:= \%/.baszerl
split into ? terms: 4
\goodbreakpoint%
         -(1/2) + 2 c   c    1 + 2 c    2 + 2 c    3 + 2 c
        4             \MATHGamma (-) \MATHGamma (-------) \MATHGamma (-------) \MATHGamma (-------)
                        2       4          4          4
Out[6]= ---------------------------------------------------
                              3/2  3/2
                             2    \MATHpi 
\endMATH 
\Seealso baszerl, baszus, Ers, PosListe, ManipulationsListe.


\Name baszus
\Description \vtab $(a)_n \to m^{-mn}(am)_{mn}/
                                 \prod _{k=1} ^{m-1} (a+k/m)_n$,\\
             $\Gamma(a) \to m^{1/2-am}(2\pi)^{(m-1)/2}\Gamma(am)/
                                 \prod _{k=1} ^{m-1} \Gamma(a+{k/m})$.
\endvtab
\vskip6pt
\hskip10pt  The parameter \hbox{\tt m} has to be entered on request.
  This operation is basically the inverse of \hbox{\tt baszerl}.
\Usage Expr/.baszus.
\Example 
\MATH
In[1]:= p[a/2,n]*p[(a+1)/2,n]
\goodbreakpoint%
         a    1 + a
Out[1]= (-)  (-----)
         2 n    2   n
\goodbreakpoint%
In[2]:= Ers[\%,baszus,%
\MATHlbrace 1%
\MATHrbrace ]
put together ? terms: 2
\goodbreakpoint%
                1 + a
        (a)    (-----)
           2 n    2   n
Out[2]= ---------------
          2 n  1   a
         2    (- + -)
               2   2 n
\goodbreakpoint%
In[3]:= ExpandAll[\%]
\goodbreakpoint%
        (a)
           2 n
Out[3]= ------
          2 n
         2
\goodbreakpoint%
In[4]:= GAMMA[(a-1)/2]*GAMMA[a/2]*p[b/3,m]*p[(b+1)/3,m]*p[(b+2)/3,m]
\goodbreakpoint%
          -1 + a    a   b    1 + b    2 + b
Out[4]= \MATHGamma (------) \MATHGamma (-) (-)  (-----)  (-----)
            2       2   3 m    3   m    3   m
\goodbreakpoint%
In[5]:= Ers[\%,baszus,%
\MATHlbrace 3%
\MATHrbrace ]
put together ? terms: 3
\goodbreakpoint%
          -1 + a    a          1 + b    2 + b
        \MATHGamma (------) \MATHGamma (-) (b)    (-----)  (-----)
            2       2     3 m    3   m    3   m
Out[5]= ---------------------------------------
                 3 m  1   b    2   b
                3    (- + -)  (- + -)
                      3   3 m  3   3 m
\goodbreakpoint%
In[6]:= ExpandAll[\%]
\goodbreakpoint%
            1    a    a
        \MATHGamma (-(-) + -) \MATHGamma (-) (b)
            2    2    2     3 m
Out[6]= -----------------------
                  3 m
                 3
\goodbreakpoint%
In[7]:= PosListe[\%]
\goodbreakpoint%
           -3 m               1    a              a
Out[7]= %
\MATHlbrace %
\MATHlbrace 3    , %
\MATHlbrace %
\MATHlbrace 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace \MATHGamma (-(-) + -), %
\MATHlbrace %
\MATHlbrace 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace \MATHGamma (-), %
\MATHlbrace %
\MATHlbrace 3%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace (b)   , %
\MATHlbrace %
\MATHlbrace 4%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
                              2    2              2               3 m
\goodbreakpoint%
In[8]:= Ers[\%\%,baszus,%
\MATHlbrace 2%
\MATHrbrace ]
put together ? terms: 2
\goodbreakpoint%
         1 - 2 (-(1/2) + a/2)                1    a
        2                     Sqrt[\MATHpi ] \MATHGamma (2 (-(-) + -)) (b)
                                             2    2      3 m
Out[8]= ----------------------------------------------------
                                 3 m
                                3
\goodbreakpoint%
In[9]:= ExpandAll[\%]
\goodbreakpoint%
        4 Sqrt[\MATHpi ] \MATHGamma (-1 + a) (b)
                               3 m
Out[9]= --------------------------
                  a  3 m
                 2  3
\endMATH 
\Seealso Ers, PosListe, ManipulationsListe.


\Name Binomialp
\Description \hbox{\tt Binomialp[n,k]} is the binomial coefficient, written in terms
  of factorial symbols (Pochhammer symbols) \hbox{\tt p}. 
\Usage Binomialp[n,k].
\Example 
\MATH
In[1]:= Binomialp[n,k]
\goodbreakpoint%
        (1 - k + n)
                   k
Out[1]= ------------
            (1)
               k
\goodbreakpoint%
In[2]:= Binomialp[6,3]
\goodbreakpoint%
        (4)
           3
Out[2]= ----
        (1)
           3
\endMATH 
\Seealso Factorialp, Multinomialp.

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

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

\Name C14
\Description Contiguous relation in form of a rule.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ {
    \Bi}\endmatrix ; {\displaystyle z}\right ]  + 
   {z    }
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
   {} _{r} F _{s} \!\left [ \matrix { a, \qAi}\\ { \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ {
    \Bi}\endmatrix ; {\displaystyle z}\right ]  - 
   {z    }
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
   {} _{r} F _{s} \!\left [ \matrix { a + 1, \qAi}\\ { \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow    {{  1 }
      \over {z }}
   {{\prodl_{i = 1}^{s}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { a, \Aiq}\\ { \Biq}\endmatrix ;
        {\displaystyle z}\right ]  -{{  
       1   }\over 
      {z }}
   {{\prodl_{i = 1}^{s}(\bi - 1)}\over
    { \prodl_{i = 1}^{r-1}(\ai - 1) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Aiq}\\ {
         \Biq}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ {
    \Bi}\endmatrix ; {\displaystyle z}\right ]  + 
   {z    }
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
  {} _{r} F _{s} \!\left [ \matrix { a, \qAi}\\ { \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ {
    \Bi}\endmatrix ; {\displaystyle z}\right ]  - 
   {z    }
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \qAi}\\ { \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow    {{  1 }
      \over {z }}
   {{\prodl_{i = 1}^{s}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { a, \Aiq}\\ { \Biq}\endmatrix ;
        {\displaystyle z}\right ]  -{{  
    1 }\over 
      {z }}
   {{    \prodl_{i = 1}^{s}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Aiq}\\ {
         \Biq}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b + 1,
    \Bi}\endmatrix ; {\displaystyle z}\right ]  + 
   {{z    }\over
       {b \left( 1 + b \right)  }}
   {{\prodl_{i = 1}^{r}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r} F _{s} \!\left [ \matrix { \qAi}\\ { b + 2, \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b - 1,
    \Bi}\endmatrix ; {\displaystyle z}\right ]  - 
   {{z    }\over
       {\left( b - 1 \right)  b }}
   {{\prodl_{i = 1}^{r}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r} F _{s} \!\left [ \matrix { \qAi}\\ { b + 1, \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{\left( b - 2 \right)  \left( b - 1 \right)  
        }\over 
     {z }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { \Aiq}\\ { b - 2, \Biq}\endmatrix ;
        {\displaystyle z}\right ] \\
  -  {{\left( b - 2 \right)  \left( b - 1 \right)  
        }\over 
     {z }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { \Aiq}\\ { b - 1, \Biq}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b + 1,
    \Bi}\endmatrix ; {\displaystyle z}\right ]  + 
   {{z    }\over
       {b \left( 1 + b \right)  }}
   {{\prodl_{i = 1}^{r}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r} F _{s} \!\left [ \matrix { \qAi}\\ { b + 2, \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b - 1,
    \Bi}\endmatrix ; {\displaystyle z}\right ]  - 
   {{z    }\over
       {\left( b - 1 \right)  b }}
   {{\prodl_{i = 1}^{r}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r} F _{s} \!\left [ \matrix { \qAi}\\ { b + 1, \qBi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{\left( b - 2 \right)  \left( b - 1 \right)  
        }\over 
     {z }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { \Aiq}\\ { b - 2, \Biq}\endmatrix ;
        {\displaystyle z}\right ]\\ - 
   {{\left( b - 2 \right)  \left( b - 1 \right)  
        }\over 
     {z }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { \Aiq}\\ { b - 1, \Biq}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{b  }\over {b - a}}
  {} _{r} F _{s} \!\left [ \matrix { a, b + 1, \Ai}\\ { \Bi}\endmatrix ;
        {\displaystyle z}\right ] + 
   {{a  }\over {a - b}}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b, \Ai}\\ { \Bi}\endmatrix ;
        {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - b - 1  \right)  
        }\over {a - 1}}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b, \Ai}\\ { \Bi}\endmatrix ;
        {\displaystyle z}\right ] + 
   {{b  }\over {a - 1}}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b + 1, \Ai}\\ {
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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 C30
\Description Contiguous relation in form of a rule.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b + 1, \Ai}\\ { \Bi}\endmatrix ;
    {\displaystyle z}\right ]  + 
   \left( 1 - a + b \right)  z 
   {{\prodl_{i = 1}^{r-2}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
  {} _{r} F _{s} \!\left [ \matrix { a, b + 1, \qAi}\\ { \qBi}\endmatrix
        ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow    {{  1 }
      \over {\left( b - a \right)  z 
       }}
   {{\prodl_{i = 1}^{s}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-2}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { a, b - 1, \Aiq}\\ { \Biq}\endmatrix ;
        {\displaystyle z}\right ]
    -{{ 1     }\over {\left( b - a \right)  z 
      }}
   {{ \prodl_{i = 1}^{s}(\bi - 1)  }\over
    {   \prodl_{i = 1}^{r-2}(\ai - 1) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b, \Aiq}\\ { \Biq}\endmatrix ;
         {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b - 1, \Ai}\\ { \Bi}\endmatrix ;
    {\displaystyle z}\right ]  + 
   \left( a + b - 1  \right)  z 
   {{\prodl_{i = 1}^{r-2}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a, b, a + b, \qAi}\\ { -1 +
        a + b, \qBi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b + 1, \Ai}\\ { \Bi}\endmatrix ;
    {\displaystyle z}\right ]  - 
   {\left( 1 + a + b \right)  z }
   {{\prodl_{i = 1}^{r-2}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a + 1, b + 1, a + b + 2,
        \qAi}\\ { a + b + 1, \qBi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)   }\over {b - 1 - a }}
  {} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ {b -1
        , \Bi}\endmatrix ; {\displaystyle z}\right ] +
     {{a  }\over {1 + a - b}}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { b, \Bi}\endmatrix
        ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)   }\over {a - 1}}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\
        { b - 1, \Bi}\endmatrix ; {\displaystyle z}\right ] 
    + {{\left( a - b \right)   }\over {a - 1}
     }
  {} _{r} F _{s} \!\left [ \matrix { a - 1,
        \Ai}\\ { b, \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - a \right)   }\over b}
  {} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { 1 +
        b, \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{a  }\over b}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { b + 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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 C40
\Description Contiguous relation in form of a rule.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { b - 1, \Bi}\endmatrix ;
    {\displaystyle z}\right ]  + 
   {{\left( b - a \right)  z 
        }\over 
     {\left( b - 1 \right)  b }}
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r} F _{s} \!\left [ \matrix { a, \qAi}\\ { b + 1, \qBi}\endmatrix
        ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { b + 1, \Bi}\endmatrix ;
    {\displaystyle z}\right ]  - 
   {{\left( b - a \right)  z 
        }\over 
     {b \left( 1 + b \right)  }}
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \qAi}\\ { b + 2,
        \qBi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow    {{\left( b - 2 \right)  \left( b - 1 \right)  
        }\over 
     {\left( b - a - 1  \right)  z 
       }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}
  {} _{r} F _{s} \!\left [ \matrix { a, \Aiq}\\ { b - 1, \Biq}\endmatrix
        ; {\displaystyle z}\right ]
  \\  -{{\left( b - 2 \right)  \left( b - 1 \right)  
        }\over 
      {\left( b - a - 1  \right)  z 
      }}
   {{ \prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {   \prodl_{i = 1}^{r-1}(\ai - 1) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Aiq}\\ { b - 2,
         \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { b + 1, \Bi}\endmatrix ;
    {\displaystyle z}\right ]  + 
   {{\left( a + b \right)  z    }\over 
     {b \left( 1 + b \right)  }}
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a, 1
        + a + b, \qAi}\\ { b + 2, a + b, \qBi}\endmatrix ; {\displaystyle
        z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { b - 1, \Bi}\endmatrix ;
    {\displaystyle z}\right ]  - 
   {{\left( a + b \right)  z    }\over 
     {\left( b - 1 \right)  b }}
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 +
        a, a + b + 1, \qAi}\\ { b + 1, a + b, \qBi}\endmatrix ; {\displaystyle
        z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, b, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - 1 \right)   }\over {a - b}}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { -1 +
        a, b, \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{\left( b - 1 \right)   }\over {b - a}}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, -1
        + b, \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\Usage Expr/.C45[n1,n2].\newline
\rm {\tt n1}, {\tt n2} are the positions of the special lower
parameters.
\Seealso C64, ContigListe, Ers, PosListe.

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

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

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

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

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

\Name C53
\Description Contiguous relation in form of a rule.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, c, \Ai}\\ { \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{b \left( c - a - 1  \right)  
        }\over 
     {\left( b - a \right)  \left( c - 1 \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, b + 1, c - 1, \Ai}\\ {
        \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{a \left( c - b - 1  \right)  
        }\over 
     {\left( a - b \right)  \left( c - 1 \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b, c - 1, \Ai}\\ {
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{b \left( c - a \right)   }\over 
     {\left( b - a \right)  c}}
  {} _{r} F _{s} \!\left [ \matrix { a, b + 1,
        \Ai}\\ { c + 1, \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{a \left( c - b \right)  
        }\over 
     {\left( a - b \right)  c}}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b, \Ai}\\ { c + 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( 1 - a + b \right)  \left( c - 1 \right)  
        }\over 
     {\left( a - 1 \right)  \left( 1 + b - c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b, \Ai}\\ { c - 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{b \left( c - a \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - b - 1  \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b + 1, \Ai}\\ { c,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, c, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( c - a \right)  
        }\over 
     {\left( b - a - 1  \right)  c}}
  {} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b - 1, c + 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{a \left( 1 - b + c \right)  
        }\over 
     {\left( 1 + a - b \right)  c}}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { b, c + 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { b, c, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( c - a \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { b - 1, c,
        \Bi}\endmatrix ; {\displaystyle z}\right ] + 
   {{\left( b - a \right)  \left( c - 1 \right)  
        }\over 
     {\left( a - 1 \right)  \left( b - c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { b, c - 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, b, c, \Bi}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - 1 \right)  \left( 1 - b + c \right)  
        }\over 
     {\left( a - b \right)  c}}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a - 1, b, c + 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( b - 1 \right)  \left( 1 - a + c \right)  
        }\over 
     {\left( b - a \right)  c}}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, b - 1, c + 1,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, 2 - a + c, b, c - b, \Ai}\\ {
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - b - 1  \right)  \left( 1 - a - b + c \right)  
        }\over 
     {\left( a - 1 \right)  \left( 1 - a + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, 1 - a + c, b, c - b, \Ai}\\
        { \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{b \left( c - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( 1 - a + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, 1 - a + c, b + 1, 1 - b + c,
        \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, c - b, \Ai}\\ {
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  
        }\over 
     {\left( a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, b, c - b, \Ai}\\
        { \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{b \left( c - b \right)  
         }\over 
      {\left( a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, c - a, b + 1, 1 - b + c, \Ai}\\
         { \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, c - b, \Ai}\\ {
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a b  }\over 
     {\left( c - a - 1  \right)  \left( c - b - 1  \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, c - a - 1 , b + 1, -1 - b
        + c, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( c - 1 \right)  \left( -1 - a - b + c \right)  
        }\over 
     {\left( c - a - 1  \right)  \left( c - b - 1  \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {{c - 1}\over 2}, a, -1
        - a + c, b, c - b - 1 , \Ai}\\ { {{c - 1}\over 2}, \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b , 2 - b + c, \Ai}\\ {
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, b - 1,1 + c - b, \Ai}\\ {
    \Bi}\endmatrix ; {\displaystyle z}\right ] \\ + 
   {\left( a - b +1\right)  \left( 1 + c \right)  
       \left( c - a - b +1\right)  z 
         }
   {{ \prodl_{i = 1}^{r-4}\ai }\over
    {\prodl_{i = 1}^{s}\bi}}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a + 1, 1 - a + c, b , 2 -
        b + c , c + 2, \qAi}\\ { c + 1, \qBi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, 2 - a + c, b, c - b, \Ai}\\ {
   \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
  {{b \left( c - b \right)  \left( a - d - 1  \right)  
       \left( 1 - a + c - d \right)  
        }\over 
     {\left( a - 1 \right)  \left( 1 - a + c \right)  
       \left( b - d \right)  \left( c - b - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, 1 - a + c, b + 1, 1 - b + c,
        \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( 1 - a + b \right)  \left( 1 - a - b + c \right)  
       \left( c - d \right)  d 
        }\over 
     {\left( a - 1 \right)  \left( 1 - a + c \right)  
       \left( b - d \right)  \left( c - b - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a - 1, 1 - a + c, b, -b +
        c, d + 1, 1 + c - d, \Ai}\\ { d, c - d, \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, 1 - a + c, b, \Ai}\\ { c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - b - 1  \right)  \left( c - a - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a, b, \Ai}\\ { c - b,
        \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{b \left( c - b - 1  \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a, b + 1, \Ai}\\ { -1 -
        b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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]:= F[%
\MATHlbrace A,B,1+C,D%
\MATHrbrace ,%
\MATHlbrace E,A+C-D,F%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck                     %
\MATHruEck 
           %
\MATHvStrich  A, B, 1 + C, D     %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich                 ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  E, A + C - D, F    %
\MATHvStrich 
           %
\MATHloEck                     %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.C64[3,1,4,2]
\goodbreakpoint%
           %
\MATHluEck                     %
\MATHruEck 
           %
\MATHvStrich  C, -1 + A, D, B    %
\MATHvStrich 
         F %
\MATHvStrich                 ; z %
\MATHvStrich  (-1 + A - D)  (C - D)
        4 3%
\MATHvStrich  A + C - D, E, F    %
\MATHvStrich              1        1
           %
\MATHloEck                     %
\MATHroEck 
Out[2]= ------------------------------------------------ + 
                         (-1 + A)  (C)
                                 1    1
 
        %
\MATHluEck                          %
\MATHruEck 
        %
\MATHvStrich  C, -1 + A, 1 + D, B     %
\MATHvStrich 
      F %
\MATHvStrich                      ; z %
\MATHvStrich  (-1 + A + C - D)  (D)
     4 3%
\MATHvStrich  -1 + A + C - D, E, F    %
\MATHvStrich                  1    1
        %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser     -----------------------------------------------------
                        (-1 + A)  (C)
                                1    1
\goodbreakpoint%
In[3]:= \%/.paufl
\goodbreakpoint%
                                %
\MATHluEck                     %
\MATHruEck 
                                %
\MATHvStrich  C, -1 + A, D, B    %
\MATHvStrich 
        (-1 + A - D) (C - D)  F %
\MATHvStrich                 ; z %
\MATHvStrich 
                             4 3%
\MATHvStrich  A + C - D, E, F    %
\MATHvStrich 
                                %
\MATHloEck                     %
\MATHroEck 
Out[3]= ---------------------------------------------- + 
                          (-1 + A) C
 
                           %
\MATHluEck                          %
\MATHruEck 
                           %
\MATHvStrich  C, -1 + A, 1 + D, B     %
\MATHvStrich 
     (-1 + A + C - D) D  F %
\MATHvStrich                      ; z %
\MATHvStrich 
                        4 3%
\MATHvStrich  -1 + A + C - D, E, F    %
\MATHvStrich 
                           %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser     -------------------------------------------------
                        (-1 + A) C
\endMATH
\rm
The third, first, and fourth upper parameters in {\tt Out[1]} are {\tt
1+C}, {\tt A}, and {\tt D}, respectively, the second lower parameter
in {\tt Out[1]} is {\tt A+C-D}. Hence {\tt C64} can be applied with
the replacements $a\to \text {{\tt 1+C}}$, $b\to \text {{\tt D}}$ and
$c\to \text {{\tt A+C}}$.
\Seealso ContigListe, Ers, PosListe.

\Name C65
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, \Ai}\\ { 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  
        }\over 
     {\left( b - 1 \right)  \left( 1 - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, b - 1, \Ai}\\ { 2
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( 1 + a - b \right)  \left( 1 - a - b + c \right)  
         }\over 
      {\left( b - 1 \right)  \left( 1 - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, c - a, b - 1, \Ai}\\ { 2 - b +
         c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, \Ai}\\ { 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  
        }\over 
     {\left( a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, b, \Ai}\\ { 1 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\
    -{{b \left( c - b \right)  
         }\over 
      {\left( a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, c - a, b + 1, \Ai}\\ { c - b,
         \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, 1 - a + c, b, \Ai}\\ { c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a b  }\over 
     {\left( c - a \right)  \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, c - a, b + 1, \Ai}\\ { 1
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{c \left( c - a - b \right)  
        }\over {\left( c - a \right)  \left( c - b \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {c\over 2}, a, c - a,
        b, \Ai}\\ { {c\over 2}, 1 - b + c, \Bi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, \Ai}\\ { 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( c - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( b - 1 \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, 1 - a + c, b - 1, \Ai}\\ {
        c - b, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( c - 1 \right)  \left( 1 - a - b + c \right)  
        }\over 
     {\left( a - 1 \right)  \left( b - 1 \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {{c - 1}\over 2}, -1 +
        a, c - a, b - 1, \Ai}\\ { {{c - 1}\over 2}, 1 - b + c,
        \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, \Ai}\\ { 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, b - 1, \Ai}\\ { 2 - b
    + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ + 
   {{\left( 1 + a - b \right)  \left( 1 + c \right)  
       \left( 1 - a - b + c \right)  z 
          }\over 
     {\left( 1 - b + c \right)  \left( 2 - b + c \right)  
       }}
   {{\prodl_{i = 1}^{r-3}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a + 1, 1 - a + c, b, c + 2,
        \qAi}\\ { 3 - b + c, c + 1, \qBi}\endmatrix ; {\displaystyle z}\right
        ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, 1 - a + c, b, \Ai}\\ { c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a, b + 1, \Ai}\\ { -1 - b +
    c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ - 
   {{\left( a - b - 1  \right)  c \left( c - a - b \right)  z 
        }\over 
     {\left( c - b - 1  \right)  \left( c - b \right)  
       }}
   {{\prodl_{i = 1}^{r-3}\ai }\over
    {\prodl_{i = 1}^{s-1}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a, 1 - a + c, b + 1, c + 1,
        \qAi}\\ { 1 - b + c, c, \qBi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, 1 - a + c, b, \Ai}\\ { c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{b \left( c - b - 1  \right)  \left( a - d - 1  \right)  
       \left( c - a - d \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right)  \left( b - d \right)  
       \left( c - b - 1  - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a, b + 1, \Ai}\\ { -1 -
        b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( 1 - a + b \right)  \left( c - a - b \right)  
       \left( c - d - 1  \right)  d 
        }\over 
     {\left( a - 1 \right)  \left( c - a \right)  \left( b - d \right)  
       \left( c - b - 1  - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a - 1, c - a, b, d + 1, c
        - d, \Ai}\\ { c - b, d, c - d - 1 , \Bi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, \Ai}\\ { 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a \left( c - a \right)  \left( b - d - 1  \right)  
       \left( 1 - b + c - d \right)  
        }\over 
     {\left( b - 1 \right)  \left( 1 - b + c \right)  
       \left( a - d \right)  \left( c - a - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, b - 1, \Ai}\\ { 2
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( b - a - 1  \right)  \left( 1 - a - b + c \right)  
       \left( c - d \right)  d 
        }\over 
     {\left( b - 1 \right)  \left( 1 - b + c \right)  
       \left( a - d \right)  \left( c - a - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a, c - a, b - 1, d + 1, 1
        + c - d, \Ai}\\ { 2 - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, a + b, \Ai}\\ { b - a + 1,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{    1   }\over 
     {\left( a + b - 1  \right)  z 
       }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-3}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, b, \Aiq}\\ {
        \Biq}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{  1 }\over 
      {\left( a + b - 1  \right)  z 
      }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {   \prodl_{i = 1}^{r-3}(\ai - 1) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a - 1, b - 1, \Aiq}\\ {
         \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c - a, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - b - 1  \right)  \left( c - a - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b, \Ai}\\ { 1 - a + c, -b +
        c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{b \left( c - b - 1  \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b + 1, \Ai}\\ { 1 - a + c,
        c - b - 1 , \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c - a, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a - 1  \right)  
        }\over 
     {\left( a - b \right)  \left( -1 - a - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b, \Ai}\\ { c - a - 1 , -b +
        c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{b \left( c - b - 1  \right)  
         }\over 
      {\left( a - b \right)  \left( -1 - a - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, b + 1, \Ai}\\ { c - a, -1 - b +
         c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c - a, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a b  }\over 
     {\left( c - a \right)  \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b + 1, \Ai}\\ { 1 - a + c,
        1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{c \left( c - a - b \right)  
        }\over {\left( c - a \right)  \left( c - b \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {c\over 2}, a, b, \Ai}\\
        { {c\over 2}, 1 - a + c, 1 - b + c, \Bi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c - a, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a - 1  \right)  \left( c - b - 1  \right)  
        }\over 
     {\left( a - 1 \right)  \left( b - 1 \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b - 1, \Ai}\\ { c - a - 1 ,
        c - b - 1 , \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( c - 2 \right)  \left( c - a - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( b - 1 \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {{c - 2}\over 2}, -1 +
        a, b - 1, \Ai}\\ { {{c - 2}\over 2}, c - a, c - b, \Bi}\endmatrix
        ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c - a, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, b - 1, \Ai}\\ { c - a - 1 , 1 - b
    + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ + 
   {{\left( 1 + a - b \right)  c \left( c - a - b \right)  z 
        }\over 
     {\left( c - a - 1  \right)  \left( c - a \right)  
       \left( c - b \right)  \left( 1 - b + c \right)  
       }}
   {{\prodl_{i = 1}^{r-2}\ai }\over
    {\prodl_{i = 1}^{s-2}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a + 1, b, c + 1, \qAi}\\ { 1
        - a + c, 2 - b + c, c, \qBi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, \Ai}\\ { c - a, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{b \left( c - b - 1  \right)  \left( a - d - 1  \right)  
       \left( c - a - d \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right)  \left( b - d \right)  
       \left( c - b - 1  - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, b + 1, \Ai}\\ { 1 - a + c,
        c - b - 1 , \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( 1 - a + b \right)  \left( c - a - b \right)  
       \left( c - d - 1  \right)  d 
        }\over 
     {\left( a - 1 \right)  \left( c - a \right)  \left( b - d \right)  
       \left( c - b - 1  - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a - 1, b, d + 1, c - d,
        \Ai}\\ { 1 - a + c, c - b, d, c - d - 1 , \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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]:= F[%
\MATHlbrace A,B,B+F-E,D%
\MATHrbrace ,%
\MATHlbrace E,F%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck                        %
\MATHruEck 
           %
\MATHvStrich  A, B, B - E + F, D    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich                    ; z %
\MATHvStrich 
        4 2%
\MATHvStrich         E, F           %
\MATHvStrich 
           %
\MATHloEck                        %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.C79[2,3,2,1,x]
\goodbreakpoint%
Out[2]= ((-1 + E) (B - E + F) (-1 + B - x) (F - x) 
 
           %
\MATHluEck                                 %
\MATHruEck 
           %
\MATHvStrich  -1 + B, 1 + B - E + F, A, D    %
\MATHvStrich 
\MATHgroesser         F %
\MATHvStrich                             ; z %
\MATHvStrich ) / 
        4 2%
\MATHvStrich         1 + F, -1 + E           %
\MATHvStrich 
           %
\MATHloEck                                 %
\MATHroEck 
 
\MATHgroesser      ((-1 + B) F (-1 + E - x) (B - E + F - x)) + 
 
\MATHgroesser     ((-B + E) (1 - E + F) (-1 + B + F - x) x 
 
           %
\MATHluEck                                               %
\MATHruEck 
           %
\MATHvStrich  -1 + B, B - E + F, 1 + x, B + F - x, A, D    %
\MATHvStrich 
\MATHgroesser         F %
\MATHvStrich                                           ; z %
\MATHvStrich ) / 
        6 4%
\MATHvStrich         1 + F, E, x, -1 + B + F - x           %
\MATHvStrich 
           %
\MATHloEck                                               %
\MATHroEck 
 
\MATHgroesser      ((-1 + B) F (-1 + E - x) (B - E + F - x))
\endMATH
\rm
The second and third upper parameters in {\tt Out[1]} are {\tt
B}, and {\tt B+F-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 $a\to \text {{\tt B}}$, $b\to \text {{\tt B+F-E}}$ and
$c\to \text {{\tt B+F}}$. The {\tt x} replaces $d$ in the right-hand
side expression.
\Seealso C79, ContigListe, Ers, PosListe.

\Name C80
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( c - b - 1  \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a - 1  \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a - 1 , \Ai}\\ { b - 1,
        c - b - 1 , \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( a - b \right)  \left( c - a - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a - 1  \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a - 1 , \Ai}\\ { b, -b +
        c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  
        }\over 
     {b \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, \Ai}\\ { b + 1, 1
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
    -{{\left( a - b \right)  \left( c - a - b \right)  
         }\over 
      {b \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b + 1, 1 - b +
         c, \Bi}\endmatrix ; {\displaystyle z}\right ]
$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, 2 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  
        }\over 
     {\left( 1 + a - b \right)  \left( 1 - a - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, \Ai}\\ { b, 2 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( b - 1 \right)  \left( 1 - b + c \right)  
         }\over 
      {\left( 1 + a - b \right)  \left( 1 - a - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b - 1, 1 - b +
         c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a \left( b - 1 \right)   }\over {\left( c - a - 1  \right)  \left( c - b \right) }
     }
  {} _{r} F _{s} \!\left [ \matrix { a + 1, -1 -
        a + c, \Ai}\\ { b - 1, 1 - b + c, \Bi}\endmatrix ; {\displaystyle
        z}\right ]\\ + {{\left( c - 1 \right)  \left( c - a - b \right)  
        }\over 
     {\left( c - a - 1  \right)  \left( c - b \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {{c - 1}\over 2}, a, -1
        - a + c, \Ai}\\ { {{c - 1}\over 2}, b, 1 - b + c, \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, \Ai}\\ { b + 1, 1 - b +
    c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ + 
   {{\left( a - b \right)  \left( 1 + c \right)  
       \left( c - a - b \right)  z 
          }\over 
     {b \left( 1 + b \right)  \left( c - b \right)  
       \left( 1 - b + c \right)  }}
   {{\prodl_{i = 1}^{r-2}\ai }\over
    {\prodl_{i = 1}^{s-2}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a + 1, 1 - a + c, c + 2,
        \qAi}\\ { b + 2, 2 - b + c, c + 1, \qBi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a - 1 , \Ai}\\ { b - 1, -1 -
    b + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ - 
   {{\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
        }\over 
     {\left( b - 1 \right)  b \left( c - b - 1  \right)  
       \left( c - b \right)  }}
   {{\prodl_{i = 1}^{r-2}\ai }\over
    {\prodl_{i = 1}^{s-2}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a, c - a, c, \qAi}\\ { 1 +
        b, 1 - b + c, c - 1, \qBi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( c - b - 1  \right)  
       \left( a - d - 1  \right)  \left( c - a - 1  - d \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a - 1  \right)  
       \left( b - d - 1  \right)  \left( c - b - 1  - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, c - a - 1 , \Ai}\\ { b - 1,
        c - b - 1 , \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( b - a \right)  \left( c - a - b \right)  
       \left( c - d - 2  \right)  d 
        }\over 
     {\left( a - 1 \right)  \left( c - a - 1  \right)  
       \left( b - d - 1  \right)  \left( c - b - 1  - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a - 1, c - a - 1 , d + 1,
        c - d - 1 , \Ai}\\ { b, c - b, d, c - d - 2 , \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, \Ai}\\ { b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a \left( c - a \right)  \left( b - d \right)  
       \left( c - b - d \right)  
        }\over 
     {b \left( c - b \right)  \left( a - d \right)  
       \left( c - a - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, 1 - a + c, \Ai}\\ { b + 1, 1
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( b - a \right)  \left( c - a - b \right)  
       \left( c - d \right)  d 
        }\over 
     {b \left( c - b \right)  \left( a - d \right)  
       \left( c - a - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a, c - a, d + 1, 1 + c - d,
        \Ai}\\ { b + 1, 1 - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, a + b, \Ai}\\ { b + 1, a + b - 1,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{\left( b - 1 \right)  b 
        }\over 
     {\left( a + b - 1  \right)  z 
       }}
   {{\prodl_{i = 1}^{s-2}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-2}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, \Aiq}\\ { b - 1,
        \Biq}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( b - 1 \right)  b 
       }\over 
      {\left( a + b - 1  \right)  z 
      }}
   {{  \prodl_{i = 1}^{s-2}(\bi - 1) }\over
    {   \prodl_{i = 1}^{r-2}(\ai - 1) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a - 1, \Aiq}\\ { b,
         \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { c - a, b, 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( c - b \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { 1 - a + c, b - 1,
        c - b, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( a - b \right)  \left( 1 - a - b + c \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { 1 - a + c, b, 1 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { 1 - a + c, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  
        }\over 
     {b \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { c - a, b + 1, 1 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( a - b \right)  \left( c - a - b \right)  
         }\over 
      {b \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { 1 - a + c, b + 1, 1 - b
         + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { c - a, b, 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a - 1  \right)  
        }\over 
     {\left( 1 + a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { c - a - 1 , b, 1 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( b - 1 \right)  \left( c - b \right)  
         }\over 
      {\left( 1 + a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { c - a, b - 1, c - b,
         \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { c - a, b, 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a \left( b - 1 \right)   }\over {\left( c - a \right)  \left( 1 - b + c \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1,
        \Ai}\\ { 1 - a + c, b - 1, 2 - b + c, \Bi}\endmatrix ; {\displaystyle
        z}\right ]\\
     + {{c \left( 1 - a - b + c \right)  
        }\over {\left( c - a \right)  \left( 1 - b + c \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {c\over 2}, a, \Ai}\\ {
        {c\over 2}, 1 - a + c, b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { 1 - a + c, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( c - b - 1  \right)  
        }\over 
     {\left( a - 1 \right)  b}}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { c - a, b + 1, -1 -
        b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( c - 1 \right)  \left( c - a - b \right)  
        }\over 
     {\left( a - 1 \right)  b}}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {{c - 1}\over 2}, -1 +
        a, \Ai}\\ { {{c - 1}\over 2}, 1 - a + c, b + 1, c - b,
        \Bi}\endmatrix ; {\displaystyle z}\right ]\longrightarrow
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { 1 - a + c, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { c - a, b + 1, 1 - b + c,
    \Bi}\endmatrix ; {\displaystyle z}\right ] \\ + 
   {{\left( a - b \right)  \left( 1 + c \right)  
       \left( c - a - b \right)  z 
          }\over 
     {b \left( 1 + b \right)  \left( c - a \right)  
       \left( 1 - a + c \right)  \left( c - b \right)  
       \left( 1 - b + c \right)  }}
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s-3}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a + 1, c + 2, \qAi}\\ { 2 -
        a + c, b + 2, 2 - b + c, c + 1, \qBi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { c - a, b, 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { 1 - a + c, b - 1, -b +
    c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ - 
   {{\left( a - b \right)  c \left( 1 - a - b + c \right)  z 
        }\over 
     {\left( b - 1 \right)  b \left( c - a \right)  
       \left( 1 - a + c \right)  \left( c - b \right)  
       \left( 1 - b + c \right)  }}
   {{\prodl_{i = 1}^{r-1}\ai }\over
    {\prodl_{i = 1}^{s-3}\bi  }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { a, c + 1, \qAi}\\ { 2 - a +
        c, b + 1, 2 - b + c, c, \qBi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { c - a, b, 1 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( c - b \right)  
       \left( a - d - 1  \right)  \left( c - a - d \right)  
        }\over 
     {\left( a - 1 \right)  \left( c - a \right)  
       \left( b - d - 1  \right)  \left( c - b - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a - 1, \Ai}\\ { 1 - a + c, b - 1,
        c - b, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( b - a \right)  \left( 1 - a - b + c \right)  
       \left( c - d - 1  \right)  d 
        }\over 
     {\left( a - 1 \right)  \left( c - a \right)  
       \left( b - d - 1  \right)  \left( c - b - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a - 1, d + 1, c - d, \Ai}\\
        { 1 - a + c, b, 1 - b + c, d, c - d - 1 , \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, \Ai}\\ { 1 - a + c, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{a \left( c - a \right)  \left( b - d \right)  
       \left( c - b - d \right)  
        }\over 
     {b \left( c - b \right)  \left( a - d \right)  
       \left( c - a - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { a + 1, \Ai}\\ { c - a, b + 1, 1 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( b - a \right)  \left( c - a - b \right)  
       \left( c - d \right)  d 
        }\over 
     {b \left( c - b \right)  \left( a - d \right)  
       \left( c - a - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { a, d + 1, 1 + c - d, \Ai}\\
        { 1 - a + c, b + 1, 1 - b + c, d, c - d, \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a + b, \Ai}\\ { a + 1, b + 1, b - a + 1,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - 1 \right)  a \left( b - 1 \right)  b 
        }\over 
     {\left( a + b - 1  \right)  z 
       }}
   {{\prodl_{i = 1}^{s-3}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { \Aiq}\\ { a - 1, b - 1,
        \Biq}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( a - 1 \right)  a \left( b - 1 \right)  b 
        }\over 
     {\left( a + b - 1  \right)  z 
       }}
   {{\prodl_{i = 1}^{s-3}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { \Aiq}\\ { a, b,
        \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, c - a, b, 2 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( b - 1 \right)  \left( 1 - b + c \right)  
        }\over 
     {a \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a + 1, 1 - a + c, b - 1, 1
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( 1 + a - b \right)  \left( 1 - a - b + c \right)  
        }\over 
     {a \left( c - a \right) }}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a + 1, 1 - a + c, b, 2 - b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, c - a, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - 1 \right)  \left( c - a - 1  \right)  
        }\over 
     {\left( a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a - 1, c - a - 1 , b, -b +
        c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( b - 1 \right)  \left( c - b - 1  \right)  
        }\over 
     {\left( a - b \right)  \left( c - a - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, c - a, b - 1, -1 - b +
        c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, c - a, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( a - 1 \right)  \left( b - 1 \right)  
        }\over 
     {\left( c - a \right)  \left( c - b \right) }}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a - 1, 1 - a + c, b - 1,
        1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( c - 1 \right)  \left( 1 - a - b + c \right)  
        }\over 
     {\left( c - a \right)  \left( c - b \right) }}
  {} _{r+1} F _{s+1} \!\left [ \matrix { 1 + {{c - 1}\over 2},
        \Ai}\\ { {{c - 1}\over 2}, a, 1 - a + c, b, 1 - b + c, \Bi}\endmatrix
        ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, 2 - a + c, b, c - b,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a - 1, 1 - a + c, b + 1, 1 - b
    + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ + 
   {{\left( a - b - 1  \right)  \left( 1 + c \right)  
       \left( 1 - a - b + c \right)  z 
          }\over 
     {\left( a - 1 \right)  a b \left( 1 + b \right)  
       \left( 1 - a + c \right)  \left( 2 - a + c \right)  
       \left( c - b \right)  \left( 1 - b + c \right)  
       }}
   {{\prodl_{i = 1}^{r}\ai }\over
    {\prodl_{i = 1}^{s-4}\bi  }}\\
  \times  {} _{r+1} F _{s+1} \!\left [ \matrix { c + 2, \qAi}\\ { a + 1, 3 -
        a + c, b + 2, 2 - b + c, c + 1, \qBi}\endmatrix ; {\displaystyle
        z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a, c - a, b, 2 - b + c,
   \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
  {{\left( b - 1 \right)  \left( 1 - b + c \right)  \left( a - d \right)  
       \left( c - a - d \right)  
        }\over 
     {a \left( c - a \right)  \left( b - d - 1  \right)  
       \left( 1 - b + c - d \right) }}
  {} _{r} F _{s} \!\left [ \matrix { \Ai}\\ { a + 1, 1 - a + c, b - 1, 1
        - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ + 
   {{\left( b - a - 1  \right)  \left( 1 - a - b + c \right)  
       \left( c - d \right)  d 
        }\over 
     {a \left( c - a \right)  \left( b - d - 1  \right)  
       \left( 1 - b + c - d \right) }}
  {} _{r+2} F _{s+2} \!\left [ \matrix { d + 1, 1 + c - d, \Ai}\\ { 1
        + a, 1 - a + c, b, 2 - b + c, d, c - d, \Bi}\endmatrix ;
        {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, c - b, c, \Ai}\\ { c - 1,
   \Bi}\endmatrix ; {\displaystyle z}\right ]  \\\longrightarrow 
  {{     1   }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-5}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a - 1, c - a - 1 , b, -b +
        c, \Aiq}\\ { \Biq}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{    1   }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-1}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-5}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, c - a, b - 1, -1 - b +
        c, \Aiq}\\ { \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C110
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { 1 + {c\over 2}, a, c - a, b, c - b,
   \Ai}\\ { {c\over 2}, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( c - b \right)  
        }\over 
     {c \left( c - a - b \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, 1 - a + c, b, 1 - b +
        c, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{a b  }\over 
     {c \left( c - a - b \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a + 1, c - a, b + 1,
        c - b, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C110[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.
\NoBlackBoxes
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, c, \Ai}\\ { 1 - b + c, -1 +
   c, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - b - 1  \right)  \left( c - b \right)  
        }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-2}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-4}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a - 1, c - a - 1 , b,
        \Aiq}\\ { c - b - 1 , \Biq}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( c - b - 1  \right)  \left( c - b \right)  
        }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-2}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-4}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, c - a, b - 1, \Aiq}\\
        { c - b, \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
\BlackBoxes
\Seealso C64, ContigListe, Ers, PosListe.

\Name C112
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { 1 + {c\over 2}, a, c - a, b, \Ai}\\ {
   {c\over 2}, 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( c - b \right)  
        }\over 
     {c \left( c - a - b \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, 1 - a + c, b, \Ai}\\ {
        c - b, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{a b  }\over 
     {c \left( c - a - b \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a + 1, c - a, b + 1,
        \Ai}\\ { 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C112[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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, c, \Ai}\\ { 1 - a + c, 1 - b + c, -1
   + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
  {{\left( c - a - 1  \right)  \left( c - a \right)  
       \left( c - b - 1  \right)  \left( c - b \right)  
        }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-3}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-3}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a - 1, b, \Aiq}\\ { -a +
        c, c - b - 1 , \Biq}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( c - a - 1  \right)  \left( c - a \right)  
       \left( c - b - 1  \right)  \left( c - b \right)  
        }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-3}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-3}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, b - 1, \Aiq}\\ { -1 -
        a + c, c - b, \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C106[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 C107
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, c, \Ai}\\ { b + 1, 1 - b + c, -1
   + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
  {{\left( b - 1 \right)  b \left( c - b - 1  \right)  
       \left( c - b \right)    
        }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-3}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-3}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { -1
        + a, c - a - 1 , \Aiq}\\ { b - 1, c - b - 1 , \Biq}\endmatrix ;
        {\displaystyle z}\right ]\\ - 
   {{\left( b - 1 \right)  b \left( c - b - 1  \right)  
       \left( c - b \right)     }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( c - a - b \right)  z 
       }}
   {{\prodl_{i = 1}^{s-3}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-3}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a,
        c - a, \Aiq}\\ { b, c - b, \Biq}\endmatrix ; {\displaystyle z}\right
        ]
\endmultline$$
\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 C114
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { 1 + {c\over 2}, a, b, \Ai}\\ { {c\over 2},
   1 - a + c, 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( c - b \right)  
        }\over 
     {c \left( c - a - b \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, b, \Ai}\\ { c - a, -b
        + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{a b  }
      \over {c \left( c - a - b \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a + 1, b + 1, \Ai}\\ {
        1 - a + c, 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C114[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 C115
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { 1 + {c\over 2}, a, c - a, \Ai}\\ { {c\over
   2}, b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( 1 - b + c \right)  
        }\over 
     {c \left( 1 - a - b + c \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, 1 - a + c, \Ai}\\ { b,
        1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{a \left( b - 1 \right)  
        }\over 
     {c \left( 1 - a - b + c \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a + 1, c - a, \Ai}\\ { -1
        + b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C115[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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c, \Ai}\\ { 1 - a + c, b, 2 - b + c, -1
   + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
  {{\left( b - 2 \right)  \left( b - 1 \right)  
       \left( c - a - 1  \right)  \left( c - a \right)  
       \left( c - b \right)  \left( 1 - b + c \right)  
        }\over 
     {\left( 1 + a - b \right)  \left( c - 1 \right)  
       \left( 1 - a - b + c \right)  z 
       }}
   {{\prodl_{i = 1}^{s-4}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-2}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a - 1, \Aiq}\\ { c - a,
        b - 2, c - b, \Biq}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( b - 2 \right)  \left( b - 1 \right)  
       \left( c - a - 1  \right)  \left( c - a \right)  
       \left( c - b \right)  \left( 1 - b + c \right)  
        }\over 
     {\left( 1 + a - b \right)  \left( c - 1 \right)  
       \left( 1 - a - b + c \right)  z 
       }}
   {{\prodl_{i = 1}^{s-4}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-2}(\ai - 1)  }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, \Aiq}\\ { c - a - 1 ,
        b - 1, 1 - b + c, \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C118
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { 1 + {c\over 2}, a, \Ai}\\ { {c\over 2}, 1 -
   a + c, b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow 
  {{\left( c - a \right)  \left( 1 - b + c \right)  
        }\over 
     {c \left( 1 - a - b + c \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a, \Ai}\\ { c - a, b, 1 -
        b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{a \left( b - 1 \right)  
        }\over 
     {c \left( 1 - a - b + c \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { a + 1, \Ai}\\ { 1 - a + c,
        b - 1, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C118[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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { c, \Ai}\\ { a, 2 - a + c, b, 2 - b + c, -1
   + c, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow   
 {{\left( a - 2 \right)  \left( a - 1 \right)  \left( b - 2 \right)  
       \left( b - 1 \right)  \left( c - a \right)  
       \left( 1 - a + c \right)  \left( c - b \right)  
       \left( 1 - b + c \right)  
        }\over 
     {\left( a - b \right)  \left( c - 1 \right)  
       \left( 2 - a - b + c \right)  z 
       }}
   {{\prodl_{i = 1}^{s-5}(\bi - 1) }\over
    {\prodl_{i = 1}^{r-1}(\ai - 1)  }}\\
\times  {} _{r-1} F _{s-1} \!\left [ \matrix { \Aiq}\\ { a - 1, 1 - a +
        c, b - 2, c - b, \Biq}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( a - 2 \right)  \left( a - 1 \right)  \left( b - 2 \right)  
        \left( b - 1 \right)  \left( c - a \right)  
        \left( 1 - a + c \right)  \left( c - b \right)  
        \left( 1 - b + c \right)  
      }\over 
      {\left( a - b \right)  \left( c - 1 \right)  
        \left( 2 - a - b + c \right)  z        }}
    {{   \prodl_{i = 1}^{s-5}(\bi - 1) }\over
     {  \prodl_{i = 1}^{r-1}(\ai - 1) }}\\
  \times  {} _{r-1} F _{s-1} \!\left [ \matrix { \Aiq}\\ { a - 2, c - a,
         b - 1, 1 - b + c, \Biq}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C120
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { 1 + {c\over 2}, \Ai}\\ { {c\over 2}, a, 2 -
   a + c, b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{\left( 1 - a + c \right)  \left( 1 - b + c \right)  
        }\over 
     {c \left( 2 - a - b + c \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { \Ai}\\ { a, 1 - a + c, b,
        1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( a - 1 \right)  \left( b - 1 \right)  
         }
       \over {c \left( 2 - a - b + c \right) }}
  {} _{r-1} F _{s-1} \!\left [ \matrix { \Ai}\\ { a - 1, 2 - a +
         c, b - 1, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\Usage Expr/.C120[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 C111
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, c - b, d + 1, 1 + c - d,
   \Ai}\\ { d, c - d, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
   {{a \left( c - a \right)  \left( b - d \right)  
       \left( c - b - d \right)  
        }\over 
     {\left( b - a \right)  \left( c - a - b \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a + 1, 1 - a + c, b, -b +
        c, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{b \left( c - b \right)  \left( a - d \right)  
        \left( c - a - d \right)           }\over 
      {\left( b - a \right)  \left( c - a - b \right)  
        \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a, c - a, b + 1, 1 - b +
         c, \Ai}\\ { \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C113
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, b, d + 1, 1 + c - d, \Ai}\\ { 1
   - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  \left( b - d \right)  
       \left( c - b - d \right)  
        }\over 
     {\left( b - a \right)  \left( c - a - b \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a + 1, 1 - a + c, b,
        \Ai}\\ { 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{b \left( c - b \right)  \left( a - d \right)  
        \left( c - a - d \right)  
         }\over 
      {\left( b - a \right)  \left( c - a - b \right)  
        \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a, c - a, b + 1, \Ai}\\
         { c - b, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C116
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, b, d + 1, 1 + c - d, \Ai}\\ { 1 - a + c,
   1 - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  \left( b - d \right)  
       \left( c - b - d \right)  
        }\over 
     {\left( b - a \right)  \left( c - a - b \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a + 1, b, \Ai}\\ { c - a,
        1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{b \left( c - b \right)  \left( a - d \right)  
        \left( c - a - d \right)  
         }\over 
      {\left( b - a \right)  \left( c - a - b \right)  
        \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a, b + 1, \Ai}\\ { 1 - a
         + c, c - b, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, c - a, d + 1, 1 + c - d, \Ai}\\ { b, 2
   - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow    {{a \left( c - a \right)  \left( b - d - 1  \right)  
       \left( 1 - b + c - d \right)  
        }\over 
     {\left( b - a - 1  \right)  \left( 1 - a - b + c \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a + 1, 1 - a + c, \Ai}\\ {
        b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( b - 1 \right)  \left( 1 - b + c \right)  
        \left( a - d \right)  \left( c - a - d \right)  
         }\over 
      {\left( b - a - 1  \right)  \left( 1 - a - b + c \right)  
        \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a, c - a, \Ai}\\ { -1 +
         b, 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C119
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { a, d + 1, 1 + c - d, \Ai}\\ { 1 - a + c, b,
   2 - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow  
  {{a \left( c - a \right)  \left( b - d - 1  \right)  
       \left( 1 - b + c - d \right)          }\over 
     {\left( b - a - 1  \right)  \left( 1 - a - b + c \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a + 1, \Ai}\\ { c - a, b,
        2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
  \\  -{{\left( b - 1 \right)  \left( 1 - b + c \right)  
        \left( a - d \right)  \left( c - a - d \right)  
         }\over 
      {\left( b - a - 1  \right)  \left( 1 - a - b + c \right)  
        \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { a, \Ai}\\ { 1 - a + c, -1
         + b, 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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 C121
\Description Contiguous relation in form of a rule.
$$\multline
{} _{r} F _{s} \!\left [ \matrix { d + 1, 1 + c - d, \Ai}\\ { a, 2 - a + c, b,
   2 - b + c, d, c - d, \Bi}\endmatrix ; {\displaystyle z}\right ] \\ \longrightarrow 
  {{\left( a - 1 \right)  \left( 1 - a + c \right)  
       \left( b - d - 1  \right)  \left( 1 - b + c - d \right)  
        }\over 
     {\left( b - a \right)  \left( 2 - a - b + c \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { \Ai}\\ { a - 1, 1 - a +
        c, b, 2 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]\\ - 
   {{\left( b - 1 \right)  \left( 1 - b + c \right)  
       \left( a - d - 1  \right)  \left( 1 - a + c - d \right)  
        }\over 
     {\left( b - a \right)  \left( 2 - a - b + c \right)  
       \left( c - d \right)  d}}
  {} _{r-2} F _{s-2} \!\left [ \matrix { \Ai}\\ { a, 2 - a + c, -1
        + b, 1 - b + c, \Bi}\endmatrix ; {\displaystyle z}\right ]
\endmultline$$
\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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= Div[a\MATHhoch n]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
         F %
\MATHvStrich       ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich    c      %
\MATHvStrich     (-a + c)
           %
\MATHloEck           %
\MATHroEck             n
Out[2]= --------------- == ---------
               n             n
              a             a  (c)
                                  n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
         F %
\MATHvStrich       ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich    c      %
\MATHvStrich     (-a + c)
           %
\MATHloEck           %
\MATHroEck             n
Out[3]= --------------- == ---------
               n             n
              a             a  (c)
                                  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, AmSTeX, AmSLaTeX, LaTeX, TeX, TeXFV.

\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*p[a+1,n]*p[c,k]/p[-b,m]/p[1-d,l]
\goodbreakpoint%
            n
        (-1)  (1 + a)  (c)
                     n    k
Out[1]= -------------------
          (-b)  (1 - d)
              m        l
\goodbreakpoint%
In[2]:= PosListe[\%]
\goodbreakpoint%
              n                                1
Out[2]= %
\MATHlbrace %
\MATHlbrace (-1) , %
\MATHlbrace %
\MATHlbrace 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace (1 + a) , %
\MATHlbrace %
\MATHlbrace 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace -----, %
\MATHlbrace %
\MATHlbrace 3%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace (c) , %
\MATHlbrace %
\MATHlbrace 4%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , 
                                 n           (-b)               k
                                                 m
 
         1
\MATHgroesser     %
\MATHlbrace --------, %
\MATHlbrace %
\MATHlbrace 5%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
      (1 - d)
             l
\goodbreakpoint%
In[3]:= Ers[\%\%,neg1,%
\MATHlbrace 5%
\MATHrbrace ]
\goodbreakpoint%
            n
        (-1)  (1 + a)  (c)  (1 - d + l)
                     n    k            -l
Out[3]= ---------------------------------
                      (-b)
                          m
\goodbreakpoint%
In[4]:= Ers[\%\%\%,neg1,%
\MATHlbrace 2,4%
\MATHrbrace ]
\goodbreakpoint%
                            n
                        (-1)
Out[4]= --------------------------------------
        (-b)  (1 - d)  (c + k)   (1 + a + n)
            m        l        -k            -n
\goodbreakpoint%
In[5]:= SUM[\%,%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck                     n
         \MATHbackslash                    (-1)
Out[5]=   \MATHgroesser   --------------------------------------
         /   (-b)  (1 - d)  (c + k)   (1 + a + n)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck     m        l        -k            -n
         k=0
\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 , 
 
        1                   1                      1
\MATHgroesser     %
\MATHlbrace -----, %
\MATHlbrace %
\MATHlbrace 1, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace --------, %
\MATHlbrace %
\MATHlbrace 1, 3%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace ---------, %
\MATHlbrace %
\MATHlbrace 1, 4%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , 
      (-b)               (1 - d)               (c + k)
          m                     l                     -k
 
            1
\MATHgroesser     %
\MATHlbrace -------------, %
\MATHlbrace %
\MATHlbrace 1, 5%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
      (1 + a + n)
                 -n
\goodbreakpoint%
In[7]:= Ers[\%\%,neg1,%
\MATHlbrace %
\MATHlbrace 1,3%
\MATHrbrace %
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty            n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck      (-1)  (1 - d + l)
         \MATHbackslash                          -l
Out[7]=   \MATHgroesser   -----------------------------
         /   (-b)  (c + k)   (1 + a + n)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck     m        -k            -n
         k=0
\goodbreakpoint%
In[8]:= Ers[\%\%\%,neg1,%
\MATHlbrace %
\MATHlbrace 1,4%
\MATHrbrace ,%
\MATHlbrace 1,5%
\MATHrbrace %
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty       n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (-1)  (1 + a)  (c)
         \MATHbackslash                 n    k
Out[8]=   \MATHgroesser   -------------------
         /     (-b)  (1 - d)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck       m        l
         k=0
\goodbreakpoint%
In[9]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[9]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[10]:= Ers[a-\MATHgroesser 1-a]
\goodbreakpoint%
            %
\MATHluEck               %
\MATHruEck     (-1 + a + c)
            %
\MATHvStrich  1 - a, -n    %
\MATHvStrich                 n
Out[10]=  F %
\MATHvStrich           ; 1 %
\MATHvStrich  == -------------
         2 1%
\MATHvStrich      c        %
\MATHvStrich         (c)
            %
\MATHloEck               %
\MATHroEck            n
\goodbreakpoint%
In[11]:= Gleichung
\goodbreakpoint%
            %
\MATHluEck               %
\MATHruEck     (-1 + a + c)
            %
\MATHvStrich  1 - a, -n    %
\MATHvStrich                 n
Out[11]=  F %
\MATHvStrich           ; 1 %
\MATHvStrich  == -------------
         2 1%
\MATHvStrich      c        %
\MATHvStrich         (c)
            %
\MATHloEck               %
\MATHroEck            n
\goodbreakpoint%
In[12]:= P
\goodbreakpoint%
In[13]:= p[a,4]+1/p[b,3]
\goodbreakpoint%
                                             1
Out[13]= a (1 + a) (2 + a) (3 + a) + -----------------
                                     b (1 + b) (2 + b)
\goodbreakpoint%
In[14]:= Ers[\%,Expand,%
\MATHlbrace 1%
\MATHrbrace ]
\goodbreakpoint%
                   2      3    4           1
Out[14]= 6 a + 11 a  + 6 a  + a  + -----------------
                                   b (1 + b) (2 + b)
\endMATH
\Seealso PosListe, ManipulationsListe, Subst.

\Name erw1
\Description \vtab $(a)_n \to (a)_{m+n}/(a+n)_m$,\\
             $(a)_n \to \Gamma(a+n)/\Gamma(a)$.
\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]:= p[a,n+1]
\goodbreakpoint%
Out[1]= (a)
           1 + n
\goodbreakpoint%
In[2]:= \%/.erw1
top-extend by: m-1
\goodbreakpoint%
            (a)
               m + n
Out[2]= -----------------
        (1 + a + n)
                   -1 + m
\goodbreakpoint%
In[3]:= p[-a+1,m]
\goodbreakpoint%
Out[3]= (1 - a)
               m
\goodbreakpoint%
In[4]:= \%/.erw1
top-extend by: Infinity
\goodbreakpoint%
        \MATHGamma (1 - a + m)
Out[4]= ------------
          \MATHGamma (1 - a)
\endMATH 
\Seealso erw2, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

\Name erw2
\Description \vtab $(a)_n \to (a-m)_{m+n}/(a-m)_m$,\\
             $\Gamma(a) \to \Gamma(a-m)\cdot (a-m)_m$.
\endvtab
\vskip6pt
\leavevmode\hskip10pt  The parameter \hbox{\tt m} has to be entered on request. 
\Usage Expr/.erw2.
\Example 
\MATH
In[1]:= p[a-1,n+1]
\goodbreakpoint%
Out[1]= (-1 + a)
                1 + n
\goodbreakpoint%
In[2]:= \%/.erw2
bottom-extend by: m-1
\goodbreakpoint%
        (a - m)
               m + n
Out[2]= -------------
        (a - m)
               -1 + m
\goodbreakpoint%
In[3]:= GAMMA[-b]
\goodbreakpoint%
Out[3]= \MATHGamma (-b)
\goodbreakpoint%
In[4]:= \%/.erw2
bottom-extend by: n
\goodbreakpoint%
Out[4]= \MATHGamma (-b - n) (-b - n)
                          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]:= (-1)\MATHhoch m*p[a,m+1]
\goodbreakpoint%
            m
Out[1]= (-1)  (a)
                 1 + m
\goodbreakpoint%
In[2]:= \%/.baszerl
split into ? terms: 2
\goodbreakpoint%
            m  1 + m  a            1 + a
Out[2]= (-1)  2      (-)          (-----)
                      2 (1 + m)/2    2   (1 + m)/2
\goodbreakpoint%
In[3]:= Ers[\%,trans,%
\MATHlbrace 3%
\MATHrbrace ]
\goodbreakpoint%
            m + (1 + m)/2  1 + m  1 + a                a   1 + m
Out[3]= (-1)              2      (-----)          (1 - - - -----)
                                    2   (1 + m)/2      2     2   (1 + m)/2
\goodbreakpoint%
In[4]:= \%/.Expandq
\goodbreakpoint%
            1/2 + (3 m)/2  1 + m  1 + a                a   1 + m
Out[4]= (-1)              2      (-----)          (1 - - - -----)
                                    2   (1 + m)/2      2     2   (1 + m)/2
\endMATH 
\Seealso SimplifyP, MinusOne, SUMExpand, Ers, PosListe.

\Name F
\Description \hbox{\tt F[List1,List2,z]} is the  hypergeometric series with 
  upper parameters \hbox{\tt List}1, lower parameters \hbox{\tt List}2, 
 and argument \hbox{\tt z}.
\Usage F[List1,List2,z].
\Example 
\MATH
In[1]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich      ; z %
\MATHvStrich 
        2 1%
\MATHvStrich   c      %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[2]:= F[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, c    %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich         ; z %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\endMATH 
\Seealso SListe, TListe, SUMRegeln, SUMF, FSUM, H, p, GAMMA, FCancel, FOrdne, FPerm, FTausche,
P,\linebreak FFormat.

\Name Factorialp
\Description \hbox{\tt Factorialp[n,k]} is the usual factorial, written in terms
  of factorial symbols (Pochhammer symbols) \hbox{\tt p}. 
\Usage Factorialp[n].
\Example 
\MATH
In[1]:= Factorialp[n]
\goodbreakpoint%
Out[1]= (1)
           n
\goodbreakpoint%
In[2]:= Factorialp[5]
\goodbreakpoint%
Out[2]= (1)
           5
\endMATH 
\Seealso Binomialp, Multinomialp.

\Name FCancel
\Description Switch that activates automatic cancelling of the upper and 
  lower parameters in \hbox{\tt F[]} and \hbox{\tt H[]}, 
  or makes it inactive, respectively. By default
  automatic cancelling is active.
\Usage FCancel.
\Example 
\MATH
In[1]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\goodbreakpoint%
In[2]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace a,c%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck       %
\MATHruEck 
           %
\MATHvStrich  b    %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich   ; z %
\MATHvStrich 
        1 1%
\MATHvStrich  c    %
\MATHvStrich 
           %
\MATHloEck       %
\MATHroEck 
\goodbreakpoint%
In[3]:= p[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace a,c%
\MATHrbrace ,n]
\goodbreakpoint%
        (b)
           n
Out[3]= ----
        (c)
           n
\goodbreakpoint%
In[4]:= GAMMA[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace a,c%
\MATHrbrace ]
\goodbreakpoint%
         %
\MATHluEck    %
\MATHruEck 
         %
\MATHvStrich  b %
\MATHvStrich 
Out[4]= \MATHGamma %
\MATHvStrich    %
\MATHvStrich 
         %
\MATHvStrich  c %
\MATHvStrich 
         %
\MATHloEck    %
\MATHroEck 
\goodbreakpoint%
In[5]:= FCancel
\goodbreakpoint%
In[6]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is inactive.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\goodbreakpoint%
In[7]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace a,c%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[7]=  F %
\MATHvStrich      ; z %
\MATHvStrich 
        2 2%
\MATHvStrich  a, c    %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[8]:= p[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace a,c%
\MATHrbrace ,n]
\goodbreakpoint%
        (b)
           n
Out[8]= ----
        (c)
           n
\goodbreakpoint%
In[9]:= GAMMA[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace a,c%
\MATHrbrace ]
\goodbreakpoint%
         %
\MATHluEck    %
\MATHruEck 
         %
\MATHvStrich  b %
\MATHvStrich 
Out[9]= \MATHGamma %
\MATHvStrich    %
\MATHvStrich 
         %
\MATHvStrich  c %
\MATHvStrich 
         %
\MATHloEck    %
\MATHroEck 
\goodbreakpoint%
In[10]:= FCancel
\goodbreakpoint%
In[11]:= hypAttributes

Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.

\endMATH 
\Seealso F, H, V, hypAttributes.

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

\Name FFormat
\Description Switch that activates  hypergeometric output, or makes 
  it inactive, respectively. By default  hypergeometric output is 
  active.
\Usage FFormat.
\Example
\MATH
In[1]:= p[a,n]/GAMMA[b/2]*F[%
\MATHlbrace c,d%
\MATHrbrace ,%
\MATHlbrace c*d%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  c, d    %
\MATHvStrich 
         F %
\MATHvStrich      ; z %
\MATHvStrich  (a)
        2 1%
\MATHvStrich  c d     %
\MATHvStrich     n
           %
\MATHloEck          %
\MATHroEck 
Out[1]= -------------------
                 b
               \MATHGamma (-)
                 2
\goodbreakpoint%
In[2]:= Tgl5402
Do you want to set values for the equation? [y|n]: n
\goodbreakpoint%
\goodbreakpoint%
Format::toobig: Expression too big for output.
                 Enter "FFormat" and retry.
\goodbreakpoint%
\goodbreakpoint%
In[3]:= FFormat
\goodbreakpoint%
In[4]:= \%\%
\goodbreakpoint%
Out[4]= F[%
\MATHlbrace -n, b, c, d, e%
\MATHrbrace , %
\MATHlbrace 1 - b - n, 1 - c - n, 1 - d - n, 
 
\MATHgroesser       -2 + 2 b + 2 c + 2 d + e + 2 n%
\MATHrbrace , 1] == 
 
                             3   b   c   d
\MATHgroesser     F[%
\MATHlbrace 1 - b - c - d - 2 n, - - - - - - - - n, 1 - c - d - n, 1 - b - d - n, 
                             2   2   2   2
 
                       -n  1   n
\MATHgroesser        1 - b - c - n, --, - - -, e, 3 - 2 b - 2 c - 2 d - e - 3 n%
\MATHrbrace , 
                       2   2   2
 
        1   b   c   d
\MATHgroesser       %
\MATHlbrace - - - - - - - - n, 1 - b - n, 1 - c - n, 1 - d - n, 
        2   2   2   2
 
                        3 n  3               3 n
\MATHgroesser        2 - b - c - d - ---, - - b - c - d - ---, 2 - b - c - d - e - 2 n, 
                         2   2                2
 
\MATHgroesser        -1 + b + c + d + e + n%
\MATHrbrace , 1] 
 
\MATHgroesser      p[%
\MATHlbrace 2 - b - c - d - e - 2 n, 3 - 2 b - 2 c - 2 d - 3 n%
\MATHrbrace , 
 
\MATHgroesser       %
\MATHlbrace 2 - b - c - d - 2 n, 3 - 2 b - 2 c - 2 d - e - 3 n%
\MATHrbrace , n]
\goodbreakpoint%
In[5]:= \%1
\goodbreakpoint%
        F[%
\MATHlbrace c, d%
\MATHrbrace , %
\MATHlbrace c d%
\MATHrbrace , z] p[a, n]
Out[5]= ---------------------------
                       b
                 GAMMA[-]
                       2
\goodbreakpoint%
In[6]:= FFormat
\goodbreakpoint%
In[7]:= \%1
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  c, d    %
\MATHvStrich 
         F %
\MATHvStrich      ; z %
\MATHvStrich  (a)
        2 1%
\MATHvStrich  c d     %
\MATHvStrich     n
           %
\MATHloEck          %
\MATHroEck 
Out[7]= -------------------
                 b
               \MATHGamma (-)
                 2
\endMATH

\Name FH
\Description Rule that transforms a \hbox{\tt F[]} into a difference of a
\hbox{\tt H[]} and a \hbox{\tt F[]}.
\Usage Expr/.FH.
\Example
\MATH
In[1]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich      ; z %
\MATHvStrich 
        2 1%
\MATHvStrich   c      %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.FH
\goodbreakpoint%
                            %
\MATHluEck                  %
\MATHruEck    (-1 + c, 0)
                            %
\MATHvStrich  2 - c, 1, 1   1 %
\MATHvStrich               1
                          F %
\MATHvStrich              ; - %
\MATHvStrich  -----------------
           %
\MATHluEck          %
\MATHruEck    3 2%
\MATHvStrich  2 - a, 2 - b  z %
\MATHvStrich  (-1 + a, -1 + b)
           %
\MATHvStrich  a, b    %
\MATHvStrich       %
\MATHloEck                  %
\MATHroEck                  1
Out[2]=  H %
\MATHvStrich      ; z %
\MATHvStrich  - ----------------------------------------
        2 2%
\MATHvStrich  c, 1    %
\MATHvStrich                       z
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[3]:= \%/.paufl
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[3]=  H %
\MATHvStrich      ; z %
\MATHvStrich 
        2 2%
\MATHvStrich  c, 1    %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[4]:= F[%
\MATHlbrace a,b,1%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, 1    %
\MATHvStrich 
Out[4]=  F %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 2%
\MATHvStrich   c, d      %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[5]:= \%/.FH
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck       %
\MATHluEck                     %
\MATHruEck  (-1 + c, -1 + d)
           %
\MATHvStrich  a, b    %
\MATHvStrich       %
\MATHvStrich  2 - c, 2 - d, 1    %
\MATHvStrich                  1
Out[5]=  H %
\MATHvStrich      ; 1 %
\MATHvStrich  -  F %
\MATHvStrich                 ; 1 %
\MATHvStrich  -----------------
        2 2%
\MATHvStrich  c, d    %
\MATHvStrich    3 2%
\MATHvStrich   2 - a, 2 - b      %
\MATHvStrich  (-1 + a, -1 + b)
           %
\MATHloEck          %
\MATHroEck       %
\MATHloEck                     %
\MATHroEck                  1
\endMATH
\Seealso F, H, HF, Ers, PosListe.

\Name FOrdne
\Description Rule that tries to order the parameters of a  
  hypergeometric series in ``well-poised" order. If the parameters
  could be paired such that the sum of each pair equals 
\hbox{\tt A}${}+1$, 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 FEinf} before applying \hbox{\tt
FOrdne}.
\Usage Expr/.FOrdne.
\Example 
\MATH
In[1]:= F[%
\MATHlbrace -n,b,1+a/2,a%
\MATHrbrace ,%
\MATHlbrace a+1-b,a/2,a+1+n%
\MATHrbrace ,z]
\goodbreakpoint%
                            a
           %
\MATHluEck      -n, b, 1 + -, a        %
\MATHruEck 
           %
\MATHvStrich                 2           %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich             a               %
\MATHvStrich 
           %
\MATHloEck  1 + a - b, -, 1 + a + n    %
\MATHroEck 
                        2
\goodbreakpoint%
In[2]:= \%/.FOrdne
\goodbreakpoint%
                        a
           %
\MATHluEck      a, 1 + -, -n, b        %
\MATHruEck 
           %
\MATHvStrich             2               %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  a                          %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a + n, 1 + a - b    %
\MATHroEck 
             2
In[3]:= F[%
\MATHlbrace b,c,1%
\MATHrbrace ,%
\MATHlbrace a+1-c,a+1-b%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck 
           %
\MATHvStrich        b, c, 1           %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich                      ; z %
\MATHvStrich 
        3 2%
\MATHvStrich  1 + a - c, 1 + a - b    %
\MATHvStrich 
           %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.FOrdne
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck 
           %
\MATHvStrich        b, c, 1           %
\MATHvStrich 
Out[4]=  F %
\MATHvStrich                      ; z %
\MATHvStrich 
        3 2%
\MATHvStrich  1 + a - c, 1 + a - b    %
\MATHvStrich 
           %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[5]:= \%/.FEinf
Add the parameter: a
\goodbreakpoint%
           %
\MATHluEck                             %
\MATHruEck 
           %
\MATHvStrich        a, b, c, 1           %
\MATHvStrich 
Out[5]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  a, 1 + a - c, 1 + a - b    %
\MATHvStrich 
           %
\MATHloEck                             %
\MATHroEck 
\goodbreakpoint%
In[6]:= \%/.FOrdne
\goodbreakpoint%
           %
\MATHluEck                             %
\MATHruEck 
           %
\MATHvStrich        a, b, c, 1           %
\MATHvStrich 
Out[6]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  1 + a - b, 1 + a - c, a    %
\MATHvStrich 
           %
\MATHloEck                             %
\MATHroEck 
\goodbreakpoint%
In[7]:= FCancel
\goodbreakpoint%
In[8]:= \%\%/.FOrdne
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck 
           %
\MATHvStrich        b, c, 1           %
\MATHvStrich 
Out[8]=  F %
\MATHvStrich                      ; z %
\MATHvStrich 
        3 2%
\MATHvStrich  1 + a - b, 1 + a - c    %
\MATHvStrich 
           %
\MATHloEck                          %
\MATHroEck 
\endMATH 
\Seealso FEinf, FPerm, FTausche, F, V, PSort, Ers, PosListe.
\Name FPerm
\Description Rule for permuting parameters in basic hypergeometric series.
\Usage Expr/.FPerm[$\langle$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. 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]:= F[%
\MATHlbrace a,b,c,d%
\MATHrbrace ,%
\MATHlbrace e,f,g%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, b, c, d    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich            ; z %
\MATHvStrich 
        4 3%
\MATHvStrich   e, f, g      %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%1/.FPerm[3,2,1,u]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  c, b, a, d    %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich            ; z %
\MATHvStrich 
        4 3%
\MATHvStrich   e, f, g      %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[3]:= \%1/.FPerm[3,2,1,l]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, b, c, d    %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich            ; z %
\MATHvStrich 
        4 3%
\MATHvStrich   g, f, e      %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%1/.FPerm[2,1,b]
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck 
           %
\MATHvStrich  a, c, b, d    %
\MATHvStrich 
Out[4]=  F %
\MATHvStrich            ; z %
\MATHvStrich 
        4 3%
\MATHvStrich   f, e, g      %
\MATHvStrich 
           %
\MATHloEck                %
\MATHroEck 
\endMATH
\Seealso FTausche, FOrdne, F, V, PSort, Ers, PosListe.


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

\Name FTausche
\Description Rule for reordering parameters in  hypergeometric series.
\Usage Expr/.FTausche[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]:= F[%
\MATHlbrace -n,b,1+a/2,a%
\MATHrbrace ,%
\MATHlbrace a+1-b,a/2,a+1+n%
\MATHrbrace ,z]
\goodbreakpoint%
                            a
           %
\MATHluEck      -n, b, 1 + -, a        %
\MATHruEck 
           %
\MATHvStrich                 2           %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich             a               %
\MATHvStrich 
           %
\MATHloEck  1 + a - b, -, 1 + a + n    %
\MATHroEck 
                        2
\goodbreakpoint%
In[2]:= \%/.FTausche[1,4,u]
\goodbreakpoint%
                        a
           %
\MATHluEck      b, 1 + -, a, -n        %
\MATHruEck 
           %
\MATHvStrich             2               %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich             a               %
\MATHvStrich 
           %
\MATHloEck  1 + a - b, -, 1 + a + n    %
\MATHroEck 
                        2
\goodbreakpoint%
In[3]:= \%/.FTausche[3,2,l]
\goodbreakpoint%
                        a
           %
\MATHluEck      b, 1 + -, a, -n        %
\MATHruEck 
           %
\MATHvStrich             2               %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich                        a    %
\MATHvStrich 
           %
\MATHloEck  1 + a - b, 1 + a + n, -    %
\MATHroEck 
                                   2
\goodbreakpoint%
In[4]:= \%/.FTausche[1,3,b]
\goodbreakpoint%
                               a
           %
\MATHluEck      b, a, -n, 1 + -        %
\MATHruEck 
           %
\MATHvStrich                    2        %
\MATHvStrich 
Out[4]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich             a               %
\MATHvStrich 
           %
\MATHloEck  1 + a + n, -, 1 + a - b    %
\MATHroEck 
                        2
\endMATH 
\Seealso FPerm, FOrdne, F, V, PSort, Ers, PosListe.

\Name GAMMA
\Description \hbox{\tt GAMMA[x]} is the Gamma function $\Gamma(x)$.
  \hbox{\tt GAMMA[List1,List2]} is also provided as the usual abbreviation for the
  quotient of Gamma functions (cf\. the example for \hbox{\tt Gzerl}). 
\Usage GAMMA[x]

   \leavevmode\hphantom{Usa}\rm or: \tt GAMMA[List1,List2].

\Example 
\MATH
In[1]:= GAMMA[2*a+1]
\goodbreakpoint%
Out[1]= \MATHGamma (1 + 2 a)
\goodbreakpoint%
In[2]:= GAMMA[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ]
\goodbreakpoint%
         %
\MATHluEck       %
\MATHruEck 
         %
\MATHvStrich  a, b %
\MATHvStrich 
Out[2]= \MATHGamma %
\MATHvStrich       %
\MATHvStrich 
         %
\MATHvStrich  c, d %
\MATHvStrich 
         %
\MATHloEck       %
\MATHroEck 
\endMATH 
\Seealso p, P, Gzerl, Gzus, FFormat.

\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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= Gleichung
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[3]:= Add[1]
\goodbreakpoint%
               %
\MATHluEck           %
\MATHruEck         (-a + c)
               %
\MATHvStrich  a, -n    %
\MATHvStrich                 n
Out[3]= 1 +  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == 1 + ---------
            2 1%
\MATHvStrich    c      %
\MATHvStrich           (c)
               %
\MATHloEck           %
\MATHroEck              n
\goodbreakpoint%
In[4]:= Gleichung
\goodbreakpoint%
               %
\MATHluEck           %
\MATHruEck         (-a + c)
               %
\MATHvStrich  a, -n    %
\MATHvStrich                 n
Out[4]= 1 +  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == 1 + ---------
            2 1%
\MATHvStrich    c      %
\MATHvStrich           (c)
               %
\MATHloEck           %
\MATHroEck              n
\goodbreakpoint%
In[5]:= LS=5
\goodbreakpoint%
Out[5]= 5
\goodbreakpoint%
In[6]:= Gleichung
\goodbreakpoint%
                 (-a + c)
                         n
Out[6]= 5 == 1 + ---------
                   (c)
                      n
\goodbreakpoint%
In[7]:= SUM[n,0,m]
\goodbreakpoint%
          m         m        m
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck      %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck     %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (-a + c)
         \MATHbackslash          \MATHbackslash         \MATHbackslash            n
Out[7]=   \MATHgroesser   5 ==   \MATHgroesser   1 +   \MATHgroesser   ---------
         /         /        /     (c)
        %
\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
        %
\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 + c)
         \MATHbackslash     \MATHbackslash          \MATHbackslash     \MATHbackslash         \MATHbackslash     \MATHbackslash            n
Out[8]=   \MATHgroesser     \MATHgroesser   5 ==   \MATHgroesser     \MATHgroesser   1 +   \MATHgroesser     \MATHgroesser   ---------
         /    /         /    /        /    /     (c)
        %
\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
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck      %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck     (-a + c)
         \MATHbackslash     \MATHbackslash          \MATHbackslash     \MATHbackslash                n
Out[9]=   \MATHgroesser     \MATHgroesser   5 ==   \MATHgroesser     \MATHgroesser   1 + ---------
         /    /         /    /         (c)
        %
\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
\endMATH 
\Seealso SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, Sub,
Hoch, GlTausche, Ers, Subst,\linebreak PSort.

\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
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck     (-a + c)  (-b + c)
           %
\MATHvStrich        a, b, -n          %
\MATHvStrich             n         n
Out[1]=  F %
\MATHvStrich                      ; 1 %
\MATHvStrich  == -------------------
        3 2%
\MATHvStrich  c, 1 + a + b - c - n    %
\MATHvStrich     (c)  (-a - b + c)
           %
\MATHloEck                          %
\MATHroEck        n             n
\goodbreakpoint%
In[2]:= Gleichung
\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]:= GlTausche
\goodbreakpoint%
        (-a + c)  (-b + c)        %
\MATHluEck                          %
\MATHruEck 
                n         n       %
\MATHvStrich        a, b, -n          %
\MATHvStrich 
Out[3]= ------------------- ==  F %
\MATHvStrich                      ; 1 %
\MATHvStrich 
        (c)  (-a - b + c)      3 2%
\MATHvStrich  c, 1 + a + b - c - n    %
\MATHvStrich 
           n             n        %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[4]:= Gleichung
\goodbreakpoint%
        (-a + c)  (-b + c)        %
\MATHluEck                          %
\MATHruEck 
                n         n       %
\MATHvStrich        a, b, -n          %
\MATHvStrich 
Out[4]= ------------------- ==  F %
\MATHvStrich                      ; 1 %
\MATHvStrich 
        (c)  (-a - b + c)      3 2%
\MATHvStrich  c, 1 + a + b - c - n    %
\MATHvStrich 
           n             n        %
\MATHloEck                          %
\MATHroEck 
\endMATH
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, 
    Sub, Hoch, Ers, Subst.

\Name GOSPER
\Description Rule that does symbolic summation using Gosper's algorithm
\cite{\GospAB}.

Here a call is made to the function {\tt Gosper} of the {\sl Mathematica} 
implementation of Gosper's and Zeilberger's algorithms written by Peter Paule and
Markus Schorn. The current version~1.1 or updates can be received via
e-mail request to \hbox{\tt peter.paule\@risc.uni-linz.ac.at}. This
implemenation provides the user with the objects
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Zb}, \hbox{\tt Gosper}, \hbox{\tt RunMode}, \hbox{\tt FileName}, 
\hbox{\tt SolAmount},
\hbox{\tt Fnk}, \hbox{\tt GoRat}, \hbox{\tt GoSol}, \hbox{\tt Cert}, 
  \hbox{\tt DegBound}, \hbox{\tt System}, \hbox{\tt SystemDimension}.
\par}
\medskip\noindent
Also within the package HYP,
all these objects work as described in the documentation of this
implementation. Therefore the user is referred to this documentation
and the description
\cite{\PaScAA} in order to learn about the
various features of these objects. 

The package HYP provides two additional objects, {\tt ZB} and
\hbox{\tt GOSPER}. The rule {\tt GOSPER} allows to apply Gosper's algorithm
directly to an expression containing a {\tt SUM} or a hypergeometric
series.
\Usage Expr/.GOSPER[]

   \leavevmode\hphantom{Usa}\rm or: \tt Expr/.GOSPER[order].

\rm   \leavevmode\hphantom{Usaor:} (Runs Gosper's algorithm trying an additional polynomial Ansatz of degree 
        'order'.)
\Example
\MATH
In[1]:= SUM[1/(k*(k+1)),%
\MATHlbrace k,1,n%
\MATHrbrace ]
\goodbreakpoint%
          n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
         \MATHbackslash        1
Out[1]=   \MATHgroesser   ---------
         /   k (1 + k)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=1
\goodbreakpoint%
In[2]:= \%/.GOSPER[]
 
    Peter Paule and Markus Schorn's implementation of the
                Gosper algorithm. (Version 1.1)
\goodbreakpoint%
                 1                          1
Out[2]= %
\MATHlbrace SUM[---------, %
\MATHlbrace k, 1, n%
\MATHrbrace ] == 1 - -----%
\MATHrbrace 
             k (1 + k)                    1 + n
In[3]:= F[%
\MATHlbrace 1,1%
\MATHrbrace ,%
\MATHlbrace 3%
\MATHrbrace ,1]/2
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  1, 1    %
\MATHvStrich 
         F %
\MATHvStrich      ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich   3      %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
Out[3]= --------------
              2
\goodbreakpoint%
In[4]:= \%/.GOSPER[]
 
    Peter Paule and Markus Schorn's implementation of the
                Gosper algorithm. (Version 1.1)
\goodbreakpoint%
                kk!
Out[4]= %
\MATHlbrace SUM[---------, %
\MATHlbrace kk, 0, \MATHinfty %
\MATHrbrace ] == 1%
\MATHrbrace 
             (2 + kk)!
\endMATH
\Seealso ZB.

\Name Gzerl
\Description Rule that splits \hbox{\tt GAMMA[List1,List2]} into a quotient of 
  products of Gamma functions.
\Usage Expr/.Gzerl.
\Example 
\MATH
In[1]:= GAMMA[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ]
\goodbreakpoint%
         %
\MATHluEck       %
\MATHruEck 
         %
\MATHvStrich  a, b %
\MATHvStrich 
Out[1]= \MATHGamma %
\MATHvStrich       %
\MATHvStrich 
         %
\MATHvStrich  c, d %
\MATHvStrich 
         %
\MATHloEck       %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.Gzerl
\goodbreakpoint%
        \MATHGamma (a) \MATHGamma (b)
Out[2]= ---------
        \MATHGamma (c) \MATHGamma (d)
\endMATH 
\Seealso paufl, pzerl, pzus, GAMMA, Gzus, Ers, PosListe.

\Name Gzus
\Description Rule that collects several Gammma functions 
  \hbox{\tt GAMMA[x$_{\text {\tt i}}$]} to an expression
\hbox{\tt GAMMA[List1,List2]}. 
\Usage Expr/.Gzus.
\Example 
\MATH
In[1]:= GAMMA[a]*GAMMA[b]/GAMMA[c]/GAMMA[d]
\goodbreakpoint%
        \MATHGamma (a) \MATHGamma (b)
Out[1]= ---------
        \MATHGamma (c) \MATHGamma (d)
\goodbreakpoint%
In[2]:= \%/.Gzus
\goodbreakpoint%
         %
\MATHluEck       %
\MATHruEck 
         %
\MATHvStrich  a, b %
\MATHvStrich 
Out[2]= \MATHGamma %
\MATHvStrich       %
\MATHvStrich 
         %
\MATHvStrich  c, d %
\MATHvStrich 
         %
\MATHloEck       %
\MATHroEck 
\endMATH 
\Seealso paufl, pzerl, pzus, GAMMA, Gzerl, Ers, PosListe.

\Name H
\Description \hbox{\tt H[List1,List2,z]} is the 
bilateral hypergeometric series with upper parameters \hbox{\tt
List}1, lower parameters \hbox{\tt List}2, and argument \hbox{\tt z}.
\Usage H[List1,List2,z].
\Example 
\MATH
In[1]:= H[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, c    %
\MATHvStrich 
Out[1]=  H %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\endMATH
\Seealso SListe, TListe, SUMRegeln, SUMH, HSUM, F, p, GAMMA, FCancel, 
FOrdne, FPerm, FTausche, P,\linebreak
FFormat.

\Name HEinf
\Description Rule that inactivates automatic cancelling in \hbox{\tt
H[]} and then
adds a parameter which has to be entered on request to the upper and
lower parameters of \hbox{\tt H[]}.
\Usage Expr/.HEinf.
\Example
\MATH
In[1]:= H[%
\MATHlbrace b,c%
\MATHrbrace ,%
\MATHlbrace 1+a-c,1+a-b%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck 
           %
\MATHvStrich          b, c            %
\MATHvStrich 
Out[1]=  H %
\MATHvStrich                      ; 1 %
\MATHvStrich 
        2 2%
\MATHvStrich  1 + a - c, 1 + a - b    %
\MATHvStrich 
           %
\MATHloEck                          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.HEinf
Add the parameter: a
\goodbreakpoint%
           %
\MATHluEck                             %
\MATHruEck 
           %
\MATHvStrich          a, b, c            %
\MATHvStrich 
Out[2]=  H %
\MATHvStrich                         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  a, 1 + a - c, 1 + a - b    %
\MATHvStrich 
           %
\MATHloEck                             %
\MATHroEck 
\endMATH
\Seealso FCancel, HOrdne, HPerm, PSort, SUMRegeln, SUMUmkehr, Ers, PosListe.

\Name HF
\Description Rule that transforms a \hbox{\tt H[]} into a sum of two
\hbox{\tt F[]}'s. 
\vskip6pt
\leavevmode\hphantom{Description: }
$ \dsize{}_r H_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; 
z\right]
\to\sum\limits _{n=-\infty} ^{m-1}\dfrac {\po{a_1}{n}\cdots\po{a_r}{n}}
{\po{b_1}{n}\cdots\po{b_s}{n}} z^n+
\sum\limits _{n=m} ^{\infty}\dfrac {\po{a_1}{n}\cdots\po{a_r}{n}}
{\po{b_1}{n}\cdots\po{b_s}{n}} z^n
$.
\vskip6pt
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.HF.
\Example
\MATH
In[1]:= H[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[1]=  H %
\MATHvStrich      ; 1 %
\MATHvStrich 
        2 2%
\MATHvStrich  c, d    %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.HF
Split at: 0
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck       %
\MATHluEck                     %
\MATHruEck  (a, b)
           %
\MATHvStrich  a, b, 1    %
\MATHvStrich       %
\MATHvStrich  2 - c, 2 - d, 1    %
\MATHvStrich        -1
Out[2]=  F %
\MATHvStrich         ; 1 %
\MATHvStrich  +  F %
\MATHvStrich                 ; 1 %
\MATHvStrich  --------
        3 2%
\MATHvStrich   c, d      %
\MATHvStrich    3 2%
\MATHvStrich   2 - a, 2 - b      %
\MATHvStrich  (c, d)
           %
\MATHloEck             %
\MATHroEck       %
\MATHloEck                     %
\MATHroEck        -1
\goodbreakpoint%
In[3]:= H[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[3]=  H %
\MATHvStrich      ; 1 %
\MATHvStrich 
        2 2%
\MATHvStrich  c, d    %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.HF
Split at: 5
\goodbreakpoint%
           %
\MATHluEck                       %
\MATHruEck  (a, b)
           %
\MATHvStrich  -3 - c, -3 - d, 1    %
\MATHvStrich        4
Out[4]=  F %
\MATHvStrich                   ; 1 %
\MATHvStrich  ------- + 
        3 2%
\MATHvStrich   -3 - a, -3 - b      %
\MATHvStrich  (c, d)
           %
\MATHloEck                       %
\MATHroEck        4
 
        %
\MATHluEck                     %
\MATHruEck  (a, b)
        %
\MATHvStrich  5 + a, 5 + b, 1    %
\MATHvStrich        5
\MATHgroesser      F %
\MATHvStrich                 ; 1 %
\MATHvStrich  -------
     3 2%
\MATHvStrich   5 + c, 5 + d      %
\MATHvStrich  (c, d)
        %
\MATHloEck                     %
\MATHroEck        5
\endMATH
\Seealso H, F, FH, Ers, PosListe.

\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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= Hoch[3]
\goodbreakpoint%
                                     3
           %
\MATHluEck           %
\MATHruEck      (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich 3            n
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich   == ----------
        2 1%
\MATHvStrich    c      %
\MATHvStrich            3
           %
\MATHloEck           %
\MATHroEck        (c)
                                 n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
                                     3
           %
\MATHluEck           %
\MATHruEck      (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich 3            n
Out[3]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich   == ----------
        2 1%
\MATHvStrich    c      %
\MATHvStrich            3
           %
\MATHloEck           %
\MATHroEck        (c)
                                 n
\endMATH 
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div,
Sub, GlTausche, Ers.

\Name HOrdne
\Description Rule that tries to order the parameters of a bilateral
hypergeometric series in ``well-poised" order. If there is an
 upper parameter of the form $-n$, where $n$ is a nonnegative integer, 
then it is put at the very last place in the upper list. 
\Usage Expr/.HOrdne.
\Example
\MATH
In[1]:= H[%
\MATHlbrace -n,b,1+a/2%
\MATHrbrace ,%
\MATHlbrace a+1-b,a/2,a+1+n%
\MATHrbrace ,z]
\goodbreakpoint%
                             a
           %
\MATHluEck       -n, b, 1 + -          %
\MATHruEck 
           %
\MATHvStrich                  2          %
\MATHvStrich 
Out[1]=  H %
\MATHvStrich                         ; z %
\MATHvStrich 
        3 3%
\MATHvStrich             a               %
\MATHvStrich 
           %
\MATHloEck  1 + a - b, -, 1 + a + n    %
\MATHroEck 
                        2
\goodbreakpoint%
In[2]:= \%/.HOrdne
\goodbreakpoint%
                      a
           %
\MATHluEck       1 + -, -n, b          %
\MATHruEck 
           %
\MATHvStrich           2                 %
\MATHvStrich 
Out[2]=  H %
\MATHvStrich                         ; z %
\MATHvStrich 
        3 3%
\MATHvStrich  a                          %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a + n, 1 + a - b    %
\MATHroEck 
             2
\endMATH
\Seealso HEinf, HPerm, H, PSort, Ers, PosListe.

\Name HPerm
\Description Rule for permuting parameters in bilateral hypergeometric 
series.
\Usage: Expr/.HPerm[$\langle$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]:= H[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, c    %
\MATHvStrich 
Out[1]=  H %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%1/.HPerm[3,2,1,u]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  c, b, a    %
\MATHvStrich 
Out[2]=  H %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[3]:= \%1/.HPerm[3,2,1,l]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, c    %
\MATHvStrich 
Out[3]=  H %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  f, e, d    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%1/.HPerm[3,2,1,b]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  c, b, a    %
\MATHvStrich 
Out[4]=  H %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  f, e, d    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\endMATH
\Seealso HOrdne, H, PSort, Ers, PosListe, FPerm.

\Name HShift
\Description Rule that shifts the summation index in a bilateral
hypergeometric series. 
\vskip6pt
\leavevmode\hphantom{Description: }
$\dsize {}_r H_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; 
z\right]
\to z^m\frac {\prod _{i=1} ^{r}(a_i)_m} {\prod _{i=1} ^{s}(b_i)_m}
{}_r H_s\!\left[\matrix a_1+m,\dots,a_r+m\\ b_1+m,\dots,b_s+m\endmatrix; 
z\right]$.
\vskip6pt
\leavevmode\hskip10pt 
    The parameter \hbox{\tt m} has to be entered on request.
\Usage Expr/.HShift.
\Example
\MATH
In[1]:= H[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, c    %
\MATHvStrich 
Out[1]=  H %
\MATHvStrich         ; 1 %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.HShift
shift by: 4
\goodbreakpoint%
           %
\MATHluEck                         %
\MATHruEck  (a, b, c)
           %
\MATHvStrich  4 + a, 4 + b, 4 + c    %
\MATHvStrich           4
Out[2]=  H %
\MATHvStrich                     ; 1 %
\MATHvStrich  ----------
        3 3%
\MATHvStrich  4 + d, 4 + e, 4 + f    %
\MATHvStrich  (d, e, f)
           %
\MATHloEck                         %
\MATHroEck           4
\endMATH
\Seealso HEinf, HPerm, H, PSort, Ers, PosListe.

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

\Name hypAttributes
\Description Shows the current setup of the session. The setup can be 
  changed by the switches \hbox{\tt P}, \hbox{\tt FCancel}, 
  \hbox{\tt AmSTeX}, \hbox{\tt AmSLaTeX}, \hbox{\tt LaTeX}, \hbox{\tt TeX}, 
  and \hbox{\tt TeXFV}. The 
  default-setup is shown in the following Example.
\Usage hypAttributes.
\Example 
\MATH
In[1]:= hypAttributes

Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.

\endMATH 
\Seealso P, FCancel, AmSTeX, AmSLaTeX, LaTeX, TeX, TeXFV.

\Name inv
\Description $\Gamma(a) \to \dfrac {\pi}{\sin(\pi a)}
\dfrac {1} {\Gamma(1-a)}$.
\Usage Expr/.inv.
\Example 
\MATH
In[1]:= GAMMA[2*a+1]
\goodbreakpoint%
Out[1]= \MATHGamma (1 + 2 a)
\goodbreakpoint%
In[2]:= \%/.inv
\goodbreakpoint%
        \MATHpi  Csc[(1 + 2 a) \MATHpi ]
Out[2]= ------------------
             \MATHGamma (-2 a)
\endMATH 
\Seealso inv, 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]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[2]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  c%
\MATHrbrace \MATHbackslash endmatrix ; %
\MATHlbrace \MATHbackslash displaystyle
   z%
\MATHrbrace \MATHbackslash right ] 
\goodbreakpoint%
In[3]:= LaTeX
\goodbreakpoint%
In[4]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[5]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[5]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b\MATHbackslash cr c%
\MATHrbrace  ; %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ] 
\endMATH 
\Seealso AmSTeX, AmSLaTeX, TeX, TeXMat, TeXFV.

\Name Limes
\Description Function for doing formal limits of  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.
\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 Vandermonde summation (see \hbox{\tt S2101}) from the
Pfaff--Saalsch\"utz summation.
\vskip10pt
\MATH
In[1]:= Sgl3201
Do you want to set values for the equation? [y|n]: n
\goodbreakpoint%
           %
\MATHluEck                          %
\MATHruEck     (-a + c)  (-b + c)
           %
\MATHvStrich        a, b, -n          %
\MATHvStrich             n         n
Out[1]=  F %
\MATHvStrich                      ; 1 %
\MATHvStrich  == -------------------
        3 2%
\MATHvStrich  c, 1 + a + b - c - n    %
\MATHvStrich     (c)  (-a - b + c)
           %
\MATHloEck                          %
\MATHroEck        n             n
\goodbreakpoint%
In[2]:= Limes[\%,b-\MATHgroesser Infinity]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\endMATH
\vskip10pt\noindent
The derivation of the Gauss summation (see \hbox{\tt S2103}) from the
Pfaff--Saalsch\"utz summation.
\vskip10pt
\MATH
In[3]:= Limes[\%\%,n-\MATHgroesser Infinity]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich     \MATHGamma (c) \MATHGamma (-a - b + c)
Out[3]=  F %
\MATHvStrich      ; 1 %
\MATHvStrich  == -------------------
        2 1%
\MATHvStrich   c      %
\MATHvStrich     \MATHGamma (-a + c) \MATHGamma (-b + c)
           %
\MATHloEck          %
\MATHroEck 
\endMATH
\Seealso AbsGreater, AbsSmaller, AbsUndetermined, MinusOne.

\Name lina1
\Description \vtab $(a)_n \to a\,(a+1)_{n-1}$,\\
             $\Gamma(a) \to (a-1)\Gamma(a-1)$.
\endvtab
\vskip6pt
\leavevmode\hskip10pt\Usage Expr/.lina1.
\Example 
\MATH
In[1]:= p[a-1,m]
\goodbreakpoint%
Out[1]= (-1 + a)
                m
\goodbreakpoint%
In[2]:= \%/.lina1
\goodbreakpoint%
Out[2]= (-1 + a) (a)
                    -1 + m
\goodbreakpoint%
In[3]:= 1/GAMMA[2*a]
\goodbreakpoint%
          1
Out[3]= ------
        \MATHGamma (2 a)
\goodbreakpoint%
In[4]:= \%/.lina1
\goodbreakpoint%
                  1
Out[4]= ----------------------
        (-1 + 2 a) \MATHGamma (-1 + 2 a)
\endMATH 
\Seealso lina2, linz, Ers, PosListe, ManipulationsListe.

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

\Name linz
\Description Rule that absorbs linear terms.
\Usage Expr/.linz.
\Example 
\MATH
In[1]:= a*p[a+1,m]/(a/2+m-1)/p[a/2,m-1]
\goodbreakpoint%
              a (1 + a)
                       m
Out[1]= ----------------------
              a       a
        (-1 + - + m) (-)
              2       2 -1 + m
\goodbreakpoint%
In[2]:= \%/.linz
\goodbreakpoint%
               (a)
                  1 + m
Out[2]= ----------------------
              a       a
        (-1 + - + m) (-)
              2       2 -1 + m
\goodbreakpoint%
In[3]:= \%/.linz
\goodbreakpoint%
        (a)
           1 + m
Out[3]= --------
           a
          (-)
           2 m
\goodbreakpoint%
In[4]:= 1/(1-b)/GAMMA[b-1]
\goodbreakpoint%
                1
Out[4]= -----------------
        (1 - b) \MATHGamma (-1 + b)
\goodbreakpoint%
In[5]:= \%/.linz
\goodbreakpoint%
           1
Out[5]= -(----)
          \MATHGamma (b)
\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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= LS
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich    c      %
\MATHvStrich 
           %
\MATHloEck           %
\MATHroEck 
\goodbreakpoint%
In[3]:= Add[1]
\goodbreakpoint%
               %
\MATHluEck           %
\MATHruEck         (-a + c)
               %
\MATHvStrich  a, -n    %
\MATHvStrich                 n
Out[3]= 1 +  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == 1 + ---------
            2 1%
\MATHvStrich    c      %
\MATHvStrich           (c)
               %
\MATHloEck           %
\MATHroEck              n
\goodbreakpoint%
In[4]:= LS
\goodbreakpoint%
               %
\MATHluEck           %
\MATHruEck 
               %
\MATHvStrich  a, -n    %
\MATHvStrich 
Out[4]= 1 +  F %
\MATHvStrich       ; 1 %
\MATHvStrich 
            2 1%
\MATHvStrich    c      %
\MATHvStrich 
               %
\MATHloEck           %
\MATHroEck 
\goodbreakpoint%
In[5]:= LS=p[a,m]
\goodbreakpoint%
Out[5]= (a)
           m
\goodbreakpoint%
In[6]:= Gleichung
\goodbreakpoint%
                    (-a + c)
                            n
Out[6]= (a)  == 1 + ---------
           m          (c)
                         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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= Mal[p[c,n]]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  (c)  == (-a + c)
        2 1%
\MATHvStrich    c      %
\MATHvStrich     n            n
           %
\MATHloEck           %
\MATHroEck 
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  (c)  == (-a + c)
        2 1%
\MATHvStrich    c      %
\MATHvStrich     n            n
           %
\MATHloEck           %
\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 
  factorial symbols and Gamma functions.
\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]:= p[a,2*n]
\goodbreakpoint%
Out[1]= (a)
           2 n
\goodbreakpoint%
In[2]:= \%/.trans
\goodbreakpoint%
            2 n
Out[2]= (-1)    (1 - a - 2 n)
                             2 n
\goodbreakpoint%
In[3]:= \%/.MinusOne
Is 2 n even, odd, or neither of both?
[e|o|n]: e
\goodbreakpoint%
Out[3]= (1 - a - 2 n)
                     2 n
\endMATH
\Seealso SimplifyP, Expandq, SUMExpand.

\Name Multinomialp
\Description: \hbox{\tt Multinomialp[n1,n2,\dots]} is the multinomial
coefficient $\binom {\sum_i n_i} {n_1,n_2,\dots}$, 
written in terms of factorial symbols (Pochhammer symbols) {\tt p}. 
\Usage Multinomial[n1,n2,\dots].
\MATH
In[1]:= Multinomialp[a,b,c]
\goodbreakpoint%
         (1)
            a + b + c
Out[1]= --------------
        (1)  (1)  (1)
           a    b    c
\goodbreakpoint%
In[2]:= Multinomialp[3,2,6]
\goodbreakpoint%
            (1)
               11
Out[2]= --------------
        (1)  (1)  (1)
           2    3    6
\endMATH
\Seealso Binomialp, Factorialp.

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

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

\Name p
\Description \hbox{\tt p[x,n]} is the factorial symbol (Pochhammer
symbol) $(x)_n$. 
  \hbox{\tt p[List1,List2,n]} is also provided as the usual abbreviation for the
  quotient of factorial symbols (see \cite{\GaRaAA}). 
\Usage p[x,n]

   \leavevmode\hphantom{Usa}\rm or: \tt p[List1,List2,n].
\Example 
\MATH
In[1]:= p[a,n]
\goodbreakpoint%
Out[1]= (a)
           n
\goodbreakpoint%
In[2]:= p[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,2*m]
\goodbreakpoint%
        (a, b)
              2 m
Out[2]= ---------
        (c, d)
              2 m
\endMATH 
\Seealso GAMMA, Binomialp, Multinomialp, Factorialp, P,
paufl, pzerl, pzus, FFormat.

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

Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
\endMATH 
\Seealso F, H, p, hypAttributes.

\Name paufl
\Description Rule that writes $(x)_n$ as the defining product
    $\prod _{i=0} ^{n-1}(x+i)$, if $n$ is an integer.
\Usage Expr/.paufl.
\Example
\MATH
In[1]:= p[a,-2]/p[b,3]*p[c,1]
\goodbreakpoint%
        (a)   (c)
           -2    1
Out[1]= ----------
           (b)
              3
\goodbreakpoint%
In[2]:= \%/.paufl
\goodbreakpoint%
                         c
Out[2]= -----------------------------------
        (-2 + a) (-1 + a) b (1 + b) (2 + b)
\goodbreakpoint%
In[3]:= F[%
\MATHlbrace a,b,c%
\MATHrbrace ,%
\MATHlbrace d,e,f%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck 
           %
\MATHvStrich  a, b, c    %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich         ; z %
\MATHvStrich 
        3 3%
\MATHvStrich  d, e, f    %
\MATHvStrich 
           %
\MATHloEck             %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.C01
\goodbreakpoint%
                 %
\MATHluEck                            %
\MATHruEck 
                 %
\MATHvStrich  1, 1 + a, 1 + b, 1 + c    %
\MATHvStrich 
            z  F %
\MATHvStrich                        ; z %
\MATHvStrich  (a)  (b)  (c)
              4 4%
\MATHvStrich  2, 1 + d, 1 + e, 1 + f    %
\MATHvStrich     1    1    1
                 %
\MATHloEck                            %
\MATHroEck 
Out[4]= 1 + -------------------------------------------------
                             (d)  (e)  (f)
                                1    1    1
\goodbreakpoint%
In[5]:= \%/.paufl
\goodbreakpoint%
                       %
\MATHluEck                            %
\MATHruEck 
                       %
\MATHvStrich  1, 1 + a, 1 + b, 1 + c    %
\MATHvStrich 
            a b c z  F %
\MATHvStrich                        ; z %
\MATHvStrich 
                    4 4%
\MATHvStrich  2, 1 + d, 1 + e, 1 + f    %
\MATHvStrich 
                       %
\MATHloEck                            %
\MATHroEck 
Out[5]= 1 + ----------------------------------------
                             d e f
\endMATH
\Seealso pzerl, pzus, p, Gzerl, Gzus, 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]:= p[a,n]/p[1,n]*SUM[p[b,k]/p[c,k+1]*a\MATHhoch k,%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
                \MATHinfty    k
              %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck a  (b)
               \MATHbackslash          k
        (a)  (  \MATHgroesser   --------)
           n   /   (c)
              %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    1 + k
               k=0
Out[1]= --------------------
                (1)
                   n
\goodbreakpoint%
In[2]:= PosListe[\%]
\goodbreakpoint%
                                          \MATHinfty    k
                                        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck a  (b)
           1                             \MATHbackslash          k
Out[2]= %
\MATHlbrace %
\MATHlbrace ----, %
\MATHlbrace %
\MATHlbrace 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace (a) , %
\MATHlbrace %
\MATHlbrace 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace   \MATHgroesser   --------, %
\MATHlbrace %
\MATHlbrace 3%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
          (1)               n            /   (c)
             n                          %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    1 + k
                                         k=0
\goodbreakpoint%
In[3]:= PosListe[\%\%,2]
\goodbreakpoint%
Out[3]= %
\MATHlbrace %
\MATHlbrace -1, %
\MATHlbrace %
\MATHlbrace 1, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace a, %
\MATHlbrace %
\MATHlbrace 2, 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace n, %
\MATHlbrace %
\MATHlbrace 2, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace %
\MATHlbrace k, 0, \MATHinfty %
\MATHrbrace , %
\MATHlbrace %
\MATHlbrace 3, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , 
 
                         k
                        a  (b)
                              k
\MATHgroesser     %
\MATHlbrace (1) , %
\MATHlbrace %
\MATHlbrace 1, 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace --------, %
\MATHlbrace %
\MATHlbrace 3, 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
         n              (c)
                           1 + k
\goodbreakpoint%
In[4]:= PosListe[\%\%\%,3]
\goodbreakpoint%
                                               k
Out[4]= %
\MATHlbrace %
\MATHlbrace 0, %
\MATHlbrace %
\MATHlbrace 3, 2, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace 1, %
\MATHlbrace %
\MATHlbrace 1, 1, 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace a , %
\MATHlbrace %
\MATHlbrace 3, 1, 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , 
 
\MATHgroesser     %
\MATHlbrace k, %
\MATHlbrace %
\MATHlbrace 3, 2, 1%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace n, %
\MATHlbrace %
\MATHlbrace 1, 1, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace \MATHinfty , %
\MATHlbrace %
\MATHlbrace 3, 2, 3%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , 
 
                              1
\MATHgroesser     %
\MATHlbrace (b) , %
\MATHlbrace %
\MATHlbrace 3, 1, 2%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace --------, %
\MATHlbrace %
\MATHlbrace 3, 1, 3%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 
         k                 (c)
                              1 + k
\endMATH 
\Seealso Ers, Subst.

\Name PSort
\Description Rule that orders the parameters of hypergeometric series 
   \hbox{\tt F[List1,List2,z]}, \hbox{\tt H[\dots]}, \hbox{\tt
V[\dots]}, of ``multiple" Pochhammer symbols
\hbox{\tt p[List1,List2,n]},
   and of ``multiple" Gamma functions \hbox{\tt GAMMA[List1,List2]} 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 Gzus} and \hbox{\tt pzus[n]}
first.
\Usage Expr/.PSort.
\Example
\MATH
In[1]:= p[%
\MATHlbrace c,c-a,b%
\MATHrbrace ,%
\MATHlbrace a-b-c,b+1%
\MATHrbrace ,n]*GAMMA[%
\MATHlbrace b-a,c%
\MATHrbrace ,%
\MATHlbrace a+b+c,-b%
\MATHrbrace ]*
        F[%
\MATHlbrace b-c,b+a,a%
\MATHrbrace ,%
\MATHlbrace b,a%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck                  %
\MATHruEck   %
\MATHluEck                %
\MATHruEck    (c, -a + c, b)
           %
\MATHvStrich  b - c, a + b    %
\MATHvStrich   %
\MATHvStrich    -a + b, c   %
\MATHvStrich                  n
Out[1]=  F %
\MATHvStrich              ; z %
\MATHvStrich  \MATHGamma %
\MATHvStrich                %
\MATHvStrich  -------------------
        2 1%
\MATHvStrich       b          %
\MATHvStrich   %
\MATHvStrich  a + b + c, -b %
\MATHvStrich  (a - b - c, 1 + b)
           %
\MATHloEck                  %
\MATHroEck   %
\MATHloEck                %
\MATHroEck                    n
\goodbreakpoint%
In[2]:= \%/.PSort
\goodbreakpoint%
           %
\MATHluEck                  %
\MATHruEck   %
\MATHluEck                %
\MATHruEck    (b, c, -a + c)
           %
\MATHvStrich  a + b, b - c    %
\MATHvStrich   %
\MATHvStrich    -a + b, c   %
\MATHvStrich                  n
Out[2]=  F %
\MATHvStrich              ; z %
\MATHvStrich  \MATHGamma %
\MATHvStrich                %
\MATHvStrich  -------------------
        2 1%
\MATHvStrich       b          %
\MATHvStrich   %
\MATHvStrich  -b, a + b + c %
\MATHvStrich  (1 + b, a - b - c)
           %
\MATHloEck                  %
\MATHroEck   %
\MATHloEck                %
\MATHroEck                    n
\endMATH
\Seealso SimplifyP, SUMExpand, FEinf, FOrdne, FPerm, FTausche, F, H, V, p, GAMMA.

\Name pzerl
\Description Rule that splits \hbox{\tt p[List1,List2,n]} into a quotient of 
products of factorial symbols (Pochhammer symbols).
\Usage Expr/.pzerl.
\Example 
\MATH
In[1]:= p[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c,d%
\MATHrbrace ,n]
\goodbreakpoint%
        (a, b)
              n
Out[1]= -------
        (c, d)
              n
\goodbreakpoint%
In[2]:= \%/.pzerl
\goodbreakpoint%
        (a)  (b)
           n    n
Out[2]= ---------
        (c)  (d)
           n    n
\endMATH 
\Seealso paufl, pzus, p, Gzerl, Gzus, Ers, PosListe.

\Name pzus
\Description Rule that collects several factorial symbols (Pochhammer
symbols) \hbox{\tt p[x$_{\text {\tt 
i}}$,n]} to
an expression\linebreak \hbox{\tt p[List1,List2,n]}. 
\Usage Expr/.pzus[n].
\Example 
\MATH
In[1]:= p[a,n]*p[b,m]/p[c,n]/p[d,m]
\goodbreakpoint%
        (a)  (b)
           n    m
Out[1]= ---------
        (c)  (d)
           n    m
\goodbreakpoint%
In[2]:= \%/.pzus[n]
\goodbreakpoint%
             (a)
                n
        (b)  ----
           m (c)
                n
Out[2]= ---------
          (d)
             m
\goodbreakpoint%
In[3]:= \%/.pzus[m]
\goodbreakpoint%
        (a)  (b)
           n    m
Out[3]= ---- ----
        (c)  (d)
           n    m
\endMATH 
\Seealso paufl, pzerl, p, Gzus, Gzerl, 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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= RS
\goodbreakpoint%
        (-a + c)
                n
Out[2]= ---------
          (c)
             n
\goodbreakpoint%
In[3]:= Add[1]
\goodbreakpoint%
               %
\MATHluEck           %
\MATHruEck         (-a + c)
               %
\MATHvStrich  a, -n    %
\MATHvStrich                 n
Out[3]= 1 +  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == 1 + ---------
            2 1%
\MATHvStrich    c      %
\MATHvStrich           (c)
               %
\MATHloEck           %
\MATHroEck              n
\goodbreakpoint%
In[4]:= RS
\goodbreakpoint%
            (-a + c)
                    n
Out[4]= 1 + ---------
              (c)
                 n
\goodbreakpoint%
In[5]:= RS=1/p[1,m]
\goodbreakpoint%
         1
Out[5]= ----
        (1)
           m
\goodbreakpoint%
In[6]:= Gleichung
\goodbreakpoint%
               %
\MATHluEck           %
\MATHruEck 
               %
\MATHvStrich  a, -n    %
\MATHvStrich      1
Out[6]= 1 +  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ----
            2 1%
\MATHvStrich    c      %
\MATHvStrich     (1)
               %
\MATHloEck           %
\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{\SlatAC}, Appendix (III.1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S1001}.
$$
{} _{1} F _{0} \!\left [ \matrix { a}\\ { -}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{\left( 1 - z \right) }^{-a}}
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2101
\Description Summation formula (\cite{\SlatAC}, (1.7.7); Appendix (III.4)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2101}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, -n}\\ { c}\endmatrix ; {\displaystyle
   1}\right ]  \longrightarrow {{({ \textstyle c-a}) _{n} }\over 
    {({ \textstyle c}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2103
\Description Summation formula (\cite{\SlatAC}, (1.7.6); Appendix (III.3)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2103}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   1}\right ]  \longrightarrow \Gamma \left [ \matrix c, c-a-b\\ c-a, 
   c-b\endmatrix \right ]
$$
\Example
\MATH
In[1]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace 2+a+b%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck               %
\MATHruEck 
           %
\MATHvStrich    a, b       %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich           ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich  2 + a + b    %
\MATHvStrich 
           %
\MATHloEck               %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.S2103
\goodbreakpoint%
         %
\MATHluEck               %
\MATHruEck 
         %
\MATHvStrich  2 + a + b, 2 %
\MATHvStrich 
Out[2]= \MATHGamma %
\MATHvStrich               %
\MATHvStrich 
         %
\MATHvStrich  2 + b, 2 + a %
\MATHvStrich 
         %
\MATHloEck               %
\MATHroEck 
\endMATH
\Seealso S3201, SListe, SumListe, Ers, PosListe.

\Name S2104
\Description Summation formula (\cite{\SlatAC}, (1.7.1.6), corrected,
(2.3.2.9); Appendix (III.5)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2104}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { 1 + a - b}\endmatrix ;
   {\displaystyle -1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix 1 + {a\over 2}, 1 + a - b\\ 1 + a, 1 + {a\over 2} -
   b\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2105
\Description Summation formula (\cite{\GaRaAA}, Ex.~1.6(i), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2105}.
$$
{} _{2} F _{1} \!\left [ \matrix { -{{n}\over 2}, {1\over 2} - {n\over 2}}\\ {
   {1\over 2} + b}\endmatrix ; {\displaystyle 1}\right ]  
\longrightarrow 
  {{{2^n} ({ \textstyle b}) _{n} }\over {({ \textstyle 2 b}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2106
\Description Summation formula (\cite{\SlatAC}, (1.5.21)) in form of a rule.
$q$-Analogues are HYPQ's \hbox{\tt S2106} and \hbox{\tt S3203}.
$$
{} _{2} V _{1} ({\displaystyle a; -}; {\displaystyle z}) 
\longrightarrow 
  {{\left( 1 - z \right) }^{-1 - a}} \left( 1 + z \right) 
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2131
\Description Summation formula (\cite{\SlatAC}, (1.7.1.8); Appendix (III.7)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2201}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, 1 - a}\\ { b}\endmatrix ; {\displaystyle
   {1\over 2}}\right ]  \longrightarrow \Gamma \left [ \matrix {b\over 2}, {1\over 2} +
   {b\over 2}\\ {a\over 2} + {b\over 2}, {1\over 2} - {a\over 2} + {b\over
   2}\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2132
\Description Summation formula (\cite{\SlatAC}, (1.7.1.9); Appendix (III.6)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2202}.
$$
{} _{2} F _{1} \!\left [ \matrix { 2 a, 2 b}\\ { {1\over 2} + a + b}\endmatrix
   ; {\displaystyle {1\over 2}}\right ]  \longrightarrow 
  \Gamma \left [ \matrix {1\over 2}, {1\over 2} + a + b\\ {1\over 2} + a,
   {1\over 2} + b\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2210
\Description Summation formula (\cite{\SlatAC}, (6.1.2.6),
$d\to-\infty$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S2210}.
$$
{} _{2} H _{2} \!\left [ \matrix { b, c}\\ { 1 + a - b, 1 + a - c}\endmatrix ;
   {\displaystyle -1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix 1 - {a\over 2}, 1 + {a\over 2}, 1 - b, 1 + a - b, 1 -
   c, 1 + a - c\\ 1 - a, 1 + a, 1 + {a\over 2} - b, 1 + {a\over 2} - c, 1 + a
   - b - c\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S2240
\Description Summation formula (\cite{\SlatAC}, (6.1.2.1),
Appendix (III.28)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S3310}.
$$
{} _{2} H _{2} \!\left [ \matrix { a, b}\\ { c, d}\endmatrix ; {\displaystyle
   1}\right ]  \longrightarrow \Gamma \left [ \matrix c, d, 1 - a, 1 - b, -1 - a - b + c +
   d\\ -a + c, -a + d, -b + c, -b + d\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3201
\Description Summation formula (\cite{\SlatAC}, (2.3.1.3); Appendix (III.2)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S3201}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { c, 1 + a + b - c -
   n}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  {{({ \textstyle c-a}) _{n}  ({ \textstyle c-b}) _{n} }\over 
    {({ \textstyle c}) _{n}  ({ \textstyle c-a-b}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Example
\MATH
In[1]:= F[%
\MATHlbrace a,b,-n%
\MATHrbrace ,%
\MATHlbrace a+b-n,1%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck                  %
\MATHruEck 
           %
\MATHvStrich    a, b, -n      %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich              ; 1 %
\MATHvStrich 
        3 2%
\MATHvStrich  a + b - n, 1    %
\MATHvStrich 
           %
\MATHloEck                  %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.S3201
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
        (a - n)  (b - n)
               n        n
Out[2]= ------------------
        (a + b - n)  (-n)
                   n     n
\goodbreakpoint%
In[3]:= \%\%/.S3201
Is n a nonnegative integer?
[y|n]:  n
Is -b a nonnegative integer?
[y|n]:  n
Is -a a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
           %
\MATHluEck                  %
\MATHruEck 
           %
\MATHvStrich    a, b, -n      %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich              ; 1 %
\MATHvStrich 
        3 2%
\MATHvStrich  a + b - n, 1    %
\MATHvStrich 
           %
\MATHloEck                  %
\MATHroEck 
\goodbreakpoint%
In[4]:= F[%
\MATHlbrace a,-m,-n%
\MATHrbrace ,%
\MATHlbrace a-m-n,1%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck                  %
\MATHruEck 
           %
\MATHvStrich   a, -m, -n      %
\MATHvStrich 
Out[4]=  F %
\MATHvStrich              ; 1 %
\MATHvStrich 
        3 2%
\MATHvStrich  a - m - n, 1    %
\MATHvStrich 
           %
\MATHloEck                  %
\MATHroEck 
\goodbreakpoint%
In[5]:= \%/.S3201
Is n a nonnegative integer?
[y|n]:  n
Is m a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
        (a - m)  (-m - n)
               m         m
Out[5]= ------------------
        (-m)  (a - m - n)
            m            m
\endMATH
\Seealso S2103, SListe, SumListe, Ers, PosListe.

\Name S3202
\Description Summation formula (\cite{\SlatAC}, (2.3.3.5); Appendix
(III.8); terminated in the first
  variable) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S3202}.
$$
{} _{3} F _{2} \!\left [ \matrix { -2 n, b, c}\\ { 1 - b - 2 n, 1 - c - 2
   n}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  {{({ \textstyle 1}) _{2 n}  ({ \textstyle b}) _{n}  ({ \textstyle c}) _{n}  
      ({ \textstyle b + c}) _{2 n} }\over 
    {({ \textstyle 1}) _{n}  ({ \textstyle b}) _{2 n}  
      ({ \textstyle c}) _{2 n}  ({ \textstyle b + c}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3204
\Description Summation formula (\cite{\GaRaAA}, Ex.~3.9, $q\to 1$) in form of a rule.
$q$-Analogues are HYPQ's \hbox{\tt S3204} and \hbox{\tt S8704}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, 1 + {{{\la}}\over 2}, b}\\ { {{{\la}}\over 2}, 1 - b + {\la}}\endmatrix ; {\displaystyle 1}\right ]  
\longrightarrow 
  \Gamma \left [ \matrix {\la}, 1 - a + {\la}, -a - 2 b + {\la}, 1 -
   b + {\la}\\ 1 + {\la}, -a + {\la}, -2 b + {\la}, 1 - a - b +
   {\la}\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3231
\Description Summation formula (\cite{\SlatAC}, (2.3.3.5); Appendix (III.8)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4301}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { 1 + a - b, 1 + a -
   c}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix 1 + a - b, 1 + a - c, 1 + {a\over 2}, 1 + {a\over 2}
   - b - c\\ 1 + a, 1 + {a\over 2} - b, 1 + {a\over 2} - c, 1 + a - b -
   c\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3232
\Description Summation formula (\cite{\SlatAC}, (2.3.3.6); Appendix (III.9)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4302}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { 1 + a - b, 1 + a +
   n}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  {{({ \textstyle 1 + a}) _{n}  ({ \textstyle 1 + {a\over 2} - b}) _{n} }\over
      {({ \textstyle 1 + {a\over 2}}) _{n}  ({ \textstyle 1 + a - b}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3233
\Description Summation formula (\cite{\SlatAC}, (2.3.3.13); Appendix (III.23)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S8702}, a terminating
$q$-analogue is HYPQ's \hbox{\tt S4303}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { {{1 + a + b}\over 2}, 2
   c}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix {1\over 2}, {1\over 2} + c, {1\over 2} + {a\over 2} +
   {b\over 2}, {1\over 2} - {a\over 2} - {b\over 2} + c\\ {1\over 2} + {a\over
   2}, {1\over 2} + {b\over 2}, {1\over 2} - {a\over 2} + c, {1\over 2} -
   {b\over 2} + c\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3234
\Description Summation formula (\cite{\SlatAC}, (2.3.3.14); Appendix (III.24)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S8703}, a terminating
$q$-analogue is HYPQ's \hbox{\tt S4304}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, 1 - a, c}\\ { d, 1 + 2 c - d}\endmatrix
   ; {\displaystyle 1}\right ]  \longrightarrow 
  {2^{1 - 2 c}} \pi  \Gamma \left [ \matrix d, 1 + 2 c - d\\ {1\over 2} +
    {a\over 2} + c - {d\over 2}, {a\over 2} + {d\over 2}, 1 - {a\over 2} + c -
    {d\over 2}, {1\over 2} - {a\over 2} + {d\over 2}\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.


\Name S3235
\Description Summation formula (\cite{\SlatAC}, (2.4.2.5); Appendix (III.16)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4308}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { 1 + a - b, 1 + 2 b -
   n}\endmatrix ; {\displaystyle 1}\right ]  
\longrightarrow 
  {{({ \textstyle 1 + {a\over 2} - b}) _{n} ({ \textstyle a - 2 b}) _{n}    
      ({ \textstyle -b}) _{n} }\over 
    {({ \textstyle {a\over 2} - b}) _{n}  ({ \textstyle 1 + a - b}) _{n}  
      ({ \textstyle -2 b}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3261
\Description Summation formula (\cite{\SlatAC}, (2.4.4.4); Appendix (III.31)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S3261}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, 1 + a + b + c -
   d}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix 1 + a - d, 1 + b - d, 1 + c - d, 1 + a + b + c - d\\
    1 - d, 1 + b + c - d, 1 + a + c - d, 1 + a + b - d\endmatrix \right ]
\\- 
    \Gamma \left [ \matrix  d-1, 1 + a - d, 1 + b - d, 1 + c - d, 1 + a + b
     + c - d\\ 1 - d, a, b, c, 2 + a + b + c - 2 d\endmatrix \right ]
   {} _{3} F _{2} \!\left [ \matrix { 1 + a - d, 1 + b - d, 1 + c - d}\\ { 2 -
     d, 2 + a + b + c - 2 d}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S3291
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.9, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4361}.\NoBlackBoxes
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, 1 + a - 2 c}\\ { 1 + a - c, 2
    b}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow
 \Gamma \left [ \matrix {1\over 2}, {1\over 2} + {a\over 2} - b, 1 + a
   - c, 1 + {a\over 2} - b - c\\ {1\over 2} + {a\over 2}, {1\over 2} - b, 1 +
   {a\over 2} - c, 1 + a - b - c\endmatrix \right ]\\
-    {} _{3} F _{2} \!\left [ \matrix { 1 - b, 1 + a - 2 b, 2 + a - 2 b - 2 c}\\
     { 2 + a - 2 b - c, 2 - 2 b}\endmatrix ; {\displaystyle 1}\right ]  
    \Gamma \left [ \matrix {1\over 2} + {a\over 2} - b, 1 + {a\over 2} - b,
     -{1\over 2} + b, 1 + a - c, 1 + {a\over 2} - b - c, {3\over 2} + {a\over
     2} - b - c\\ {1\over 2} + {a\over 2}, {a\over 2}, {1\over 2} - b, {1\over
     2} + {a\over 2} - c, 1 + {a\over 2} - c, 2 + a - 2 b - c\endmatrix \right
     ]
\endmultline$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.
\par\BlackBoxes

\Name S3340
\Description Summation formula (\cite{\SlatAC}, (6.1.2.6),
Appendix (III.30)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4410}.
$$
\multline
{} _{3} H _{3} \!\left [ \matrix { b, c, d}\\ { 1 + a - b, 1 + a - c, 1 + a -
   d}\endmatrix ; {\displaystyle 1}\right ]  \\
\longrightarrow 
  \Gamma \left [ \matrix 1 - {a\over 2}, 1 + {a\over 2}, 1 - b, 1 + a - b, 1 -
   c, 1 + a - c, 1 - d, 1 + a - d, 1 + {{3 a}\over 2} - b - c - d\\ 1 - a, 1
   + a, 1 + {a\over 2} - b, 1 + {a\over 2} - c, 1 + a - b - c, 1 + {a\over 2}
   - d, 1 + a - b - d, 1 + a - c - d\endmatrix \right ]
\endmultline
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4306
\Description Summation formula (\cite{\SlatAC}, (2.4.2.6); Appendix
(III.17)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4306}.
$$
{} _{4} F _{3} \!\left [ \matrix { a, 1 + {a\over 2}, b, -n}\\ { {a\over 2}, 1
   + a - b, 1 + 2 b - n}\endmatrix ; {\displaystyle 1}\right ]  
\longrightarrow 
  {{({ \textstyle a - 2 b, -b}) _{n}}\over  {({ \textstyle 1 + a - b, -2 b})
   _{n}}} 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4307
\Description Summation formula (\cite{\SlatAC}, (2.3.4.6); Appendix
(III.10)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S4307}.
$$
{} _{4} V _{3} ({\displaystyle a; b, c}; {\displaystyle -1}) 
\longrightarrow 
  \Gamma\left[ \matrix 1 + a - b, 1 + a - c \\ 
    1 + a, 1 + a - b - c\endmatrix \right]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4331
\Description Summation formula (\cite{\SlatAC}, Appendix
(III.22)) in form of a rule.
$q$-Analogues are HYPQ's \hbox{\tt S5401} and \hbox{\tt S5501}.
$$
{} _{4} V _{3} ({\displaystyle a; b, c}; {\displaystyle 1}) 
\longrightarrow 
  \Gamma \left [ \matrix 1 + a - b, 1 + a - c, {1\over 2} + {a\over 2},
   {1\over 2} + {a\over 2} - b - c\\ 1 + a, 1 + a - b - c, {1\over 2} +
   {a\over 2} - b, {1\over 2} + {a\over 2} - c\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S4332
\Description Summation formula (\cite{\SlatAC}, (2.4.2.7); Appendix
(III.18)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S5402}.
$$
{} _{4} F _{3} \!\left [ \matrix { a, 1 + {a\over 2}, b, -n}\\ { {a\over 2}, 1
   + a - b, 2 + 2 b - n}\endmatrix ; {\displaystyle 1}\right ]  
\longrightarrow 
  {{({ \textstyle {1\over 2} + {a\over 2} - b}) _{n}
   ({ \textstyle -1 + a - 2 b}) _{n}  ({ \textstyle -1 - b}) _{n}  
       }\over 
    {({ \textstyle -{1\over 2} + {a\over 2} - b}) _{n}  
      ({ \textstyle 1 + a - b}) _{n} ({ \textstyle -1 - 2 b}) _{n}  }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S5431
\Description Summation formula (\cite{\SlatAC}, (2.3.4.5); Appendix (III.12)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S6501}.
$$
{} _{5} V _{4} ({\displaystyle a; b, c, d}; {\displaystyle 1}) \longrightarrow 
  \Gamma \left [ \matrix 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - b - c - d\\
   1 + a, 1 + a - b - c, 1 + a - b - d, 1 + a - c - d\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S5432
\Description Summation formula (\cite{\SlatAC}, (2.3.4.6); Appendix (III.13)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S6502}.
$$
{} _{5} V _{4} ({\displaystyle a; b, c, -n}; {\displaystyle 1}) \longrightarrow 
  {{({ \textstyle 1 + a}) _{n}  ({ \textstyle 1 + a - b - c}) _{n} }\over 
    {({ \textstyle 1 + a - b}) _{n}  ({ \textstyle 1 + a - c}) _{n} }}
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S5540
\Description Summation formula (\cite{\SlatAC}, (6.1.2.5),
Appendix (III.29)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S6610}.
$$
\multline
{} _{5} H _{5} \!\left [ \matrix { 1 + {a\over 2}, b, c, d, e}\\ { {a\over 2},
   1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e}\endmatrix ; {\displaystyle
   1}\right ]  \\
\longrightarrow \Gamma \left [ \matrix 1 - b, 1 + a - b, 1 - c, 1 + a - c, 1
   - d, 1 + a - d, 1 - e, 1 + a - e, 1 + 2 a - b - c - d - e\\ 1 - a, 1 + a,
   1 + a - b - c, 1 + a - b - d, 1 + a - c - d, 1 + a - b - e, 1 + a - c - e,
   1 + a - d - e\endmatrix \right ]
\endmultline
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S6531
\Description Summation formula (\cite{\VeJaAH}, (1.8)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S8702}.
$$
{} _{6} V _{5} ({\displaystyle -{1\over 2} + {a\over 2} + {b\over 2} + c; a,
   b, c, {1\over 2} + {a\over 2} + {b\over 2} - c}; {\displaystyle -1}) 
\longrightarrow 
  \Gamma \left [ \matrix {1\over 2}, {1\over 2} + {a\over 2} + {b\over 2},
   {1\over 2} + c, {1\over 2} + {a\over 2} - {b\over 2} + c, {1\over 2} -
   {a\over 2} + {b\over 2} + c\\ {1\over 2} + {a\over 2}, {1\over 2} + {b\over
   2}, {1\over 2} - {a\over 2} + c, {1\over 2} - {b\over 2} + c, {1\over 2} +
   {a\over 2} + {b\over 2} + c\endmatrix \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S6532
\Description Summation formula (\cite{\SlatAC}, Appendix (III.27),
corrected; \cite{\VeJaAH}, (1.7)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S8703}.
$$
{} _{6} V _{5} ({\displaystyle a; b, 1 - b, d, 1 - d}; {\displaystyle -1}) 
\longrightarrow 
  {2^{2 b}} \Gamma \left [ \matrix 1 + a - b, a + b, 1 + a - d, 1 + {a\over 2}
    + {b\over 2} - {d\over 2}, {1\over 2} + {a\over 2} + {b\over 2} + {d\over
    2}, a + d\\ a, 1 + a, 1 + a + b - d, 1 + {a\over 2} - {b\over 2} - {d\over
    2}, {1\over 2} + {a\over 2} - {b\over 2} + {d\over 2}, a + b + d\endmatrix
    \right ]
$$
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S7631
\Description Summation formula (\cite{\SlatAC}, (2.3.4.4); Appendix (III.14)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S8701}.
$$
{} _{7} V _{6} ({\displaystyle a; b, c, d, 1 + 2 a - b - c - d + n, -n};
   {\displaystyle 1}) \longrightarrow {{({ \textstyle 1 + a}) _{n}  
      ({ \textstyle 1 + a - b - c}) _{n}  ({ \textstyle 1 + a - b - d}) _{n}  
      ({ \textstyle 1 + a - c - d}) _{n} }\over 
    {({ \textstyle 1 + a - b}) _{n}  ({ \textstyle 1 + a - c}) _{n}  
      ({ \textstyle 1 + a - d}) _{n}  ({ \textstyle 1 + a - b - c - d}) _{n} }
    }
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S7632
\Description Summation formula (\cite{\SlatAC}, (2.4.1.5); Appendix
(III.19)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S10901}.
$$
{} _{7} V _{6} ({\displaystyle a; {b}, {1\over 2} + {b}, a - 2b,
   1 + 2 a - 2b + n, -n}; {\displaystyle 1}) 
\longrightarrow 
  {{({ \textstyle 1 + a, 1 + 2 a - 4 b}) _{n}}\over  {({ \textstyle 1 + a -
   2b, 1 + 2 a - 2b}) _{n}}} 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, SListe, SumListe, Ers, PosListe.

\Name S7691
\Description Summation formula (\cite{\SlatAC}, (4.2.3.8); Appendix (III.32)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt S8761}.
$$\align
{} _{7} V &_{6} ( a; b, c, d, e, 1 + 2 a - b - c - d - e;
   {\displaystyle 1}) \\
&\longrightarrow \Gamma \left [ \matrix 1 + a - c, 1 + a - d, 1 + a -
    e, -a + b + c + d + e, -a + b + c, -a + b + d, -a + b + e, 1 + a - c - d -
    e\\ 1 + a, -a + b, 1 + a - c - d, 1 + a - c - e, -a + b + d + e, 1 + a - d
    - e, -a + b + c + e, -a + b + c + d\endmatrix \right ] \\
& \quad -  \Gamma \left [ \matrix a - b, 1 + a - c, 1 + a - d, 1 + a - e, -a + b + c
     + d + e, -a + b + c, -a + b + d, -a + b + e, \\
 1 + a, c, d, e, 1 + 2 a - b - c - d - e, -a + b, 1 + b - c, 1 + b - d,
    \endmatrix \right .\\
&\hskip1cm
 \left.\matrix  1 + a - c - d - e, 1 - a + 2 b\\  1 + b - e,
   -2 a + 2 b + c + d + e\endmatrix\right]
   {} _{7} V _{6} ({\displaystyle -a + 2 b; b, -a + b + c, -a + b + d, -a + b
     + e, 1 + a - c - d - e}; {\displaystyle 1}) 
\endalign$$
\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[p[a,n],filename]
\goodbreakpoint%
In[3]:= !type filename.m
A[1]:=
p[a, n]
\goodbreakpoint%
In[3]:= !type filename.tex
A[1]:=
(%
\MATHlbrace  \MATHbackslash textstyle a%
\MATHrbrace ) \MATHtief %
\MATHlbrace n%
\MATHrbrace  
\goodbreakpoint%
In[3]:= SchreibeZahl=4
\goodbreakpoint%
Out[3]= 4
\goodbreakpoint%
In[4]:= TeXMat[p[a,n],filename]
\goodbreakpoint%
In[5]:= !type filename.m
A[1]:=
p[a, n]
A[5]:=
p[a, n]
\goodbreakpoint%
In[5]:= !type filename.tex
A[1]:=
(%
\MATHlbrace  \MATHbackslash textstyle a%
\MATHrbrace ) \MATHtief %
\MATHlbrace n%
\MATHrbrace  
A[5]:=
(%
\MATHlbrace  \MATHbackslash textstyle a%
\MATHrbrace ) \MATHtief %
\MATHlbrace n%
\MATHrbrace  
\endMATH 
\Seealso TeXMat.

\Name Sgl1001
\Description Summation formula (\cite{\SlatAC}, Appendix (III.1)) in form of an equation.
It is the same summation as that in \hbox{\tt S1001}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2101
\Description Summation formula (\cite{\SlatAC}, Appendix (III.4)) in form of an equation.
It is the same summation as that in \hbox{\tt S2101}.
\Example 
\MATH
In[1]:= %
\MATHlbrace a,c%
\MATHrbrace 
\goodbreakpoint%
Out[1]= %
\MATHlbrace a, c%
\MATHrbrace 
\goodbreakpoint%
In[2]:= Sgl2101
Do you want to set values for the equation? [y|n]: n
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[3]:= Sgl2101
Do you want to set values for the equation? [y|n]: y
a=2*a
c=-b
n=n
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck     (-2 a - b)
           %
\MATHvStrich  2 a, -n    %
\MATHvStrich               n
Out[3]=  F %
\MATHvStrich         ; 1 %
\MATHvStrich  == -----------
        2 1%
\MATHvStrich    -b       %
\MATHvStrich        (-b)
           %
\MATHloEck             %
\MATHroEck            n
\goodbreakpoint%
In[4]:= %
\MATHlbrace a,c%
\MATHrbrace 
\goodbreakpoint%
Out[4]= %
\MATHlbrace a, c%
\MATHrbrace 
\goodbreakpoint%
In[5]:= a=2*b
\goodbreakpoint%
Out[5]= 2 b
\goodbreakpoint%
In[6]:= 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
\goodbreakpoint%
           %
\MATHluEck             %
\MATHruEck     (-2 b + c)
           %
\MATHvStrich  2 b, -n    %
\MATHvStrich               n
Out[6]=  F %
\MATHvStrich         ; 1 %
\MATHvStrich  == -----------
        2 1%
\MATHvStrich     c       %
\MATHvStrich        (c)
           %
\MATHloEck             %
\MATHroEck           n
\goodbreakpoint%
In[7]:= %
\MATHlbrace a,c%
\MATHrbrace 
\goodbreakpoint%
Out[7]= %
\MATHlbrace 2 b, c%
\MATHrbrace 
\goodbreakpoint%
In[8]:= 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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[8]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[9]:= %
\MATHlbrace a,c%
\MATHrbrace 
\goodbreakpoint%
Out[9]= %
\MATHlbrace a, c%
\MATHrbrace 
\goodbreakpoint%
In[10]:= c=3*e
\goodbreakpoint%
Out[10]= 3 e
\goodbreakpoint%
In[11]:= 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]: yv
a=w+2
c=d
n=n
\goodbreakpoint%
            %
\MATHluEck               %
\MATHruEck     (-2 + d - w)
            %
\MATHvStrich  2 + w, -n    %
\MATHvStrich                 n
Out[11]=  F %
\MATHvStrich           ; 1 %
\MATHvStrich  == -------------
         2 1%
\MATHvStrich      d        %
\MATHvStrich         (d)
            %
\MATHloEck               %
\MATHroEck            n
\goodbreakpoint%
In[12]:= %
\MATHlbrace a,c%
\MATHrbrace 
\goodbreakpoint%
Out[12]= %
\MATHlbrace a, c%
\MATHrbrace 
\endMATH 
\Seealso SumListe\$gl, Gleichung.

\Name Sgl2103
\Description Summation formula (\cite{\SlatAC}, Appendix (III.3)) 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{\SlatAC}, Appendix (III.5)) 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), $q\to 1$) 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{\SlatAC}, (1.5.21)) in form of an equation.
It is the same summation as that in \hbox{\tt S2106}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2131
\Description Summation formula (\cite{\SlatAC}, Appendix (III.7)) in form of an equation.
It is the same summation as that in \hbox{\tt S2131}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2132
\Description Summation formula (\cite{\SlatAC}, Appendix (III.6)) in form of an equation.
It is the same summation as that in \hbox{\tt S2132}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2210
\Description Summation formula (\cite{\SlatAC}, (6.1.2.6),
$d\to-\infty$) in form of an equation.
It is the same summation as that in \hbox{\tt S2210}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl2240
\Description Summation formula (\cite{\SlatAC}, (6.1.2.1), 
Appendix (III.28)) in form of an equation.
It is the same summation as that in \hbox{\tt S3310}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3201
\Description Summation formula (\cite{\SlatAC}, Appendix (III.2)) 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{\SlatAC}, Appendix (III.8), terminated in the first
  variable) in form of an equation.
It is the same summation as that in \hbox{\tt S3202}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3204
\Description Summation formula (\cite{\GaRaAA}, Ex.~3.9, $q\to 1$) in form of an equation.
It is the same summation as that in \hbox{\tt S3204}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3231
\Description Summation formula (\cite{\SlatAC}, Appendix (III.8)) in form of an equation.
It is the same summation as that in \hbox{\tt S3231}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3232
\Description Summation formula (\cite{\SlatAC}, Appendix (III.9)) in form of an equation.
It is the same summation as that in \hbox{\tt S3232}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3233
\Description Summation formula (\cite{\SlatAC}, Appendix (III.23)) in form of an equation.
It is the same summation as that in \hbox{\tt S3233}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3234
\Description Summation formula (\cite{\SlatAC}, Appendix (III.24)) in form of an equation.
It is the same summation as that in \hbox{\tt S3234}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3235
\Description Summation formula (\cite{\SlatAC}, (2.4.2.5); Appendix (III.16))
in form of an equation.
It is the same summation as that in \hbox{\tt S3235}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3261
\Description Summation formula (\cite{\SlatAC}, Appendix (III.31)) in form of an equation.
It is the same summation as that in \hbox{\tt S3261}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3291
\Description Summation formula (\cite{\GaRaAA}, Ex.~2.9, $q\to 1$) in form of an equation.
It is the same summation as that in \hbox{\tt S3291}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl3340
\Description Summation formula (\cite{\SlatAC}, (6.1.2.6),
Appendix (III.30)) in form of an equation.
It is the same summation as that in \hbox{\tt S4410}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4306
\Description Summation formula (\cite{\SlatAC}, (2.4.2.6); Appendix
(III.17)) 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{\SlatAC}, (2.3.4.6); Appendix
(III.10)) in form of an equation.
It is the same summation as that in \hbox{\tt S4307}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4331
\Description Summation formula (\cite{\SlatAC}, Appendix
(III.22)) in form of an equation.
It is the same summation as that in \hbox{\tt S4331}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl4332
\Description Summation formula (\cite{\SlatAC}, (2.4.2.7); Appendix
(III.18)) in form of an equation.
It is the same summation as that in \hbox{\tt S4332}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl5431
\Description Summation formula (\cite{\SlatAC}, Appendix (III.12)) in form of an equation.
It is the same summation as that in \hbox{\tt S5431}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl5432
\Description Summation formula (\cite{\SlatAC}, Appendix (III.13)) in form of an equation.
It is the same summation as that in \hbox{\tt S5432}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl5540
\Description Summation formula (\cite{\SlatAC}, (6.1.2.5),
Appendix (III.29)) in form of an equation.
It is the same summation as that in \hbox{\tt S6610}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl6531
\Description Summation formula (\cite{\VeJaAH}, (1.8)) in form of an equation.
It is the same summation as that in \hbox{\tt S6531}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl6532
\Description Summation formula (\cite{\SlatAC}, Appendix (III.27),
corrected; \cite{\VeJaAH}, (1.7)) in form of an equation.
It is the same summation as that in \hbox{\tt S6532}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl7631
\Description Summation formula (\cite{\SlatAC}, Appendix (III.14)) in form of an equation.
It is the same summation as that in \hbox{\tt S7631}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl7632
\Description Summation formula (\cite{\SlatAC}, (2.4.1.5); Appendix
(III.19)) in form of an equation.
It is the same summation as that in \hbox{\tt S7632}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name Sgl7691
\Description Summation formula (\cite{\SlatAC}, Appendix (III.32)) in form of an equation.
It is the same summation as that in \hbox{\tt S7691}.
\Seealso Sgl2101, SumListe\$gl, Gleichung.

\Name SimplifyP
\Description Rule that simplifies arguments in \hbox{\tt p},
\hbox{\tt GAMMA}, \hbox{\tt F}, \hbox{\tt H}, \hbox{\tt V}, \hbox{\tt SUM}, and
    expands exponents in powers.
\Usage Expr/.SimplifyP.
\Example
\MATH
In[1]:= p[a-(d-c)/2,(k-l)*2]/GAMMA[(f-g)\MATHhoch 2]*F[%
\MATHlbrace -(d+c)/2,-n%
\MATHrbrace ,%
\MATHlbrace -d/2-c/2%
\MATHrbrace ,
        (t-1)/(1-t)]
\goodbreakpoint%
             -c - d
           %
\MATHluEck  ------, -n         %
\MATHruEck 
           %
\MATHvStrich    2         -1 + t %
\MATHvStrich       c - d
         F %
\MATHvStrich            ; ------ %
\MATHvStrich  (a + -----)
        2 1%
\MATHvStrich    -c   d    1 - t  %
\MATHvStrich         2   2 (k - l)
           %
\MATHloEck    -- - -           %
\MATHroEck 
               2    2
Out[1]= ----------------------------------------------
                                  2
                         \MATHGamma ((f - g) )
\goodbreakpoint%
In[2]:= \%/.SimplifyP
Is c/2 + d/2 a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
           %
\MATHluEck         %
\MATHruEck 
           %
\MATHvStrich  -n     %
\MATHvStrich       c   d
         F %
\MATHvStrich    ; -1 %
\MATHvStrich  (a + - - -)
        1 0%
\MATHvStrich  -      %
\MATHvStrich       2   2 2 k - 2 l
           %
\MATHloEck         %
\MATHroEck 
Out[2]= ----------------------------------
                   2            2
                \MATHGamma (f  - 2 f g + g )
\goodbreakpoint%
In[3]:= Simplify[\%1]
\goodbreakpoint%
             -c - d
           %
\MATHluEck  ------, -n     %
\MATHruEck 
           %
\MATHvStrich    2            %
\MATHvStrich       c   d
         F %
\MATHvStrich            ; -1 %
\MATHvStrich  (a + - - -)
        2 1%
\MATHvStrich    -c   d       %
\MATHvStrich       2   2 2 (k - l)
           %
\MATHloEck    -- - -       %
\MATHroEck 
               2    2
Out[3]= ------------------------------------------
                                2
                       \MATHGamma ((f - g) )
\endMATH
\Seealso Expandq, MinusOne, SUMExpand, PSort.

\Name SListe
\Description Rule that gives for a hypergeometric series a list of 
   applicable summation formulas. Each entry of this list has the format 
   \hbox{\tt $\{$S$\langle$number$\rangle$$\}$}, 
where \hbox{\tt S$\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 FOrdne} 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]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich      ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich   c      %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.SListe
Is -b a nonnegative integer?
[y|n]:  n
Is -a a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
Be sure to apply "FOrdne" before using the following information!
\goodbreakpoint%
Out[2]= %
\MATHlbrace %
\MATHlbrace S2103%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[3]:= F[%
\MATHlbrace a,-n%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich 
        2 1%
\MATHvStrich    c      %
\MATHvStrich 
           %
\MATHloEck           %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.SListe
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
Be sure to apply "FOrdne" before using the following information!
\goodbreakpoint%
Out[4]= %
\MATHlbrace %
\MATHlbrace S2101%
\MATHrbrace , %
\MATHlbrace S2103%
\MATHrbrace %
\MATHrbrace 
\endMATH
\vskip10pt\noindent
Now we consider two examples illustrating the note above. Though none of the 
implemented summations can be applied, both series can be summed, the
first by a special case of Dougall's sum, the second by a special
case of the very well-poised $_5F_4$ sum. These facts are also observed by 
using this package.
\vskip10pt
\MATH
In[5]:= F[%
\MATHlbrace -n,b,c,1-b-c-n/2,-n%
\MATHrbrace ,%
\MATHlbrace 1-b-n,1-c-n,b+c-n/2,1%
\MATHrbrace ,1]
\goodbreakpoint%
                                      n
           %
\MATHluEck     -n, b, c, 1 - b - c - -, -n        %
\MATHruEck 
           %
\MATHvStrich                           2            %
\MATHvStrich 
Out[5]=  F %
\MATHvStrich                                    ; 1 %
\MATHvStrich 
        5 4%
\MATHvStrich                                n       %
\MATHvStrich 
           %
\MATHloEck  1 - b - n, 1 - c - n, b + c - -, 1    %
\MATHroEck 
                                           2
\goodbreakpoint%
In[6]:= \%/.SListe
\goodbreakpoint%
Out[6]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[7]:= Sgl7631
Do you want to set values for the equation? [y|n]: y
a=-n
b=b
c=c
d=-n/2
n=n
\goodbreakpoint%
                                      n
           %
\MATHluEck     -n, b, c, 1 - b - c - -, -n        %
\MATHruEck 
           %
\MATHvStrich                           2            %
\MATHvStrich 
Out[7]=  F %
\MATHvStrich                                    ; 1 %
\MATHvStrich  == 
        5 4%
\MATHvStrich                                n       %
\MATHvStrich 
           %
\MATHloEck  1 - b - n, 1 - c - n, b + c - -, 1    %
\MATHroEck 
                                           2
 
                                        n            n
     (1 - n)  (1 - b - c - n)  (1 - b - -)  (1 - c - -)
            n                n          2 n          2 n
\MATHgroesser     ---------------------------------------------------
                                    n                n
     (1 - b - n)  (1 - c - n)  (1 - -)  (1 - b - c - -)
                n            n      2 n              2 n
\goodbreakpoint%
In[8]:= F[%
\MATHlbrace 1+a/2,1,b,c%
\MATHrbrace ,%
\MATHlbrace a/2,1+a-b,1+a-c%
\MATHrbrace ,1]
\goodbreakpoint%
                     a
           %
\MATHluEck      1 + -, 1, b, c         %
\MATHruEck 
           %
\MATHvStrich          2                  %
\MATHvStrich 
Out[8]=  F %
\MATHvStrich                         ; 1 %
\MATHvStrich 
        4 3%
\MATHvStrich  a                          %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - c    %
\MATHroEck 
             2
\goodbreakpoint%
In[9]:= \%/.SListe
\goodbreakpoint%
Out[9]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[10]:= \%\%/.FEinf
Add the parameter: a
\goodbreakpoint%
                         a
            %
\MATHluEck      a, 1 + -, 1, b, c         %
\MATHruEck 
            %
\MATHvStrich             2                  %
\MATHvStrich 
Out[10]=  F %
\MATHvStrich                            ; 1 %
\MATHvStrich 
         5 4%
\MATHvStrich     a                          %
\MATHvStrich 
            %
\MATHloEck  a, -, 1 + a - b, 1 + a - c    %
\MATHroEck 
                 2
\goodbreakpoint%
In[11]:= \%/.SListe
Is -c a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
Be sure to apply "FOrdne" before using the following information!
\goodbreakpoint%
Out[11]= %
\MATHlbrace %
\MATHlbrace S5431%
\MATHrbrace %
\MATHrbrace 
\endMATH
\Seealso TListe, FPerm, FTausche, 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
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck     (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich             n
Out[1]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  == ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich       (c)
           %
\MATHloEck           %
\MATHroEck          n
\goodbreakpoint%
In[2]:= Sub[p[a,n]/p[c-a,n]]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck      (a)            (a)        (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich         n              n               n
Out[2]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  - --------- == -(---------) + ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich    (-a + c)       (-a + c)       (c)
           %
\MATHloEck           %
\MATHroEck            n              n         n
\goodbreakpoint%
In[3]:= Gleichung
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck      (a)            (a)        (-a + c)
           %
\MATHvStrich  a, -n    %
\MATHvStrich         n              n               n
Out[3]=  F %
\MATHvStrich       ; 1 %
\MATHvStrich  - --------- == -(---------) + ---------
        2 1%
\MATHvStrich    c      %
\MATHvStrich    (-a + c)       (-a + c)       (c)
           %
\MATHloEck           %
\MATHroEck            n              n         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[p[a,k]/p[b,k]/k!,%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck  (a)
         \MATHbackslash        k
Out[1]=   \MATHgroesser   -------
         /   k! (b)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck       k
         k=0
\goodbreakpoint%
In[2]:= Subst[\%,%
\MATHlbrace 1%
\MATHrbrace ,p[a,k]/p[b,k],F[%
\MATHlbrace b-a,-k%
\MATHrbrace ,%
\MATHlbrace b%
\MATHrbrace ,1]]
\goodbreakpoint%
                %
\MATHluEck                %
\MATHruEck 
                %
\MATHvStrich  -a + b, -k    %
\MATHvStrich 
          \MATHinfty    F %
\MATHvStrich            ; 1 %
\MATHvStrich 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 2 1%
\MATHvStrich      b         %
\MATHvStrich 
         \MATHbackslash       %
\MATHloEck                %
\MATHroEck 
Out[2]=   \MATHgroesser   --------------------
         /            k!
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\goodbreakpoint%
In[3]:= Sgl2101
Do you want to set values for the equation? [y|n]: y
a=b-a
c=b
n=k
\goodbreakpoint%
           %
\MATHluEck                %
\MATHruEck     (a)
           %
\MATHvStrich  -a + b, -k    %
\MATHvStrich        k
Out[3]=  F %
\MATHvStrich            ; 1 %
\MATHvStrich  == ----
        2 1%
\MATHvStrich      b         %
\MATHvStrich     (b)
           %
\MATHloEck                %
\MATHroEck        k
\goodbreakpoint%
In[4]:= GlTausche
\goodbreakpoint%
        (a)        %
\MATHluEck                %
\MATHruEck 
           k       %
\MATHvStrich  -a + b, -k    %
\MATHvStrich 
Out[4]= ---- ==  F %
\MATHvStrich            ; 1 %
\MATHvStrich 
        (b)     2 1%
\MATHvStrich      b         %
\MATHvStrich 
           k       %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[5]:= Gleichung
\goodbreakpoint%
        (a)        %
\MATHluEck                %
\MATHruEck 
           k       %
\MATHvStrich  -a + b, -k    %
\MATHvStrich 
Out[5]= ---- ==  F %
\MATHvStrich            ; 1 %
\MATHvStrich 
        (b)     2 1%
\MATHvStrich      b         %
\MATHvStrich 
           k       %
\MATHloEck                %
\MATHroEck 
\goodbreakpoint%
In[6]:= Subst[\%1,%
\MATHlbrace 1%
\MATHrbrace ]
\goodbreakpoint%
                %
\MATHluEck                %
\MATHruEck 
                %
\MATHvStrich  -a + b, -k    %
\MATHvStrich 
          \MATHinfty    F %
\MATHvStrich            ; 1 %
\MATHvStrich 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 2 1%
\MATHvStrich      b         %
\MATHvStrich 
         \MATHbackslash       %
\MATHloEck                %
\MATHroEck 
Out[6]=   \MATHgroesser   --------------------
         /            k!
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck 
         k=0
\endMATH
\Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, GlTausche, Ers,
    PosListe.

\Name SUM
\Description This is HYP'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 SUMF} and \hbox{\tt
SUMInfinity}.
\Seealso SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMSammle, 
SUMTausche, SUMF.

\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 SUMF
\Description Rule that transforms a \hbox{\tt SUM[]} into  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/.SUMF.
\Example 
\MATH
In[1]:= SUM[p[-n,k]/p[1,k]*a\MATHhoch k,%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty    k
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck a  (-n)
         \MATHbackslash           k
Out[1]=   \MATHgroesser   --------
         /     (1)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck      k
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMF
\goodbreakpoint%
           %
\MATHluEck        %
\MATHruEck 
           %
\MATHvStrich  -n    %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich    ; a %
\MATHvStrich 
        1 0%
\MATHvStrich  -     %
\MATHvStrich 
           %
\MATHloEck        %
\MATHroEck 
\goodbreakpoint%
In[3]:= SUM[(k+2)*p[%
\MATHlbrace -n,a%
\MATHrbrace ,%
\MATHlbrace b,c,1%
\MATHrbrace ,k+1]*z\MATHhoch k,%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck             (-n, a)
         \MATHbackslash             k         1 + k
Out[3]=   \MATHgroesser   (2 + k) z  --------------
         /              (b, c, 1)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck                     1 + k
         k=0
\goodbreakpoint%
In[4]:= \%/.SUMF
\goodbreakpoint%
             %
\MATHluEck                        %
\MATHruEck 
             %
\MATHvStrich  3, 1 + a, 1 - n, 1    %
\MATHvStrich 
        2  F %
\MATHvStrich                    ; z %
\MATHvStrich  (a)  (-n)
          4 4%
\MATHvStrich  2, 2, 1 + b, 1 + c    %
\MATHvStrich     1     1
             %
\MATHloEck                        %
\MATHroEck 
Out[4]= -----------------------------------------
                     (1)  (b)  (c)
                        1    1    1
\endMATH 
\Seealso SUM, F, SUMRegeln, SUMH, SUMInfinity, FSUM, Ers, PosListe.

\Name SUMH
\Description Rule that transforms a bilateral \hbox{\tt SUM[]} into  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 SUMF} is applied.
\Usage Expr/.SUMH.
\Example 
\MATH
In[1]:= SUM[p[a,k]/p[b,k],%
\MATHlbrace k,-Infinity,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (a)
         \MATHbackslash       k
Out[1]=   \MATHgroesser   ----
         /   (b)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    k
        k=-\MATHinfty 
\goodbreakpoint%
In[2]:= \%/.SUMH
\goodbreakpoint%
           %
\MATHluEck       %
\MATHruEck 
           %
\MATHvStrich  a    %
\MATHvStrich 
Out[2]=  H %
\MATHvStrich   ; 1 %
\MATHvStrich 
        1 1%
\MATHvStrich  b    %
\MATHvStrich 
           %
\MATHloEck       %
\MATHroEck 
\endMATH
\Seealso SUM, H, SUMRegeln, SUMF, SUMInfinity, HSUM, Ers, PosListe.

\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[p[-n,k]/p[1,k]*a\MATHhoch k,%
\MATHlbrace k,0,n%
\MATHrbrace ]
\goodbreakpoint%
          n   k
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck a  (-n)
         \MATHbackslash           k
Out[1]=   \MATHgroesser   --------
         /     (1)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck      k
         k=0
\goodbreakpoint%
In[2]:= \%/.SUMInfinity
\goodbreakpoint%
          \MATHinfty    k
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck a  (-n)
         \MATHbackslash           k
Out[2]=   \MATHgroesser   --------
         /     (1)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck      k
         k=0
\endMATH 
\Seealso SUM, SUMF, SUMH, 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 SUMRegeln
\Description Rule that transforms the expressions in a \hbox{\tt
SUM[]} into a
form that could also be expressed in hypergeometric notation.
This is useful, if you want to convert a \hbox{\tt SUM[]} into hypergeometric
notation but without using the \hbox{\tt F[]}-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[Binomial[n,i]*Binomial[m,k-i],%
\MATHlbrace i,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (        ) (   )
         \MATHbackslash    (   m    ) ( n )
Out[1]=   \MATHgroesser   (        ) (   )
         /   ( -i + k ) ( i )
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck (        ) (   )
         i=0
\goodbreakpoint%
In[2]:= \%/.SUMRegeln
\goodbreakpoint%
                        \MATHinfty 
                      %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck    (-k)  (-n)
                       \MATHbackslash           i     i
        (1 - k + m)  (  \MATHgroesser   -----------------)
                   k   /   (1)  (1 - k + m)
                      %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    i            i
                       i=0
Out[2]= -------------------------------------
                        (1)
                           k
\endMATH 
\Seealso SUM, F, H, SUMF, SUMH, FSUM, HSUM, 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]:= p[a,n]/p[b,m]*(-1)\MATHhoch n*SUM[1/p[1,k],%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
                      \MATHinfty 
                    %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck 
            n        \MATHbackslash     1
        (-1)  (a)  (  \MATHgroesser   ----)
                 n   /   (1)
                    %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    k
                     k=0
Out[1]= ----------------------
                 (b)
                    m
\goodbreakpoint%
In[2]:= \%/.SUMSammle
\goodbreakpoint%
          \MATHinfty       n
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (-1)  (a)
         \MATHbackslash             n
Out[2]=   \MATHgroesser   ----------
         /   (1)  (b)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    k    m
         k=0
\endMATH 
\Seealso SUM, SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, pzus, Gzus, Ers, 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, SUMShift, 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[Binomial[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 (       )
         \MATHbackslash     \MATHbackslash    (   n   )
Out[1]=   \MATHgroesser     \MATHgroesser   (       )
         /    /   ( k + l )
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck (       )
         l=0  k=0
\goodbreakpoint%
In[2]:= \%/.SUMTausche
\goodbreakpoint%
         n1   n2
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (       )
         \MATHbackslash     \MATHbackslash    (   n   )
Out[2]=   \MATHgroesser     \MATHgroesser   (       )
         /    /   ( k + l )
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck (       )
         k=0  l=0
\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 F[]}.
\Usage Expr/.SUMUmkehr.
\Example 
\MATH
In[1]:= F[%
\MATHlbrace a,-n%
\MATHrbrace ,%
\MATHlbrace b%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck           %
\MATHruEck 
           %
\MATHvStrich  a, -n    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich       ; z %
\MATHvStrich 
        2 1%
\MATHvStrich    b      %
\MATHvStrich 
           %
\MATHloEck           %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.SUMUmkehr
Is -a a nonnegative integer?
[y|n]:  n
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
                    %
\MATHluEck                   %
\MATHruEck  (a)
            n  n    %
\MATHvStrich  -n, 1 - b - n  1 %
\MATHvStrich     n
Out[2]= (-1)  z   F %
\MATHvStrich               ; - %
\MATHvStrich  ----
                 2 1%
\MATHvStrich    1 - a - n    z %
\MATHvStrich  (b)
                    %
\MATHloEck                   %
\MATHroEck     n
\goodbreakpoint%
In[3]:= SUM[p[-n,k]/p[1,k],%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (-n)
         \MATHbackslash        k
Out[3]=   \MATHgroesser   -----
         /   (1)
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    k
         k=0
\goodbreakpoint%
In[4]:= \%/.SUMUmkehr
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
           n
         %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (-n)
          \MATHbackslash        k
        (  \MATHgroesser   -----) (-n)
          /   (1)        n
         %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck    k
          k=0
Out[4]= ------------------
               (1)
                  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 T2103
\Description Transformation formula (\cite{\SlatAC}, (1.3.15)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2103}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{\left( 1 - z \right) }^{c-a-b}} 
   {} _{2} F _{1} \!\left [ \matrix { c-a, c-b}\\ { c}\endmatrix ;
    {\displaystyle z}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2104
\Description Transformation formula (\cite{\SlatAC}, (1.7.1.3)) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2104}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{{\left( 1 - z \right) }^{-a}}}
 {{{} _{2} F _{1} \!\left [ \matrix { a, c-b}\\ {
      c}\endmatrix ; {\displaystyle -{z\over {1 - z}}}\right ] }
    }
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2106
\Description Transformation formula (\cite{\SlatAC}, (1.7.1.3), sum
reversed at the right-hand side) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2106}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, -n}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {z^n}
  {{  ({ \textstyle c-a}) _{n} }\over {({ \textstyle c}) _{n} }}
  {} _{2} F _{1} \!\left [ \matrix { -n, 1 - c - n}\\ {
       1 + a - c - n}\endmatrix ; {\displaystyle -{{1 - z}\over z}}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2107
\Description Transformation formula (\cite{\SlatAC}, (1.8.10),
terminating form) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2107}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, -n}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow 
{{    ({ \textstyle c-a}) _{n} }\over {({ \textstyle c}) _{n} }}
{} _{2} F _{1} \!\left [ \matrix { -n, a}\\ { 1 + a - c -
       n}\endmatrix ; {\displaystyle 1 - z}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2110  
\Description Transformation formula (\cite{\RaVeAA}, (3.2)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2110}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { 1 + a - b}\endmatrix ;
   {\displaystyle z}\right ]  
\longrightarrow 
  {{{{\left( 1 +z \right) }^{-a}}}}
  {{  {} _{2} F _{1} \!\left [ \matrix { {a\over 2}, {1\over 2} + {a\over
       2}}\\ { 1 + a - b}\endmatrix ; {\displaystyle {{4 z}\over {{{\left( 1
       + z \right) }^2}}}}\right ] }}
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2112 
\Description Transformation formula (\cite{\RaVeAA}, (5.10)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2112}.
$$
{} _{2} F _{1} \!\left [ \matrix { {a}, {1\over 2} + {a}}\\ {
   {1\over 2} + b}\endmatrix ; {\displaystyle {z^2}}\right ]  
\longrightarrow 
{{{\left( 1 - z \right) }^{-2a}}}
  {{{} _{2} F _{1} \!\left [ \matrix { 2a, b}\\ { 2 b}\endmatrix ;
      {\displaystyle {{2 z}\over {-1 + z}}}\right ] }
    }
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2131
\Description Transformation formula (\cite{\SlatAC}, (1.8.10),
terminating form, sum reversed at the right-hand side) 
    in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3202}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, -n}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{\left( 1 - z \right) }^n} 
  {{    ({ \textstyle a}) _{n} }\over 
   {({ \textstyle c}) _{n} }}
 {} _{2} F _{1} \!\left [ \matrix { -n, c-a}\\ { 1 - a - n}\endmatrix
       ; {\displaystyle {1\over {1 - z}}}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2132
\Description Transformation formula (\cite{\SlatAC}, (2.5.7)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4307}.
$$
{{{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { {1\over 2} + a + b}\endmatrix ;
      {\displaystyle z}\right ] }^2} \longrightarrow 
  {} _{3} F _{2} \!\left [ \matrix { 2 a, 2 b, a + b}\\ { 2 a + 2 b, {1\over
   2} + a + b}\endmatrix ; {\displaystyle z}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2133 
\Description Transformation formula (\cite{\RaVeAA}, (5.12)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2202}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { {{1\over 2} + a + b}
   }\endmatrix ; {\displaystyle {{{z^2}}\over {-1 + {z^2}}}}\right ]  
\longrightarrow 
{{(1-z)}^a\over {(1+z)^a}}
   {} _{2} F _{1} \!\left [ \matrix { 2 a, a + b}\\ { 2 a + 2 b}\endmatrix
    ; {\displaystyle {{2 z}\over {1 + z}}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2134 
\Description Transformation formula (\cite{\BailAA}, Ex. 4.(iii), p. 97, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3211}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { {1\over 2} + {a\over 2} + {b\over
   2}}\endmatrix ; {\displaystyle z}\right ] \longrightarrow
  {} _{2} F _{1} \!\left [ \matrix { {a\over 2}, {b\over 2}}\\ { {1\over 2} +
   {a\over 2} + {b\over 2}}\endmatrix ; {\displaystyle 4 z\left( 1 - z \right)
    }\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2135 
\Description Transformation formula (\cite{\BailAA}, Ex. 4.(iii), p. 97) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3212}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { {1\over 2} + a + b}\endmatrix ;
   {\displaystyle z}\right ] \longrightarrow
  {} _{2} F _{1} \!\left [ \matrix { 2 a, 2 b}\\ { {1\over 2} + a +
   b}\endmatrix ; {\displaystyle {{1 - {\sqrt{1 - z}}}\over 2}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2136 
\Description Transformation formula (\cite{\RaVeAA}, (5.10), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3213}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { 2 b}\endmatrix ; {\displaystyle
   z}\right ] \longrightarrow
\left( {2\over{2-z}}\right)^a
{ {} _{2} F _{1} \!\left [ \matrix { {a\over 2},
       {1\over 2} + {a\over 2}}\\ { {1\over 2} + b}\endmatrix ; {\displaystyle
       {{{z^2}}\over {{{\left( z - 2 \right) }^2}}}}\right ] }
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2137 
\Description Transformation formula (\cite{\RaVeAA}, (5.12), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3214}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { 2 b}\endmatrix ; {\displaystyle
   z}\right ] \longrightarrow
(1-z)^{-{a\over 2}}
      {} _{2} F _{1} \!\left [ \matrix { {a\over 2}, -{{a }\over 2}+b}\\ {
       {{1 }\over 2}+b}\endmatrix ; {\displaystyle {{{z^2}}\over
       {4 \left( z - 1 \right) }}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2138 
\Description Transformation formula (\cite{\RaVeAA}, (3.31), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3215}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, 1 - a}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow {{\left( 1 - z \right) }^{c-1}} 
   {} _{2} F _{1} \!\left [ \matrix { {c\over 2}-{{a}\over 2},{a\over
   2}+{c\over 2}- {{1}\over 2}}\\ 
  { c}\endmatrix ; {\displaystyle 4 z\left( 1 - z \right)  }\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2139 
\Description Transformation formula (\cite{\RaVeAA}, (3.31)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3216}.
$$
{} _{2} F _{1} \!\left [ \matrix { c, -{1\over 2} + a - c}\\ { a}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow 
   {\left({{ 1 + {\sqrt{1 - z}}  }\over 2}\right)^{1 - a}} 
   {} _{2} F _{1} \!\left [ \matrix { a - 2 c, 1 - a + 2 c}\\ { a}\endmatrix
    ; {\displaystyle {{1 - {\sqrt{1 - z}}}\over 2}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2140 
\Description Transformation formula (\cite{\RaVeAA}, (3.2), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8707}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, {1\over 2} + a}\\ { b}\endmatrix ;
   {\displaystyle {{4 z}\over {{{\left( 1 + z \right) }^2}}}}\right ]  \longrightarrow 
  {{\left( 1 + z \right) }^{2 a}} 
   {} _{2} F _{1} \!\left [ \matrix { 2 a, 1 + 2 a - b}\\ { b}\endmatrix ;
    {\displaystyle z}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2141 
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8708}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, {1\over 2} + a}\\ { b}\endmatrix ;
   {\displaystyle {{4 z}\over {{{\left( 1 + z \right) }^2}}}}\right ]  \longrightarrow 
  {{{\left( 1 + z \right) }^{2 a}}\over {1-z}}
      {} _{3} V _{2} ({\displaystyle -1 + 2 a; 2 a - b}; {\displaystyle z})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2163 
\Description Transformation formula (\cite{\GaRaAA}, Ex 3.8, $q\to 1$; 
\cite{\SlatAC}, pp. 36/37) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T2163}.
$$
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  
\longrightarrow {{{\left( 1 - z \right) }^{-b}}}
  {{\Gamma \left [ \matrix a - b, c\\ a, -b + c\endmatrix \right ]  
       {} _{2} F _{1} \!\left [ \matrix { b, -a + c}\\ { 1 - a +
        b}\endmatrix ; {\displaystyle {1\over {1 - z}}}\right ]}}  + 
{{{\left( 1 - z \right) }^{-a}}}
   {{\Gamma \left [ \matrix -a + b, c\\ b, -a + c\endmatrix \right ]  
       {} _{2} F _{1} \!\left [ \matrix { a, -b + c}\\ { 1 + a - b}\endmatrix ;
        {\displaystyle {1\over {1 - z}}}\right ]}}
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2191
\Description Transformation formula (\cite{\SlatAC}, (1.8.10)) in form of a rule.
$$\multline
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]  \longrightarrow  {{\left( 1 - z \right) }^{c-a-b}} 
    \Gamma \left [ \matrix c, a + b - c\\ a, b\endmatrix \right ] 
    {} _{2} F _{1} \!\left [ \matrix { c-b,c -a}\\ { 1 - a - b +
     c}\endmatrix ; {\displaystyle 1 - z}\right ]  \\
 + \Gamma \left [ \matrix c, c-a-b\\ c-b, c-a\endmatrix \right ]
   {} _{2} F _{1} \!\left [ \matrix { a, b}\\ { 1 + a + b - c}\endmatrix ;
     {\displaystyle 1 - z}\right ]  
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T2192 
\Description Transformation formula (\cite{\GaRaAA}, Ex 3.8, $q\to 1$, 
reversed; \cite{\SlatAC}, pp. 36/37) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3269}.
$$\multline
{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ] \longrightarrow
{{z^{-a}}}
{{       \Gamma \left [ \matrix 1 + a - c, 1 + b - c\\ 1 - c, 1 + a + b -
        c\endmatrix \right ]
{} _{2} F _{1} \!\left [ \matrix { a, 1 + a - c}\\ { 1 + a
        + b - c}\endmatrix ; {\displaystyle  {{z-1}\over z}}\right ]  
}}  \\ - 
   {z^{1 - c}}
 \Gamma \left [ \matrix 1 + a - c, 1 + b - c, -1 + c\\ a, b, 1 -
     c\endmatrix \right ]
    {} _{2} F _{1} \!\left [ \matrix { 1 + a - c, 1 + b - c}\\ { 2
     - c}\endmatrix ; {\displaystyle z}\right ]  
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3204
\Description Transformation formula (\cite{\BailAA}, Ex.~7, p.~98) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3204}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
   \Gamma \left [ \matrix e, -a - b - c + d + e\\ -a + e, -b - c + d +
    e\endmatrix \right ]
  {} _{3} F _{2} \!\left [ \matrix { a, -b + d, -c + d}\\ { d, -b - c + d +
    e}\endmatrix ; {\displaystyle 1}\right ]  
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3205
\Description Transformation formula (\cite{\SlatAC}, (2.3.3.7)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3205}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
   \Gamma \left [ \matrix d, e, -a - b - c + d + e\\ b, -a - b + d + e, -b - c
    + d + e\endmatrix \right ]
  {} _{3} F _{2} \!\left [ \matrix { -b + d, -b + e, -a - b - c + d + e}\\ {
    -a - b + d + e, -b - c + d + e}\endmatrix ; {\displaystyle 1}\right ]  
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3206
\Description Transformation formula (\cite{\BailAA}, Ex.~7, p.~98, terminating form) in 
    form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3206}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
{{      ({ \textstyle -a - b + d + e}) _{n} }\over {({ \textstyle e}) _{n} }}  
{} _{3} F _{2} \!\left [ \matrix { -n, -a + d, -b + d}\\ { d, -a - b + d +
       e}\endmatrix ; {\displaystyle 1}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3207
\Description Transformation formula (\cite{\GaRaAA} (3.1.1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3207}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
{{ ({ \textstyle -b + e}) _{n} }
     \over {({ \textstyle e}) _{n} }}  
{} _{3} F _{2} \!\left [ \matrix { -n, b, -a + d}\\ { d, 1 + b - e -
       n}\endmatrix ; {\displaystyle 1}\right ] 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3217
\Description Transformation formula (\cite{\BailAA} 4.4(2), reversed) 
in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3217}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
\Gamma \left [ \matrix -a + d + e, -a - b - c + d +
    e\\ -a - b + d + e, -a - c + d + e\endmatrix \right ]
  {} _{6} V _{5} ({\displaystyle -1 - a + d + e; -a + e, -a + d, b, c};
    {\displaystyle -1}) 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3231
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.21), $q\uparrow1$) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4305}.
$$
{} _{3} F _{2} \!\left [ \matrix { 2 a, 2 b, c}\\ { {1\over 2} + a + b,
   d}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  {} _{4} F _{3} \!\left [ \matrix { a, b, c, -c + d}\\ { {1\over 2} + a + b,
   {d\over 2}, {{1 + d}\over 2}}\endmatrix ; {\displaystyle 1}\right ] 
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3232
\Description Transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.97,
reversed, first form) 
in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5469}.
A terminating $q$-analogue is HYPQ's \hbox{\tt T5404}.
$$
{} _{3} F _{2} \!\left [ \matrix { {a\over 2}, {{1 + a}\over 2}, 1 + a - b -
   c}\\ { 1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle {{-4\,z}\over
   {{{\left( 1 - z \right) }^2}}}}\right ]  \longrightarrow  
  {{\left( 1 - z \right) }^a}\,{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ {
    1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle z}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3233
\Description Transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.97,
reversed, second form) 
in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5469}.
A terminating $q$-analogue is HYPQ's \hbox{\tt T5404}.
$$
{} _{3} F _{2} \!\left [ \matrix { {a\over 2}, {{1 + a}\over 2}, 1 + a - b -
   c}\\ { 1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle {{-4\,z}\over
   {{{\left( 1 - z \right) }^2}}}}\right ]  \longrightarrow  
  {{\left( 1 - \frac {1} {z} \right) }^a}\,{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ {
    1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle \frac {1} {z}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3234
\Description Transformation formula (\cite{\SlatAC}, (2.5.7), reversed) 
in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5405}.
$$
{} _{3} F _{2} \!\left [ \matrix { 2\,a, 2\,b, a + b}\\ { 2\,a + 2\,b, {1\over
   2} + a + b}\endmatrix ; {\displaystyle z}\right ]  \longrightarrow  
  {{{} _{2} F _{1} \!\left [ \matrix { a, b}\\ { {1\over 2} + a + b}\endmatrix
      ; {\displaystyle z}\right ] }^2}
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3235 
\Description Transformation formula (\cite{\GaRaAA}, Ex 3.4, $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4201}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { d, 2 b}\endmatrix ;
   {\displaystyle 2}\right ] \longrightarrow
{ {   ({ \textstyle -a + d}) _{n} }\over {({ \textstyle d}) _{n} }}
  {} _{4} F _{3} \!\left [ \matrix { {a\over 2}, {1\over 2} + {a\over 2},
       {1\over 2} - {n\over 2}, {{-n}\over 2}}\\ { {1\over 2} + {a\over 2} -
       {d\over 2} - {n\over 2}, 1 + {a\over 2} - {d\over 2} - {n\over 2},
       {1\over 2} + b}\endmatrix ; {\displaystyle 1}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3236 
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4311}.
$$
{} _{3} V _{2} ({\displaystyle a; b}; {\displaystyle z}) 
\longrightarrow 
  {{\left( 1 - z \right) }\over {{\left( 1 + z \right) }^{1 +a}}} 
   {} _{2} F _{1} \!\left [ \matrix { {1\over 2} + {a\over 2}, 1 + {a\over
    2}}\\ { 1 + a - b}\endmatrix ; {\displaystyle {{4 z}\over {{{\left( 1 + z
    \right) }^2}}}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3237 
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5403}.
$$
{} _{3} F _{2} \!\left [ \matrix { x, y, -n}\\ { b, a}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
{{({ \textstyle a - x, a - y}) _{n}} \over  {({
    \textstyle a, a - x - y}) _{n}}} 
  {} _{6} V _{5} ({\displaystyle -a - n + x + y; 1 - a - b - n + x + y, x, y,
    -n}; {\displaystyle -1}) 
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3238 
\Description Transformation formula (\cite{\BailAA}, Ex. 6, p. 97) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5462}.
$$
{} _{3} F _{2} \!\left [ \matrix { {1\over 2} + {a\over 2}, 1 + {a\over 2}, -1
   - a + b + c}\\ { b, c}\endmatrix ; {\displaystyle -{{4 z}\over {{{\left(
   1-z \right) }^2}}}}\right ]  \longrightarrow 
{{{\left( 1 - z \right) }^{1 + a}}\over{(1+z)}}
      {} _{4} V _{3} ({\displaystyle a; 1 + a - b, 1 + a - c}; {\displaystyle
       z})
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3239
\Description Transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.~97) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3265}.
A terminating $q$-analogue is HYPQ's \hbox{\tt T3209}.
$$
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { 1+a-b,1+a-c}\endmatrix ;
   {\displaystyle z}\right ]  \longrightarrow {{{\left( 1 - z \right) }^{-a}}}
  {} _{3} F _{2} \!\left [ \matrix { {a\over 2}, {{1 + a}\over 2}, 1 + a - b
      - c}\\ { 1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle {{-4 z}\over
      {{{\left( 1 - z \right) }^2}}}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3240 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8711}.
$$
{} _{3} F _{2} \!\left [ \matrix { c, b, d}\\ { a, a - b + d}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
\Gamma \left [ \matrix 2 a, 2 a - 2 b - c, a - b +
    d, a - c + d\\ 2 a - 2 b, 2 a - c, a + d, a - b - c + d\endmatrix \right ]
  {} _{7} V _{6} ({\displaystyle -{1\over 2} + a; b, {c\over 2}, {1\over 2} +
    {c\over 2}, {a\over 2} - {d\over 2}, {1\over 2} + {a\over 2} - {d\over
    2}}; {\displaystyle 1}) 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3261
\Description Transformation formula (\cite{\GaRaAA}, (III.33), $q\uparrow1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3261}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
    \Gamma \left [ \matrix c, 1 + c - d, 1 - a, -b - c + e\\ -b + e, -c + e, 1
     - a + c, 1 - d\endmatrix \right ] 
  {} _{3} F _{2} \!\left [ \matrix { c, -a + d, 1 + c - e}\\ { 1 - a + c, 1 +
     b + c - e}\endmatrix ; {\displaystyle 1}\right ]  
\\-     \Gamma \left [ \matrix -1 + d, e, 1 + b - d, 1 + c - d, 1 - a, -b - c + e,
     1 + b + c - e\\ 1 - d, 1 - d + e, b, c, -a + d, -1 - b - c + d + e, 2 + b
     + c - d - e\endmatrix \right ]
  {} _{3} F _{2} \!\left [ \matrix { 1 + a - d, 1 + b - d, 1 + c - d}\\ { 2 -
     d, 1 - d + e}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3262
\Description Transformation formula (\cite{\SlatAC}, (4.3.4.2)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3262}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow
    \Gamma \left [ \matrix e, -b - c + e\\ -b + e, -c + e\endmatrix \right ] 
  {} _{3} F _{2} \!\left [ \matrix { -a + d, b, c}\\ { d, 1 + b + c -
     e}\endmatrix ; {\displaystyle 1}\right ]  \\
+    \Gamma \left [ \matrix d,e, b + c - e, -a - b - c + d + e\\ -a + d,
   b,c, -b -  c + d + e\endmatrix \right ]
   {} _{3} F _{2} \!\left [ \matrix { -b + e, -c + e, -a - b - c + d + e}\\ {
     -b - c + d + e, 1 - b - c + e}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3263 
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.33), 
$q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3263}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix 1 - c, d, -a - b + d, 1 + a - e\\ 1 + a - c, -a +
     d, -b + d, 1 - e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { 1 + a - d, -c + e, a}\\ { 1 + a + b - d,
     1 + a - c}\endmatrix ; {\displaystyle 1}\right ] \\ + 
   \Gamma \left [ \matrix 1 - c, a + b - d, d, 1 + a - e, e, -a - b - c + d +
     e\\ a, b, 1 - b - c + d, 1 + a + b - d - e, -c + e, -a - b + d +
     e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { 1 - b, -a - b - c + d + e, -b + d}\\ { 1
     - a - b + d, 1 - b - c + d}\endmatrix ; {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3264 
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.34), 
$q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3264}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  \longrightarrow 
  \Gamma \left [ \matrix 1 + b - e, 1 + c - e\\ 1 - e, 1 + b + c -
     e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { -a + d, b, c}\\ { d, 1 + b + c -
     e}\endmatrix ; {\displaystyle 1}\right ] \\ - \Gamma \left [ \matrix d, 1 + a - e, 1 + b - e, 1 + c - e, -1
     + e\\ a, b, c, 1 - e, 1 + d - e\endmatrix \right ]  {} _{3} F _{2} \!\left [ \matrix { 1 + c - e, 1 +
     b - e, 1 + a - e}\\ { 1 + d - e, 2 - e}\endmatrix ; {\displaystyle
     1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3267 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.6, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3267}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  
\longrightarrow 
  \Gamma \left [ \matrix d, 1 + a - e, 1 + b - e, 1 + c - e\\ -a + d, 1 - e,
     1 + a + b - e, 1 + a + c - e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { a, 1 + a - e, 1 + a + b + c - d - e}\\ {
     1 + a + b - e, 1 + a + c - e}\endmatrix ; {\displaystyle 1}\right ]\\ - 
   \Gamma \left [ \matrix d, 1 + a - e, 1 + b - e, 1 + c - e, -1 + e\\ a, b,
     c, 1 - e, 1 + d - e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { 1 + a - e, 1 + b - e, 1 + c - e}\\ { 2 -
     e, 1 + d - e}\endmatrix ; {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T3268 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.6, 
$q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T3268}.
$$\multline
{} _{3} F _{2} \!\left [ \matrix { a, b, c}\\ { d, e}\endmatrix ;
   {\displaystyle 1}\right ]  
\longrightarrow 
  \Gamma \left [ \matrix a - b, d, e, -a - b - c + d + e\\ a, -b + d, -b +
     e, -a - c + d + e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { b, -a + d, -a + e}\\ { 1 - a + b, -a - c
     + d + e}\endmatrix ; {\displaystyle 1}\right ] \\+ 
   \Gamma \left [ \matrix -a + b, d, e, -a - b - c + d + e\\ b, -a + d, -a +
     e, -b - c + d + e\endmatrix \right ]  
    {} _{3} F _{2} \!\left [ \matrix { a, -b + d, -b + e}\\ { -b - c + d + e, 1
     + a - b}\endmatrix ; {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4301
\Description Transformation formula (\cite{\SlatAC}, (4.3.5.1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4301}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, -n}\\ { e, f, 1 + a + b + c - e -
   f - n}\endmatrix ; {\displaystyle 1}\right ] \\ \longrightarrow 
{{({ \textstyle -a + e, -a + f}) _{n}}\over  {({ \textstyle e, f}) _{n}}}
  {} _{4} F _{3} \!\left [ \matrix { -n, a, 1 + a + c - e - f - n, 1 + a + b -
    e - f - n}\\ { 1 + a + b + c - e - f - n, 1 + a - e - n, 1 + a - f -
    n}\endmatrix ; {\displaystyle 1}\right ]   
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4302
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.16), $q\uparrow1$) in 
    form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4302}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, -n}\\ { e, f, 1 + a + b + c - e -
   f - n}\endmatrix ; {\displaystyle 1}\right ] \\ \longrightarrow 
{{({ \textstyle a, -a - b + e + f, -a - c + e + f}) _{n}}\over
     {({ \textstyle e, f, -a - b - c + e + f}) _{n}}}
  {} _{4} F _{3} \!\left [ \matrix { -n, -a + e, -a + f, -a - b - c + e + f}\\
    { -a - b + e + f, -a - c + e + f, 1 - a - n}\endmatrix ; {\displaystyle
    1}\right ]   
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4303
\Description Transformation formula (\cite{\SlatAC}, (2.4.1.1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4303}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, -n}\\ { e, f, 1 + a + b + c - e -
   f - n}\endmatrix ; {\displaystyle 1}\right ] \\ \longrightarrow 
{{({ \textstyle -a - b + e + f, -a - c + e + f})
    _{n}}\over  {({ \textstyle -a + e + f, -a - b - c + e + f}) _{n}}}
  {} _{7} V _{6} ({\displaystyle -1 - a + e + f; -a + f, -a + e, b, c, -n};
    {\displaystyle 1})  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4304
\Description Transformation formula (\cite{\SlatAC}, (4.3.6.4)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4304}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, -n}\\ { e, f, 1 + a + b + c - e -
   f - n}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
\Gamma \left [ \matrix -b - c + e + f + n, -a - c + e +
    f + n,\\ -c + e + f + n, -b + e + f + n,\endmatrix \right .\\
\left.\matrix  -a - b + e + f + n, e + f + n\\
   -a + e + f + n, -a - b - c + e + f + n\endmatrix\right]
  {} _{7} V _{6} ({\displaystyle -1 + e + f + n; a, b, c, e + n, f + n};
    {\displaystyle 1}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4306
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.21), $q\uparrow1$, reversed) 
    in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4306}.
$$
{} _{4} F _{3} \!\left [ \matrix { a, b, c, d}\\ { {1\over 2} + a + b, {{c +
   d}\over 2}, {{1 + c + d}\over 2}}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
  {} _{3} F _{2} \!\left [ \matrix { 2 a, 2 b, c}\\ { {1\over 2} + a + b, c +
   d}\endmatrix ; {\displaystyle 1}\right ] 
$$
provided both series terminate.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4309 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(i), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4309}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, d}\\ { 1 + a - b, 1 + a - c, 1 + a
   - d}\endmatrix ; {\displaystyle 1}\right ] 
\longrightarrow 
  \Gamma \left [ \matrix 3 + 2 a - 2 b - 2 c - 2 d, 2 + 2 a - b -
    c - d\\ 3 + 3 a - 2 b - 2 c - 2 d, 2 + a - b - c - d\endmatrix \right
    ] \\ {} _{7} V _{6} ({\displaystyle 1 + 2 a - b - c - d; {a\over 2}, {1\over 2}
    + {a\over 2}, 1 + a - c - d, 1 + a - b - d, 1 + a - b - c}; {\displaystyle
    1})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4310 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(ii), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4310}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, d}\\ { 1 + a - b, 1 + a - c, 1 + a
   - d}\endmatrix ; {\displaystyle -1}\right ]  \\
\longrightarrow 
  \Gamma \left [ \matrix 2 + 2 a - b - c - d, 1 + {a\over 2}\\ 1 + a, 2 +
    {{3 a}\over 2} - b - c - d\endmatrix \right ] 
   {} _{6} V _{5} ({\displaystyle 1 + 2 a - b - c - d; {a\over 2}, 1 + a - c -
    d, 1 + a - b - d, 1 + a - b - c}; {\displaystyle -1})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4312 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.4, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4312}.
$$
{} _{4} F _{3} \!\left [ \matrix { {a\over 2}, {1\over 2} + {a\over 2},
   {1\over 2} - {n\over 2}, -{{n}\over 2}}\\ { {d\over 2}, {1\over 2} +
   {d\over 2}, {1\over 2} + b}\endmatrix ; {\displaystyle 1}\right ]  
\longrightarrow 
   {{({ \textstyle d-a}) _{n}}\over  {({ \textstyle d}) _{n}}} 
  {} _{3} F _{2} \!\left [ \matrix { a, b, -n}\\ { 1 + a - d - n,
    2 b}\endmatrix ; {\displaystyle 2}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4313 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 8.15, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4313}.\NoBlackBoxes
$$\multline
  {} _{4} F _{3} \!\left [ \matrix { a, b, c, d}\\ { 1 - a + b, 1 - a + c, 1
    - a + d}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow
    \Gamma \left [ \matrix 1 - d, a + b - d, a + c - d, 1 + b + c - d\\ a - d,
     1 + b - d, 1 + c - d, a + b + c - d\endmatrix \right ]\\
\times
   {} _{9} F _{8} \!\left [ \matrix { b + c - d, 1 + \frac{b}{2} +
     \frac{c}{2} - \frac{d}{2}, \frac{1}{2} - \frac{a}{2} + \frac{b}{2} +
     \frac{c}{2} - \frac{d}{2}, 1 - \frac{a}{2} + \frac{b}{2} + \frac{c}{2} -
     \frac{d}{2}, a + b - d, a + c - d, a, b, c}\\ { \frac{b}{2} + \frac{c}{2}
     - \frac{d}{2}, \frac{1}{2} + \frac{a}{2} + \frac{b}{2} + \frac{c}{2} -
     \frac{d}{2}, \frac{a}{2} + \frac{b}{2} + \frac{c}{2} - \frac{d}{2}, 1 - a
     + c, 1 - a + b, 1 - a + b + c - d, 1 + c - d, 1 + b - d}\endmatrix ;
     {\displaystyle 1}\right ] 
\endmultline$$
provided at least one of $a,b,c$ is a non-negative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4331 
\Description Transformation formula (\cite{\BailAA}, Ex. 6, p. 97, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5461}.
$$
{} _{4} V _{3} ({\displaystyle a; b, c}; {\displaystyle z}) \longrightarrow 
  {{\left( 1 + z \right)  }\over{{\left( 1 - z \right) }^{a+1}}}
   {} _{3} F _{2} \!\left [ \matrix { {1\over 2} + {a\over 2}, 1 + {a\over 2},
    1 + a - b - c}\\ { 1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle
    -{{4 z}\over {{{\left( 1-z \right) }^2}}}}\right ] 
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4332 
\Description Transformation formula (\cite{\BailAA}, 4.6(1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10904}.
$$
{} _{4} F _{3} \!\left [ \matrix { b, x, y, -n}\\ { a - x, a - y, a +
   n}\endmatrix ; {\displaystyle 1}\right ]  \longrightarrow 
{{({ \textstyle a, a - x - y}) _{n}} \over  {({ \textstyle a - x, a - y})
    _{n}}}
  {} _{5} F _{4} \!\left [ \matrix { x, y, {a\over 2} - {b\over 2}, {1\over 2}
    + {a\over 2} - {b\over 2}, -n}\\ { a - b, {a\over 2}, {1\over 2} + {a\over
    2}, 1 - a - n + x + y}\endmatrix ; {\displaystyle 1}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4362 
\Description Transformation formula (\cite{\SlatAC}, (2.4.4.3), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T4362}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, b, c, d}\\ { e, f, 1 + a + b + c + d - e
   - f}\endmatrix ; {\displaystyle 1}\right ] \\ 
\hskip-5cm\longrightarrow 
  \Gamma \left [ \matrix -a + e + f, -a - b - c + e + f,
     -a - b - d + e + f, -a - c - d + e + f\\ -a - b + e + f, -a - c + e + f,
     -a - d + e + f, -a - b - c - d + e + f\endmatrix \right ] \\
{} _{7} V _{6} ({\displaystyle -1 - a + e + f; -a + f, -a + e, b, c, d};
     {\displaystyle 1}) \\ - 
   \Gamma \left [ \matrix e, a + b + c + d - e - f, f, -a - b - c + e + f, -a
     - b - d + e + f, -a - c - d + e + f, -b - c - d + e + f\\ a, b, c, d, -a
     - b - c - d + e + f, -a - b - c - d + 2 e + f, -a - b - c - d + e + 2
     f\endmatrix \right ]  \\
    {} _{4} F _{3} \!\left [ \matrix { -a - b - c + e + f, -a - b - d + e + f,
     -a - c - d + e + f, -b - c - d + e + f}\\ { -a - b - c - d + 2 e + f, -a
     - b - c - d + e + 2 f, 1 - a - b - c - d + e + f}\endmatrix ;
     {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T4391 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5465}.
$$\multline
{} _{4} F _{3} \!\left [ \matrix { a, c, d, e}\\ { b, 2 a, 1 - 2 b + c + d +
   e}\endmatrix ; {\displaystyle 1}\right ] \\
\hskip-3cm\longrightarrow
    \Gamma \left [ \matrix b, {1\over 2} + b, b - {c\over 2} - {d\over 2},
     {1\over 2} + b - {c\over 2} - {d\over 2}, b - {c\over 2} - {e\over 2},
     {1\over 2} + b - {c\over 2} - {e\over 2}, b - {d\over 2} - {e\over 2},
     {1\over 2} + b - {d\over 2} - {e\over 2}\\ b - {c\over 2}, {1\over 2} + b
     - {c\over 2}, b - {d\over 2}, {1\over 2} + b - {d\over 2}, b - {e\over
     2}, {1\over 2} + b - {e\over 2}, b - {c\over 2} - {d\over 2} - {e\over
     2}, {1\over 2} + b - {c\over 2} - {d\over 2} - {e\over 2}\endmatrix
     \right ]\\
  {} _{9} V _{8} ({\displaystyle -{1\over 2} + b; -a + b, {c\over 2}, {1\over
     2} + {c\over 2}, {d\over 2}, {1\over 2} + {d\over 2}, {e\over 2}, {1\over
     2} + {e\over 2}}; {\displaystyle 1}) \\
 -     \Gamma \left [ \matrix {1\over 2} + a, b, b - {c\over 2} - {d\over 2},
     {1\over 2} + b - {c\over 2} - {d\over 2}, a + 2 b - c - d - e, b -
     {c\over 2} - {e\over 2}, {1\over 2} + b - {c\over 2} - {e\over 2}, b -
     {d\over 2} - {e\over 2}, {1\over 2} + b - {d\over 2} - {e\over 2},
   \\ {c\over 2}, {1\over 2} + {c\over 2}, {d\over
     2}, {1\over 2} + {d\over 2}, 3 b - c - d - e, b - {c\over 2} - {d\over
     2} - {e\over 2}, {1\over 2} + b - {c\over 2} - {d\over 2} - {e\over 2}, a
     + b - {c\over 2} - {d\over 2} - {e\over 2}, {1\over 2} + a + b - {c\over
     2} - {d\over 2} - {e\over 2}, \endmatrix \right .\\
  \left. \matrix  -b +
     {c\over 2} + {d\over 2} + {e\over 2}, {1\over 2} - b + {c\over 2} +
     {d\over 2} + {e\over 2}\\ {e\over 2}, {1\over 2} + {e\over
     2}\endmatrix \right]
{} _{4} F _{3} \!\left [ \matrix { a + 2 b - c - d - e, 2 b
     - c - d, 2 b - c - e, 2 b - d - e}\\ { 3 b - c - d - e, 2 a + 2 b -
     c - d - e, 1 + 2 b - c - d - e}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5401
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.4)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5401}.
$$\multline
{} _{5} F _{4} \!\left [ \matrix { a, b, c, d, -n}\\ { 1 + a - b, 1 + a - c, 1
   + a - d, -2 - 2 a + 2 b + 2 c + 2 d - n}\endmatrix ; {\displaystyle
   1}\right ] \\
 \longrightarrow  {{({ \textstyle 2 + a - b - c - d, 3 + 3 a - 2 b - 2 c - 2 d}) _{n}}\over 
    {({ \textstyle 2 + 2 a - b - c - d, 3 + 2 a - 2 b - 2 c - 2 d}) _{n}}}
  {} _{9} V _{8} ( 1 + 2 a - b - c - d; 1 + a - c
    - d,\\ 1 + a - b - d, 1 + a - b - c, {a\over 2}, {1\over 2} + {a\over 2}, 3
    + 3 a - 2 b - 2 c - 2 d + n, -n; {\displaystyle 1}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5402
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.26), $q\uparrow1$) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5402}.
$$\multline
{} _{5} F _{4} \!\left [ \matrix { -n, b, c, d, e}\\ { 1 - b - n, 1 - c - n, 1
   - d - n, -2 + 2 b + 2 c + 2 d + e + 2 n}\endmatrix ; {\displaystyle
   1}\right ] \\
\longrightarrow{{({ \textstyle 2 - b - c - d - e - 2 n, 3 - 2 b - 2 c - 2 d - 3 n})
    _{n}}\over  {({ \textstyle 2 - b - c - d - 2 n, 3 - 2 b - 2 c - 2 d - e -
    3 n}) _{n}}} 
  {} _{9} V _{8} (1 - b - c - d - 2 n; 1 - c - d
    - n,\\ 1 - b - d - n, 1 - b - c - n, {{-n}\over 2}, {1\over 2} - {n\over 2},
    e, 3 - 2 b - 2 c - 2 d - e - 3 n; {\displaystyle 1}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5403 
\Description Transformation formula (\cite{\BailAA}, 4.6(1), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5403}.
$$
{} _{5} F _{4} \!\left [ \matrix { x, y, a, {1\over 2} + a, -n}\\ { 2 a, b,
   {1\over 2} + b, 1 - 2 b - n + x + y}\endmatrix ; {\displaystyle 1}\right ]
   \longrightarrow
{{({ \textstyle 2 b - x, 2 b - y}) _{n}} \over  {({ \textstyle 2 b, 2 b
    - x - y}) _{n}}}
{} _{4} F _{3} \!\left [ \matrix { -2 a + 2 b, x, y, -n}\\ { 2 b - x,
    2 b - y, 2 b + n}\endmatrix ; {\displaystyle 1}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T5468 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.25, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T5468}.
$$\multline
{} _{5} F _{4} \!\left [ \matrix { a, b, c, d, e}\\ { 1 + a - b, 1 + a - c, 1
    + a - d, -2 - 2a + 2b + 2c + 2d + e}\endmatrix ; {\displaystyle
    1}\right ] \\ + 
    \Gamma \left [ \matrix 1 + a - b, 1 + a - c, 3 + 2a - 2b - 2c -
     2d, 1 + a - d, 3 + 3a - 2b - 2c - 2d - e, 3 + 2a - b - 2c -
     2d - e, \\ a, b, c, d, 4 + 3a - 2b - 2c -
     3d - e, 4 + 3a - 2b - 3c - 2d - e, \endmatrix \right ]\\
  \left.\matrix 3 + 2a - 2b - c - 2d - e, 3 + 2a - 2b - 2c - d - e,
     -3 - 2a + 2b + 2c + 2d + e\\ 4 + 3a - 3b - 2c -
     2d - e, 3 + 2a - 2b - 2c - 2d - e, e\endmatrix\right.\\
{} _{5} F _{4} \!\left [ \matrix { 3 + 2a - 2b - 2c -
     2d, 3 + 3a - 2b - 2c - 2d - e, 3 + 2a - b - 2c - 2d - e,
     }\\ { 4 +
     2a - 2b - 2c - 2d - e, 4 + 3a - 3b - 2c - 2d - e, }\endmatrix\right. \\
   \left.\matrix 3 + 2a - 2b - c - 2d - e, 3 + 2a - 2b - 2c - d - e\\
   4 + 3a - 2b - 3c - 2d - e, 4 + 3a - 2b - 2c - 3d -
     e\endmatrix ; {\displaystyle 1}\right ] \\
\hskip-3cm\longrightarrow 
    \Gamma \left [ \matrix 3 + 2a - 2b - 2c - 2d, 2 + 2a - b - c -
     d, 3 + 3a - 2b - 2c - 2d - e, 2 + a - b - c - d - e\\ 3 + 3a -
     2b - 2c - 2d, 2 + a - b - c - d, 3 + 2a - 2b - 2c - 2d - e,
     2 + 2a - b - c - d - e\endmatrix \right ]\\
  {} _{9} V _{8} ({\displaystyle 1 + 2a - b - c - d; {a\over 2}, {1\over 2}
     + {a\over 2}, 1 + a - c - d, 1 + a - b - d, 1 + a - b - c, e, 3 + 3a -
     2b - 2c - 2d - e}; {\displaystyle 1})\\
 +     \Gamma \left [ \matrix 1 + a - b, 1 + a - c, 3 + 2a - 2b - 2c -
     2d, 1 + a - d, 6 + 4a - 3b - 3c - 3d - 2e, 3 + 3a - 2b -
     2c - 2d - e, \\ a, 1 +
     a - b - c, 1 + a - b - d, 1 + a - c - d, 7 + 5a - 4b - 4c - 4d -
     2e, \endmatrix \right
     ]\\
   \left.\matrix 3 + 2a - b - 2c - 2d - e, 3 + 2a - 2b - c -
     2d - e, 3 + 2a - 2b - 2c - d - e, -2 - a + b + c + d + e\\
   3 + 2a - 2b - 2c - 2d - e, 3 + 2a - b - c - 2d - e, 3 +
     2a - b - 2c - d - e, 3 + 2a - 2b - c - d - e, e\endmatrix\right]\\
   {} _{9} V _{8} ({\displaystyle 5 + 4a - 3b - 3c - 3d - 2e; 2 +
     {{3a}\over 2} - b - c - d - e, {5\over 2} + {{3a}\over 2} - b - c - d
     - e, 2 + a - b - c - d, }\\
  {3 + 3a - 2b - 2c - 2d - e, 3 + 2a - b -
     2c - 2d - e, 3 + 2a - 2b - c - 2d - e, 3 + 2a - 2b - 2c -
     d - e}; {\displaystyle 1})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T6501 
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.3)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T6501}.
$$\multline
{} _{6} F _{5} \!\left [ \matrix { a, 1 + {a\over 2}, b, c, d, -n}\\ { {a\over
   2}, 1 + a - b, 1 + a - c, 1 + a - d, -1 - 2 a + 2 b + 2 c + 2 d -
   n}\endmatrix ; {\displaystyle 1}\right ]  \\
\hskip-10cm\longrightarrow 
   {{({ \textstyle 2 + 3 a - 2 b - 2 c - 2 d, 1 + a - b - c - d})
    _{n}}\over  {({ \textstyle 2 + 2 a - 2 b - 2 c - 2 d, 2 + 2 a - b - c
    - d}) _{n}}} \\
  {} _{9} V _{8} ({\displaystyle 1 + 2 a - b - c - d; 1 + a - c - d, 1 + a -
    b - d, 1 + a - b - c, 1 + {a\over 2}, {1\over 2} + {a\over 2}, 2 + 3 a -
    2 b - 2 c - 2 d + n, -n}; {\displaystyle 1}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T6531
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.5)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T7601}.
$$\multline
{} _{6} F _{5} \!\left [ \matrix { a, 1 + {a\over 2}, b, c, d, -n}\\ { {a\over
   2}, 1 + a - b, 1 + a - c, 1 + a - d, -2 a + 2 b + 2 c + 2 d - n}\endmatrix
   ; {\displaystyle 1}\right ] \\ 
\longrightarrow 
{{\left( 1 + 3 a - 2 b - 2 c - 2 d + 2 n \right)  }\over
 {(1 + 3 a - 2 b - 2 c - 2 d)}}  
      {{({ \textstyle a - b - c - d, 1 + 3 a - 2 b - 2 c - 2 d}) _{n}}\over 
       {({ \textstyle 2 + 2 a - b - c - d, 1 + 2 a - 2 b - 2 c - 2 d}) _{n}}}
 {} _{9} V _{8} ( 1 + 2 a - b - c - d;\\ 
 1 + a - c - d, 1 + a
       - b - d, 1 + a - b - c, {1\over 2} + {a\over 2}, 1 + {a\over 2}, 1 + 3
       a - 2 b - 2 c - 2 d + n, -n; {\displaystyle 1}) 
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T6532 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(ii), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8706}.\NoBlackBoxes
$$
{} _{6} V _{5} ({\displaystyle a; b, c, d, 1 + 2 a - 2 b - c - d};
   {\displaystyle -1}) 
\longrightarrow \Gamma \left [ \matrix 1 + 2 b, 1 + a - b\\ 1 + b, 1 + a\endmatrix \right
    ]  
   {} _{4} F _{3} \!\left [ \matrix { 2 b, -a + 2 b +
    c, -a + 2 b + d, 1 + a - c - d}\\ { 1 + a - c, 1 + a - d, -a + 2 b + c +
    d}\endmatrix ; {\displaystyle -1}\right ]
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.
\par\BlackBoxes

\Name T6533 
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10904}.
$$
{} _{6} V _{5} ({\displaystyle a; b, x, y, -n}; {\displaystyle -1}) 
\longrightarrow 
   {{({ \textstyle 1 + a, 1 + a - x - y}) _{n}}\over  {({ \textstyle 1 + a -
    x, 1 + a - y}) _{n}}} 
  {} _{3} F _{2} \!\left [ \matrix { -n, x, y}\\ { -a - n + x + y, 1 + a -
    b}\endmatrix ; {\displaystyle 1}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T6534
\Description Transformation formula (\cite{\BailAA}, 4.4(2)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T7701}.
$$
{} _{6} V _{5} ({\displaystyle a; b, c, d, e}; {\displaystyle -1}) \longrightarrow 
   \Gamma \left [ \matrix 1 + a - d, 1 + a - e\\ 1 + a, 1 + a - d -
    e\endmatrix \right ]
  {} _{3} F _{2} \!\left [ \matrix { 1 + a - b - c, d, e}\\ { 1 + a - b, 1 + a
    - c}\endmatrix ; {\displaystyle 1}\right ]  
$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7631
\Description Transformation formula (\cite{\SlatAC}, (2.4.1.1), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8701}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, c, d, e, f}; {\displaystyle 1})
\longrightarrow    \Gamma \left [ \matrix 1 + a - d, 1 + a - e, 1 + a - f, 1 + a - d - e - f\\
    1 + a, 1 + a - d - e, 1 + a - d - f, 1 + a - e - f\endmatrix \right ]\\
  {} _{4} F _{3} \!\left [ \matrix { 1 + a - b - c, d, e, f}\\ { 1 + a - b, 1
    + a - c, -a + d + e + f}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
provided the $_7F_6$ series converges and the $_4F_3$ series
terminates.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7632
\Description Transformation formula (\cite{\SlatAC}, (2.4.1.1), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8702}.
$$
{} _{7} V _{6} ({\displaystyle a; b, c, d, e, -n}; {\displaystyle 1}) \longrightarrow 
   {{({ \textstyle 1 + a, 1 + a - d - e}) _{n}}\over  {({ \textstyle 1 + a -
    d, 1 + a - e}) _{n}}} 
  {} _{4} F _{3} \!\left [ \matrix { 1 + a - b - c, d, e, -n}\\ { 1 + a - b, 1
    + a - c, -a + d + e - n}\endmatrix ; {\displaystyle 1}\right ]  
$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7633
\Description Transformation formula (\cite{\SlatAC}, (4.3.6.4), reversed) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8703}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, c, d, e, 1 + a - e + n}; {\displaystyle
   1}) \\
  \longrightarrow   \Gamma \left [ \matrix 1 + a - d, 1 + a - c, 1 + a - b, 1 + a - b - c - d\\
    1 + a - c - d, 1 + a - b - d, 1 + a - b - c, 1 + a\endmatrix \right ]
 {} _{4} F _{3} \!\left [ \matrix { b, c, d, -n}\\ { 1 + a - e, -a + b
    + c + d, e - n}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7634
\Description Transformation formula (\cite{\BailAA}, 7.5.(1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8704}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, c, d, e, f}; {\displaystyle 1}) \longrightarrow 
   \Gamma \left [ \matrix 1 + a - e, 1 + a - f, 2 + 2 a - b - c - d, 2 + 2 a -
    b - c - d - e - f\\ 1 + a, 1 + a - e - f, 2 + 2 a - b - c - d - e, 2 + 2 a
    - b - c - d - f\endmatrix \right ]\\
  {} _{7} V _{6} ({\displaystyle 1 + 2 a - b - c - d; 1 + a - c - d, 1 + a - b
    - d, 1 + a - b - c, e, f}; {\displaystyle 1}) 
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7635
\Description Transformation formula (\cite{\BailAA}, 7.5.(2)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8705}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, c, d, e, f}; {\displaystyle 1})
\longrightarrow \Gamma \left [ \matrix 1 + a - c, 1 + a - d, 1 + a - e,
    1 + a - f, 3 + 3 a - 2 b - c - d - e - f,\\ 1
    + a, b, 2 + 2 a - b - d - e - f, 2 + 2 a - b - c - e - f,
   2 + 2 a - b - c  - d - f,\endmatrix \right .\\
\left.\matrix  2 + 2 a - b - c - d - e -
  f\\  2 + 2 a - b - c - d - e\endmatrix\right]
  {} _{7} V _{6} ( 2 + 3 a - 2 b - c - d - e - f; 1 + a - b - c,
    1 + a - b - d,\\
 1 + a - b - e, 1 + a - b - f, 2 + 2 a - b - c - d - e - f;
    {\displaystyle 1}) 
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7636 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8710}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, c, {1\over 2} + c, d, {1\over 2} + d};
   {\displaystyle 1}) \\
\longrightarrow \Gamma \left [ \matrix 1 + 2 a - 2 b, 1 + 2 a - 2 c, 1 + 2 a - 2 d, 1
    + 2 a - b - 2 c - 2 d\\ 1 + 2 a, 1 + 2 a - 2 b - 2 c, 1 + 2 a - b
    - 2 d, 1 + 2 a - 2 c - 2 d\endmatrix \right ]  
   {} _{3} F _{2} \!\left [ \matrix { 2 c, b, {1\over 2}
    + a - 2 d}\\ { 1 + 2 a - b - 2 d, {1\over 2} + a}\endmatrix ;
    {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7637 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(i), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10902}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, {1\over 2} + b, c, d, 1 + 2 a - 2 b - c
   - d}; {\displaystyle 1}) \\
\longrightarrow \Gamma \left [ \matrix 1 + a - 2 b, 1 + 2 a - 2 b\\ 1 + a, 1 + 2 a -
    4 b\endmatrix \right ]  
   {} _{4} F _{3} \!\left [ \matrix { 2 b, -a +
    2 b + c, -a + 2 b + d, 1 + a - c - d}\\ { 1 + a - c, 1 + a - d, -a +
    2 b + c + d}\endmatrix ; {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7691
\Description Transformation formula (\cite{\SlatAC}, (2.4.4.3)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8761}.
$$\align
{} _{7} V& _{6} ({\displaystyle a; b, c, d, e, f}; {\displaystyle 1})
\\&\longrightarrow   \Gamma \left [ \matrix 1 + a - d, 1 + a - e, 1 + a - f, 1 + a - d - e -
     f\\ 1 + a, 1 + a - d - e, 1 + a - d - f, 1 + a - e - f\endmatrix \right ]
  {} _{4} F _{3} \!\left [ \matrix { 1 + a - b - c, d, e, f}\\ { 1 + a - b, 1
     + a - c, -a + d + e + f}\endmatrix ; {\displaystyle 1}\right ] \\
&\quad \quad   + \Gamma \left [ \matrix 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 +
     a - f, 2 + 2 a - b - c - d - e - f, -1 - a + d + e + f\\ 1 + a, 1 + a - b
     - c, d, e, f, 2 + 2 a - b - d - e - f, 2 + 2 a - c - d - e - f\endmatrix
     \right ]\\
&\hskip3cm {} _{4} F _{3} \!\left [ \matrix { 1 + a - d - e, 1 + a - d - f, 1 + a
     - e - f, 2 + 2 a - b - c - d - e - f}\\ { 2 + 2 a - b - d - e - f, 2 + 2
     a - c - d - e - f, 2 + a - d - e - f}\endmatrix ; {\displaystyle 1}\right
     ] 
\endalign$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7692
\Description Transformation formula (\cite{\SlatAC}, (4.3.7.8)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8762}.
$$\align
{} _{7} V _{6} ( a;& b, c, d, e, f; {\displaystyle 1})
\\&\longrightarrow    -   \Gamma \left [ \matrix a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a -
       f, -a + b + d, -a + b + e, -a+b+f,-a+d+e+f,
    \\ 1 + a, -a + b, 1 + b - c, 1 + b - d, 1 + b
       - e, 1 + b - f, d, e, f, 1 + a - b - c, -2a+b+d+e+f,
     \endmatrix \right . \\
&\hskip1cm \left.\matrix a+1-d-e-f,
    1-c,1-a+2b\\ 1+2a-b-d-e-f\endmatrix\right]
  {} _{7} V _{6} ({\displaystyle -a + 2 b; b, -a + b + c, -a + b + d,
       -a + b + e, -a + b + f}; 1)
\\ &\quad \quad  +     \Gamma \left [ \matrix 1 + a - d, 1 + a - e, 1 + a - f, 1 + a - d - e - f,
     1 - c, 1 - c + e + f, -a + b + e, -a + b + f\\ 1 + a, 1 + a - d - e, 1 +
     a - d - f, 1 + a - e - f, 1 - c + e, 1 - c + f, -a + b + e + f\endmatrix
     \right ]\\
&\hskip1cm
   {} _{7} V _{6} ({\displaystyle -c + e + f; 1 + a - b - c, 1 + a - c - d, -a
     + e + f, e, f}; 1) 
\endalign$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7693 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.15, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8764}.
$$\multline
{} _{7} V _{6} ({\displaystyle 2 a; a + b, a + c, a + d, a + e, a + f};
   {\displaystyle 1}) \\
\longrightarrow - \Gamma \left [ \matrix a - b, 1 + 2 b, -b + c, b + c, 1 - a - d, 1 + a
       - d, 1 - a - e, 1 + a - e, 1 - a - f, 1 + a - f\\ 1 + 2 a, -a + b, -a
       + c, a + c, 1 - b - d, 1 + b - d, 1 - b - e, 1 + b - e, 1 - b - f, 1 +
       b - f\endmatrix \right ] \\
      {} _{7} V _{6} ({\displaystyle 2 b; a + b, b
       + c, b + d, b + e, b + f}; {\displaystyle 1})\\  - 
   \Gamma \left [ \matrix a - c, b - c, b + c, 1 + 2 c,
     1 - a - d, 1 + a - d, 1 - a - e, 1 + a - e, 1 - a - f, 1 + a - f\\ 1 +
     2 a, -a + b, a + b, -a + c, 1 - c - d, 1 + c - d, 1 - c - e, 1 + c - e,
     1 - c - f, 1 + c - f\endmatrix \right ] \\ {} _{7} V _{6} ({\displaystyle 2 c; a + c, b + c, c + d, c + e, c + f};
     {\displaystyle 1})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7694 
\Description Transformation formula (\cite{\SlatAC}, (4.3.7.8), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8763}.
$$\multline
{} _{7} V _{6} ({\displaystyle a; b, c, d, e, f}; {\displaystyle 1})
\\ \hskip-1cm\longrightarrow 
  \Gamma \left [ \matrix 1 + a - b, 1 - d, 1 + a - e, -a + c + e, 1 + a - f,
     1 + a - b - e - f, -a + c + f, 1 - d + e + f\\ 1 + a, -a + c, 1 + a - b -
     e, 1 - d + e, 1 + a - b - f, 1 + a - e - f, 1 - d + f, -a + c + e +
     f\endmatrix \right ] \\
    {} _{7} V _{6} ({\displaystyle -d + e + f; 1 + a - b - d, -a + e + f, 1 + a
     - c - d, e, f}; {\displaystyle 1}) \\ + 
    \Gamma \left [ \matrix 1 + a - b, 1 + a - c, 1 - d, 1 + a - d, 1 + a - e,
     -a + c + e, 1 + a - f, 3 + 2 a - 2 b - d - e - f,\\ 
    1 + a, b, 1 - b + c, 1 + a - c - d,
     2 + a - b - d - e, e, 2 + a - b - d - f,\endmatrix \right.\\
   \left.\matrix  2 + 2 a - b - c - d - e
     - f, -a + c + f, -1 - a + b + e + f\\  2 + 2 a - b - c - e - f, 2 + 2 a
     - b - d - e - f, f, -1 - 2 a + b + c + e + f\endmatrix\right]\\
{} _{7} V _{6} ({\displaystyle 2 + 2 a - 2 b - d -
     e - f; 1 + a - b - d, 1 - b, 2 + 2 a - b - c - d - e - f, 1 + a - b - f,
     1 + a - b - e}; {\displaystyle 1})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T7740
\Description Transformation formula (\cite{\GaRaAA}, (5.6.1); Appendix
(III.38), $q\to1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T8810}.
$$\multline
{} _{7} H _{7} \!\left [ \matrix { 1 + {a\over 2}, b, c, d, e, f, g}\\ {
   {a\over 2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a - f, 1 + a -
   g}\endmatrix ; {\displaystyle 1}\right ]  \\
\longrightarrow 
    \Gamma \left [ \matrix 1 - b, 1 + a - b, 1 - c, 1 + a - c, 1 - d, 1 + a -
     d, 1 - e, 1 + a - e, 1 - f, 1 + a - f, 1 - a + 2 f, \\
     1 - a, 1 + a, 1 + a - b - f, 1 + a - c - f, 1 + a - d - f, 1 + a - e - f,
     1 - b + f, 1 - c + f, 1 - d + f,  \endmatrix \right.
\\
\left. \matrix -f + g, -a + f + g\\1 - e + f,-a + g, g\endmatrix\right]
  {} _{7} V _{6} ({\displaystyle -a + 2 f; -a + b + f, -a + c + f, -a + d +
     f, -a + e + f, -a + f + g}; {\displaystyle 1}) \\
    +
    \Gamma \left [ \matrix 1 - b, 1 + a - b, 1 - c, 1 + a - c, 1 - d, 1 + a -
     d, 1 - e, 1 + a - e, 1 - g, 1 + a - g, f - g, \\
     1 - a, 1 + a, -a + f, f, 1 + a - b - g, 1 + a - c - g, 1 + a - d - g, 1 +
     a - e - g, 1 - b + g, 1 - c + g, \endmatrix \right.
\\
\left. \matrix -a + f + g, 1 - a + 2 g\\1 - d + g, 1 - e + g\endmatrix\right]
 {} _{7} V _{6} ({\displaystyle -a + 2 g; -a + b + g, -a + c + g, -a + d
     + g, -a + e + g, -a + f + g}; {\displaystyle 1}) 
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8731 
\Description Transformation formula (\cite{\RaVeAA}, (7.7), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10906}.
$$\multline
{} _{8} V _{7} ({\displaystyle 2 a + n; c, d, e, {1\over 2} + a + n, 1 + 4 a
   - c - d - e + n, -n}; {\displaystyle -1}) \\
\hskip-5cm\longrightarrow 
 {{({ \textstyle 1 + 2 a - c, 1 + 2 a - d, 1 + 2 a - e, 1 + 2 a -
    c - d - e}) _{n}}\over  {({ \textstyle 1 + 2 a, 1 + 2 a - c - d, 1 +
    2 a - c - e, 1 + 2 a - d - e}) _{n}}} \\
  {} _{11} V _{10} ({\displaystyle a; -n, {c\over 2}, {1\over 2} + {c\over 2},
    {d\over 2}, {1\over 2} + {d\over 2}, {e\over 2}, {1\over 2} + {e\over 2},
    {1\over 2} + 2 a - {c\over 2} - {d\over 2} - {e\over 2} + {n\over 2}, 1 +
    2 a - {c\over 2} - {d\over 2} - {e\over 2} + {n\over 2}}; {\displaystyle
    1})
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T8732 
\Description Transformation formula (\cite{\RaVeAA}, (7.8), $q\to 1$, reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10907}.
$$\multline
{} _{8} V _{7} ({\displaystyle 2 a - e; {1\over 2} + a - e, c, d, e, 1 + 4 a
   - c - d - e + n, -n}; {\displaystyle -1}) \\
\hskip-5cm\longrightarrow 
 {{({ \textstyle 1 + 2 a - c, 1 + 2 a - d, 1 + 2 a -
    e, 1 + 2 a - c - d - e}) _{n}}\over  {({ \textstyle 1 + 2 a, 1 + 2 a -
    c - d, 1 + 2 a - c - e, 1 + 2 a - d - e}) _{n}}} \\
  {} _{11} V _{10} ({\displaystyle a; e, {c\over 2}, {1\over 2} + {c\over 2},
    {d\over 2}, {1\over 2} + {d\over 2}, {1\over 2} + 2 a - {c\over 2} -
    {d\over 2} - {e\over 2} + {n\over 2}, 1 + 2 a - {c\over 2} - {d\over 2} -
    {e\over 2} + {n\over 2}, {1\over 2} - {n\over 2}, {{-n}\over 2}};
    {\displaystyle 1})
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9831
\Description Transformation formula (\cite{\SlatAC}, (2.4.4.1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10901}.
$$\align
{} _{9}& V _{8} ( a; b, c, d, e, f, 2 + 3 a - b - c - d - e - f +
   n, -n; {\displaystyle 1}) \\
&\longrightarrow 
{{({ \textstyle 1 + a, 1 + a - e - f, 2 + 2 a - b - c -
    d - e, 2 + 2 a - b - c - d - f}) _{n}}\over  {({ \textstyle 1 + a - e, 1 +
    a - f, 2 + 2 a - b - c - d - e - f, 2 + 2 a - b - c - d}) _{n}}}\\
&\hskip10pt  {} _{9} V _{8} ({\displaystyle 1 + 2 a - b - c - d; 1 + a - c - d, 1 + a - b
    - d, 1 + a - b - c, e, f, 2 + 3 a - b - c - d - e - f + n, -n};
    {\displaystyle 1}) 
\endalign$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9832
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.4), reversed) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121101}.
$$\multline
{} _{9} V _{8} ({\displaystyle a; b, c, d, {1\over 2} + a - {{b + c + d}\over
   2}, 1 + a - {{b + c + d}\over 2}, b + c + d + n, -n}; {\displaystyle 1})\\ 
\longrightarrow  {{({ \textstyle 1 + a, -1 - 2 a + 2 b + 2 c + 2 d}) _{n}}\over  {({
    \textstyle -a + b + c + d, b + c + d}) _{n}}} 
  {} _{5} F _{4} \!\left [ \matrix { 1 + 2 a - b - c - d, 1 + a - c - d, 1 + a
    - b - d, 1 + a - b - c, -n}\\ { 1 + a - b, 1 + a - c, 1 + a - d, 2 + 2 a -
    2 b - 2 c - 2 d - n}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9833
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.26), $q\uparrow1$, reversed) 
    in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121102}.
$$\multline
{} _{9} V _{8} ({\displaystyle {{-1 + b + c + d - n}\over 2}; b, c, d,
   {{-n}\over 2}, {1\over 2} - {n\over 2}, e, b + c + d - e}; {\displaystyle
   1}) \longrightarrow
{{({ \textstyle {1\over 2} + {b\over 2} + {c\over 2} + {d\over 2} - {n\over
    2}, b + c + d - e}) _{n}}\over  {({ \textstyle {1\over 2} + {b\over 2} +
    {c\over 2} + {d\over 2} - e - {n\over 2}, b + c + d}) _{n}}}\\
 {} _{5} F _{4} \!\left [ \matrix { -n, {1\over 2} + {b\over 2} -
    {c\over 2} - {d\over 2} - {n\over 2}, {1\over 2} - {b\over 2} + {c\over 2}
    - {d\over 2} - {n\over 2}, {1\over 2} - {b\over 2} - {c\over 2} + {d\over
    2} - {n\over 2}, e}\\ { {1\over 2} - {b\over 2} + {c\over 2} + {d\over 2}
    - {n\over 2}, {1\over 2} + {b\over 2} - {c\over 2} + {d\over 2} - {n\over
    2}, {1\over 2} + {b\over 2} + {c\over 2} - {d\over 2} - {n\over 2}, 1 - b
    - c - d + e - n}\endmatrix ; {\displaystyle 1}\right ]   
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9834
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.5), reversed) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121103}.
$$\align
{} _{9} V _{8} ( a;& b, c, d, 1 + a - {{b + c + d}\over 2},
   {3\over 2} + a - {{b + c + d}\over 2}, -2 + b + c + d + n, -n;
   {\displaystyle 1}) \\
&\longrightarrow {{\left( -2 + b + c + d \right)  }\over {(-2 + b + c + d + 2 n)}}
      {{({ \textstyle 1 + a, -3 - 2 a + 2 b + 2 c + 2 d}) _{n}}\over  {({
       \textstyle -2 - a + b + c + d, -2 + b + c + d}) _{n}}} \\
&\hskip1cm
  {} _{6} F _{5} \!\left [ \matrix { 1 + 2 a - b - c - d, {3\over 2} + a -
       {b\over 2} - {c\over 2} - {d\over 2}, 1 + a - c - d, 1 + a - b - d, 1 +
       a - b - c, -n}\\ { {1\over 2} + a - {b\over 2} - {c\over 2} - {d\over
       2}, 1 + a - b, 1 + a - c, 1 + a - d, 4 + 2 a - 2 b - 2 c - 2 d -
       n}\endmatrix ; {\displaystyle 1}\right ]  
\endalign$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9835 
\Description Transformation formula (\cite{\BailAA}, 7.6(1)) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10903}.
$$\multline
{} _{9} V _{8} ({\displaystyle a; b, c, d, e, f, 2 + 3a - b - c - d - e - f
   + n, -n}; {\displaystyle 1}) \\
\hskip-4cm\longrightarrow 
{{({ \textstyle 1 + a, b, 1 + a - c - e, 1 + a - d -
    e, 1 + a - e - f, -1 - 2a + b + c + d + f - n}) _{n}} \over  {({
    \textstyle 1 + a - c, 1 + a - d, 1 + a - e, b - e, 1 + a - f, -1 - 2a +
    b + c + d + e + f - n}) _{n}}} \\
  {} _{9} V _{8} ({\displaystyle -b + e - n; e, 1 + a - b - c, 1 + a - b - d,
    1 + a - b - f, -1 - 2a + c + d + e + f - n, -a + e - n, -n};
    {\displaystyle 1})
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9836 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.21(iii), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10905}.
$$\multline
{} _{9} F _{8} \!\left [ \matrix { a, 1 + {a\over 2}, b, c, a - b - c, -A + C
   - n, -B - C - n, -A + B - n, -n}\\ { {a\over 2}, 1 + a - b, 1 + a - c, 1 +
   b + c, -C - n, -A + B + C - n, -B - n, -A - n}\endmatrix ; {\displaystyle
   1}\right ] \\ 
\hskip-4cm\longrightarrow {{({ \textstyle 1 + c, 1 + A - B, 1 + A - C, 1 + B + C, 1 + a, 1 + b, 1 + a
    - b - c}) _{n}}\over  {({ \textstyle 1 + b + c, 1 + A, 1 + B, 1 + A - B -
    C, 1 + C, 1 + a - b, 1 + a - c}) _{n}}} \\
{} _{9} F _{8} \!\left [ \matrix { A, 1 + {A\over 2}, B, C, A
    - B - C, -a + c - n, -b - c - n, -a + b - n, -n}\\ { {A\over 2}, 1 + A -
    B, 1 + A - C, 1 + B + C, -c - n, -a + b + c - n, -b - n, -a - n}\endmatrix
    ; {\displaystyle 1}\right ]  \\
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9837 
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.3), reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121104}.
$$\multline
{} _{9} V _{8} ({\displaystyle a; c, d, 1 + 2 a - b - c - d, 1 + {b\over 2},
   {1\over 2} + {b\over 2}, 2 a - b + n, -n}; {\displaystyle 1}) \\
\longrightarrow 
   {{({ \textstyle 1 + a, 2 a - 2 b}) _{n}}\over  {({ \textstyle a - b, 2 a
    - b}) _{n}}} 
  {} _{6} F _{5} \!\left [ \matrix { b, 1 + {b\over 2}, -a + b + c, -a + b +
    d, 1 + a - c - d, -n}\\ { {b\over 2}, 1 + a - c, 1 + a - d, -a + b + c +
    d, 1 - 2 a + 2 b - n}\endmatrix ; {\displaystyle 1}\right ]  
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9838
\Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15, $q\to1$, 
reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121107}.
$$\multline
 {} _{9} F _{8} \!\left [ \matrix { d, 1 + \frac{d}{2}, \frac{1}{2} -
    \frac{a}{2} + \frac{d}{2}, 1 - \frac{a}{2} + \frac{d}{2}, a - c + d, a - b
    + d, a, b, c}\\ { \frac{d}{2}, \frac{1}{2} + \frac{a}{2} + \frac{d}{2},
    \frac{a}{2} + \frac{d}{2}, 1 - a + c, 1 - a + b, 1 - a + d, 1 - b + d, 1 -
    c + d}\endmatrix ; {\displaystyle 1}\right ]  \\
\longrightarrow
    \Gamma \left [ \matrix a - b - c + d, 1 - c + d, 1 - b + d, a + d\\ 1 - b
     - c + d, a - c + d, a - b + d, 1 + d\endmatrix \right ]
   {} _{4} F _{3} \!\left [ \matrix { a, b, c, b + c - d}\\ { 1 - a + b, 1 -
     a + c, 1 - a + b + c - d}\endmatrix ; {\displaystyle 1}\right ] \,
\endmultline$$
provided at least one of $a,b,c$ is a non-negative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9891
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.39), $q\uparrow1$) in form 
    of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10961}.
$$\align
{} _{9} V& _{8} (a; b, c, d, e, f, g, 2 + 3 a - b - c - d - e -
   f - g; {\displaystyle 1}) \\
&\longrightarrow \Gamma \left [ \matrix 2 + 2 a - c - d - e, -1 - 2 a +
     b + c + d + e, 1 + a - f, 1 + a - g,\\ 1 + a, -a + b, 2 + 2 a
     - c - d - e - f, 2 + 2 a - c - d - e - g,\endmatrix
     \right .\\
&\hskip0.5cm \left.\matrix  -1 - 2 a + b + c + d + e + f + g, -a
     + b + f, -a + b + g, 2 + 2 a - c - d - e - f - g\\
    -a + b + f + g, -1 - 2 a + b +
     c + d + e + f, -1 - 2 a + b + c + d + e + g, 1 + a - f - g
     \endmatrix\right] \\
&\quad \quad  {} _{9} V _{8} ({\displaystyle 1 + 2 a - c - d - e; b, 1 + a - d - e, 1 + a
     - c - e, 1 + a - c - d, f, g, 2 + 3 a - b - c - d - e - f - g};
     {\displaystyle 1}) \\
&\quad  -\Gamma \left [ \matrix 1 - a + 2 b, a - b, 1 + a - c,
     1 + a - d, 1 + a - e, 1 + a - f, 1 + a - g,
  -1 - 2 a + b + c + d + e + f + g, \\ 
  1 + a, -a + b, c, d, e, f, g, 2 + 3 a - b - c - d -
     e - f - g,\endmatrix \right . \\
&\hskip0.5cm \left.\matrix  
   -a + b + c, -a + b + d, -a + b + e, -a + b + f, -a + b + g, 2 + 2 a
     - c - d - e - f - g\\  1 + b - c, 1 + b - d, 1 + b - e, 1 + b - f, 1 + b - g, -1 - 3
     a + 2 b + c + d + e + f + g\endmatrix\right]\\
&\quad \quad  {} _{9} V _{8} ({\displaystyle -a + 2 b; b, -a + b + c, -a + b
     + d, -a + b + e, -a + b + f, -a + b + g, 2 + 2 a - c - d - e - f - g};
     {\displaystyle 1}) \\
&\quad +     \Gamma \left [ \matrix -2 a + 2 b + c + d + e, 1 + 2 a - b - c - d - e, 1
     + a - c, 1 + a - d, 1 + a - e, 1 + a - f,\\ 1 + a, -a + b, f, g, 2 + 3 a - b - c - d -
     e - f - g, 1 + b - f, 1 + b - g,\endmatrix \right .\\
&\hskip0.5cm \left.\matrix  1 + a - g, -1 - 2 a + b + c + d
     + e + f + g, -a + b + c, -a + b + d, -a + b + e, \\ 
    -1 - 3 a + 2 b + c + d + e + f + g, 1 +
     a - d - e, 1 + a - c - e, 1 + a - c - d,\endmatrix\right.\\
&\hskip0.5cm\left. \matrix -a + b + f, -a + b + g,
     2 + 2 a - c - d - e - f - g\\  -a + b + d + e, -a + b + c + e,
     -a + b + c + d\endmatrix\right]
   {} _{9} V _{8} ( -1 - 2 a + 2 b + c + d + e; b, -a + b + c,
     -a + b + d, \\
&\hskip2cm-a + b + e, -1 - 2 a + b + c + d + e + f, -1 - 2 a + b + c +
     d + e + g, 1 + a - f - g; {\displaystyle 1}) 
\endalign$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.


\Name T9892 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10962}.
$$\multline
{} _{9} V _{8} ({\displaystyle a; b, c, {1\over 2} + c, d, {1\over 2} + d, e,
   {1\over 2} + e}; {\displaystyle 1}) \\
\longrightarrow 
  \Gamma \left [ \matrix {1\over 2} + a - c, 1 + a - c, {1\over 2} + a - d,
     1 + a - d, {1\over 2} + a - e, 1 + a - e, {1\over 2} + a - c - d - e, 1 +
     a - c - d - e\\ {1\over 2} + a, 1 + a, {1\over 2} + a - c - d, 1 + a - c
     - d, {1\over 2} + a - c - e, 1 + a - c - e, {1\over 2} + a - d - e, 1 + a
     - d - e\endmatrix \right ]  \\
    {} _{4} F _{3} \!\left [ \matrix { {1\over 2} + a - b, 2 c, 2 d, 2 e}\\ {
     {1\over 2} + a, 1 + 2 a - 2 b, -2 a + 2 c + 2 d + 2 e}\endmatrix ;
     {\displaystyle 1}\right ] \\ + 
   \Gamma \left [ \matrix 1 + a - b, {1\over 2} + a - c, 1 + a
     - c, {1\over 2} + a - d, 1 + a - d, {3\over 2} + 3 a - b - 2 c - 2 d -
     2 e, \\ 1 + a, c, {1\over 2} + c, d, {1\over 2} + d, {3\over 2} + 3 a
     - 2 c - 2 d - 2 e, 1 + 2 a - b - c - d - e,\endmatrix \right .\\
\left.\matrix {1\over 2} + a - e, 1 + a - e, -{1\over 2} - a + c + d + e, -a + c
     + d + e\\  {3\over 2} + 2 a - b - c
     - d - e, e, {1\over 2} + e\endmatrix \right]\\  
 {} _{4} F _{3} \!\left [ \matrix { {3\over 2} + 3 a - b - 2 c - 2 d -
     2 e, 1 + 2 a - 2 c - 2 d, 1 + 2 a - 2 c - 2 e, 1 + 2 a - 2 d -
     2 e}\\ { {3\over 2} + 3 a - 2 c - 2 d - 2 e, 2 + 4 a - 2 b - 2 c
     - 2 d - 2 e, 2 + 2 a - 2 c - 2 d - 2 e}\endmatrix ; {\displaystyle
     1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9893 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.30, $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T10963}.
$$\multline
{} _{9} V _{8} ({\displaystyle a; b, c, d, e, f, g, 2 + 3 a - b - c - d - e -
    f - g}; {\displaystyle 1}) \\ + 
   \Gamma \left [ \matrix a - b, 1 - a + 2 b, 1 + a -
     c, -a + b + c, 1 + a - d, -a + b + d, 1 + a - e, -a + b + e, 1 + a - f,
     \\ 1 + a, -a + b, 1 + b - c, c, 1 + b - d, d,
     1 + b - e, e, 1 + b - f, f, 1 + b - g, \endmatrix \right .\\
  \left.\matrix -a + b + f, 1 + a - g, 2 + 2 a - c - d - e - f - g, -a + b + g, -1 -
     2 a + b + c + d + e + f + g\\2 + 3 a - b - c - d - e - f - g,
     g, -1 - 3 a + 2 b + c + d + e + f + g\endmatrix  \right] \\ 
 {} _{9} V _{8} ({\displaystyle -a + 2 b; b, -a + b + c, -a + b + d, -a + b
     + e, -a + b + f, -a + b + g, 2 + 2 a - c - d - e - f - g};
     {\displaystyle 1}) \\ 
\hskip-1cm\longrightarrow 
  \Gamma \left [ \matrix 1 + a - c, -a + b + c, 1 + a - d, -a + b + d, 1 + a
     - e, -a + b + e, 1 + a - f, -a + b + f,\\ 1 + a, -a + b, 1 + b - g, 2 + 2 a - c
     - d - e - g, 2 + 2 a - c - d - f - g, 2 + 2 a - c - e - f - g, 
  \endmatrix \right .\\
  \left.\matrix  3 + 3 a - c - d - e - f - 2 g,
     2 + 2 a - c - d - e - f - g, \\ 2 + 2 a
     - d - e - f - g, g, -1 - 2 a + b + c + d + e + g, -1 - 2 a + b + c + d
     + f + g, \endmatrix\right.\\
  \left.\matrix  -1 - 2 a + b + c + d + e + f + g, -2 -
     3 a + b + c + d + e + f + 2 g\\
    -1 - 2 a + b + c + e + f + g, -1 - 2 a + b + d + e + f +
     g   \endmatrix\right]\\
    {} _{9} V _{8} ({\displaystyle 2 + 3 a - c - d - e - f - 2 g; b, 1 + a - c
     - g, 1 + a - d - g, 1 + a - e - g, 1 + a - f - g,}\\
   2 + 2 a - c - d - e -
     f - g, 2 + 3 a - b - c - d - e - f - g; {\displaystyle 1}) \\ 
+ \Gamma \left [ \matrix 1 + a - c, -a + b + c, 1 + a - d, -a + b + d, 1 + a
     - e, -a + b + e, 1 + a - f, -a + b + f, \\ 1
     + a, -a + b, 1 + a - c - g, 1 + a - d - g, 1 + a - e - g, 1 + a - f - g,
     2 + 3 a - b - c - d - e - f - g, \endmatrix
     \right .\\
  \left.\matrix 2 + 3 a - b - c - d - e - f -
     2 g, 1 + a - g, -a + b + g, -1 - 3 a + 2 b + c + d + e + f + 2 g\\
  -a + b + c + g, -a + b + d + g, -a + b
     + e + g, -a + b + f + g, -1 - 3 a + 2 b + c + d + e + f + g
   \endmatrix \right]\\
    {} _{9} V _{8} ({\displaystyle -2 - 3 a + 2 b +
     c + d + e + f + 2 g; b, -1 - 2 a + b + d + e + f + g, -1 - 2 a + b + c
     + e + f + g,}\\ -1 - 2 a + b + c + d + f + g,
   -1 - 2 a + b + c + d + e +     g, -a + b + g, g; {\displaystyle 1})
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T9894 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.25, $q\to 1$, 
reversed) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121161}.
\NoBlackBoxes
$$\multline
{} _{9} V _{8} ({\displaystyle a; b, {1\over 2} + b, c, d, 1 + 2 a - 2 b - c
    - d, e, 1 + 2 a - 2 b - e}; {\displaystyle 1}) \\ + 
   \Gamma \left[ \matrix{ \textstyle 1 + a - 2 b},  
       {  1 + 2 a - 2 b},  
       {  1 + a - c},  {  1 + a - d},  
       {  -a + 2 b + c + d},  
       {  3 + 3 a - 4 b - 2 e},  
       {  1 + a - e},  
       {  1 + a - 2 b + c - e},  \\
     { \textstyle 1 + a},  {  2 b},  
       {  c},  
       {  1 + 2 a - 2 b - c - d},  
       {  d},  
       {  3 + 4 a - 6 b - 2 e},  
       {  1 + a - 2 b - e},  
  \endmatrix\right.\\
  \left.\matrix      {  2 + 3 a - 4 b - c - d - e},  
       {  1 + a - 2 b + d - e},  
       {  -1 - a + 2 b + e} \\      {  2 + 2 a - 2 b - c - e},  
       {  2 + 2 a - 2 b - d - e},  
       {  1 + c + d - e},  {  e} \endmatrix \right]\\
 {} _{9} V _{8} ({\displaystyle 2 + 3 a - 4 b - 2 e; 1 + a - b - e,
        {3\over 2} + a - b - e, 1 + a - 2 b, 1 + 2 a - 2 b - e, 1 + a -
        2 b + c - e, 1 + a - 2 b + d - e,}\\
  2 + 3 a - 4 b - c - d - e;
        {\displaystyle 1}) \\ 
\hskip-10cm\longrightarrow 
  \Gamma \left [ \matrix 1 + a - 2 b, 1 + 2 a - 2 b, 1 + a - e, 1 + 2 a
     - 4 b - e\\ 1 + a, 1 + 2 a - 4 b, 1 + a - 2 b - e, 1 + 2 a - 2 b -
     e\endmatrix \right ]  \\
    {} _{5} F _{4} \!\left [ \matrix { 2 b, -a + 2 b + c, -a + 2 b + d, 1 + a
     - c - d, e}\\ { 1 + a - c, 1 + a - d, -a + 2 b + c + d, -2 a + 4 b +
     e}\endmatrix ; {\displaystyle 1}\right ] \\ 
+ \Gamma \left [ \matrix 1 + a - 2 b, 1 + 2 a - 2 b, 1 + a - c, 1 + a -
     d, -a + 2 b + c + d, 1 + a - e, 1 + a - 2 b + c - e,\\ 1 + a, 2 b, -a +
     2 b + c, 1 + a - c - d, -a + 2 b + d, 1 + a - 2 b - e, \endmatrix
     \right .\\
  \left.\matrix  2 + 3 a - 4 b -
     c - d - e, 1 + a - 2 b + d - e, -1 - 2 a + 4 b + e\\
  2 + 3 a - 4 b
     - c - e, 2 + 3 a - 4 b - d - e, 1 + a - 2 b + c + d - e, e
   \endmatrix \right]\\
    {} _{5} F _{4} \!\left [ \matrix { 1 + 2 a -
     4 b, 1 + 2 a - 2 b - e, 1 + a - 2 b + c - e, 1 + a - 2 b + d - e, 2
     + 3 a - 4 b - c - d - e}\\ { 2 + 2 a - 4 b - e, 2 + 3 a - 4 b - c -
     e, 2 + 3 a - 4 b - d - e, 1 + a - 2 b + c + d - e}\endmatrix ;
     {\displaystyle 1}\right ]
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.
\par\BlackBoxes

\Name T9940
\Description Transformation formula (\cite{\GaRaAA}, (5.6.3); Appendix
(III.40), $q\to1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T101010}.
$$\multline
{} _{9} H _{9} \!\left [ \matrix { 1 + {a\over 2}, b, c, d, e, f, g, h, k}\\ {
   {a\over 2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a - f, 1 + a -
   g, 1 + a - h, 1 + a - k}\endmatrix ; {\displaystyle 1}\right ] 
\\
\longrightarrow
    \Gamma \left [ \matrix 1 - b, 1 + a - b, 1 - c, 1 + a - c, 1 - d, 1 + a -
     d, 1 - e, 1 + a - e, 1 - f, 1 + a - f, 1 - g, 1 + a - g, \\ 
    1 - a, 1 + a, 1 + a - b - g, 1 + a
     - c - g, 1 + a - d - g, 1 + a - e - g, 1 + a - f - g,\endmatrix \right
     .\\
\left. \matrix  1 - a + 2 g, -g
     + h, -a + g + h, -g + k, -a + g + k\\
      1 - b + g, 1 - c +
     g, 1 - d + g, 1 - e + g, 1 - f + g, h, -a + h, k, -a + k
   \endmatrix\right] 
\\
\times  {} _{9} V _{8} ({\displaystyle -a + 2 g; -a + b + g, -a + c + g, -a + d +
     g, -a + e + g, -a + f + g, -a + g + h, -a + g + k}; {\displaystyle 1})
\\
 +\Gamma \left [ \matrix 1 - b, 1 + a - b, 1 - c, 1 + a - c, 1 - d, 1
     + a - d, 1 - e, 1 + a - e, 1 - f, 1 + a - f, 1 - h, 1 + a - h,\\ 
     1 - a, 1 + a, g, -a + g, 1 +
     a - b - h, 1 + a - c - h, 1 + a - d - h, 1 + a - e - h, 1 + a - f - h, 
    \endmatrix
     \right .\\
\left. \matrix  g - h, -a
     + g + h, 1 - a + 2 h, -h + k, -a + h + k\\
    1 - b + h, 1 - c + h, 1 - d + h, 1 - e + h, 1 - f + h, k, -a + k
   \endmatrix\right] 
\\
\times
 {} _{9} V _{8} ({\displaystyle -a + 2 h; -a + b + h, -a + c + h, -a
     + d + h, -a + e + h, -a + f + h, -a + g + h, -a + h + k}; {\displaystyle
     1}) \\
 +\Gamma \left [ \matrix 1 - b, 1 + a - b, 1 - c, 1 + a
     - c, 1 - d, 1 + a - d, 1 - e, 1 + a - e, 1 - f, 1 + a - f, 1 - k, 1 + a -
     k,\\ 1 - a, 1 + a, g,
     -a + g, h, -a + h, 1 + a - b - k, 1 + a - c - k, 1 + a - d - k, 1 + a - e
     - k, 1 + a - f - k,\endmatrix \right .\\
\left. \matrix  g - k, h - k, -a + g + k, -a + h + k, 1 - a + 2 k\\
      1 - b + k, 1 - c + k, 1 - d + k, 1 - e + k, 1 - f +
     k\endmatrix\right] 
\\
\times {} _{9} V _{8} ({\displaystyle -a + 2 k; -a + b + k, -a + c +
     k, -a + d + k, -a + e + k, -a + f + k, -a + h + k, -a + g + k};
     {\displaystyle 1}) 
\endmultline$$
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T111031 
\Description Transformation formula (\cite{\RaVeAA}, (7.8), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121105}.
$$\multline
{} _{11} V _{10} ({\displaystyle a; c, {1\over 2} + c, d, {1\over 2} + d, e,
   {1\over 2} + e, {1\over 2} + 2 a - c - d - e + {n\over 2}, 1 + 2 a - c -
   d - e + {n\over 2}, -n}; {\displaystyle 1}) \\
\longrightarrow 
   {{({ \textstyle 1 + 2 a, 1 + 2 a - 2 c - 2 d, 1 + 2 a - 2 c - 2 e, 1
    + 2 a - 2 d - 2 e}) _{n}}\over  {({ \textstyle 1 + 2 a - 2 c, 1 +
    2 a - 2 d, 1 + 2 a - 2 e, 1 + 2 a - 2 c - 2 d - 2 e}) _{n}}} \\
  {} _{8} V _{7} ({\displaystyle 2 a + n; 2 c, 2 d, 2 e, {1\over 2} + a +
    n, 1 + 4 a - 2 c - 2 d - 2 e + n, -n}; {\displaystyle -1})
\endmultline$$
where $n$ is a nonnegative integer.
\Seealso S2103, S3201, TListe, TransListe, Ers, PosListe.

\Name T111032 
\Description Transformation formula (\cite{\RaVeAA}, (7.7), $q\to 1$) in form of a rule.
A $q$-analogue is HYPQ's \hbox{\tt T121106}.
$$\multline
{} _{11} V _{10} ({\displaystyle a; 2 e, c, {1\over 2} + c, d, {1\over 2} +
   d, {1\over 2} + 2 a - c - d - e + {n\over 2}, 1 + 2 a - c - d - e +
   {n\over 2}, {1\over 2} - {n\over 2}, {{-n}\over 2}}; {\displaystyle 1})\\ 
\longrightarrow 
   {{({ \textstyle 1 + 2 a, 1 + 2 a - 2 c - 2 d, 1 + 2 a - 2 c - 2 e, 1
    + 2 a - 2 d - 2 e}) _{n}}\over  {({ \textstyle 1 + 2 a - 2 c, 1 +
    2 a - 2 d, 1 + 2 a - 2 e, 1 + 2 a - 2 c - 2 d - 2 e}) _{n}}} \\
  {} _{8} V _{7} ({\displaystyle 2 a - 2 e; {1\over 2} + a - 2 e, 2 c,
    2 d, 2 e, 1 + 4 a - 2 c - 2 d - 2 e + n, -n}; {\displaystyle -1})
\endmultline$$
where $n$ is a nonnegative integer.
\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]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[2]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  c%
\MATHrbrace \MATHbackslash endmatrix ; %
\MATHlbrace \MATHbackslash displaystyle
   z%
\MATHrbrace \MATHbackslash right ] 
\goodbreakpoint%
In[3]:= TeX
\goodbreakpoint%
In[4]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[5]:= TeXForm[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]]
\goodbreakpoint%
Out[5]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b\MATHbackslash cr c%
\MATHrbrace  ; %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace  \MATHbackslash right ] 
\endMATH 
\Seealso AmSTeX, AmSLaTeX, LaTeX, TeXMat, TeXFV.

\Name TeXFV
\Description Switch that toggles between writing very well-poised 
  hypergeometric series in terms of \hbox{\tt V} and in terms of \hbox{\tt F}, respectively,
  when written in TeXForm. By default very well-poised hypergeometric series 
  are written in terms of \hbox{\tt V}.
\Usage TeXFV.
\Example 
\MATH
In[1]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= F[%
\MATHlbrace a,1+a/2,b,c%
\MATHrbrace ,%
\MATHlbrace a/2,a+1-b,a+1-b%
\MATHrbrace ,z]
\goodbreakpoint%
                        a
           %
\MATHluEck      a, 1 + -, b, c         %
\MATHruEck 
           %
\MATHvStrich             2               %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  a                          %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - b    %
\MATHroEck 
             2
\goodbreakpoint%
In[3]:= Reset
\goodbreakpoint%
Out[0]= 0
\goodbreakpoint%
In[1]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[2]:= F[%
\MATHlbrace a,1+a/2,b,c%
\MATHrbrace ,%
\MATHlbrace a/2,a+1-b,a+1-c%
\MATHrbrace ,z]
\goodbreakpoint%
                        a
           %
\MATHluEck      a, 1 + -, b, c         %
\MATHruEck 
           %
\MATHvStrich             2               %
\MATHvStrich 
Out[2]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  a                          %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - c    %
\MATHroEck 
             2
\goodbreakpoint%
In[3]:= TeXForm[\%]
\goodbreakpoint%
Out[3]//TeXForm= %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 4%
\MATHrbrace  V \MATHtief %
\MATHlbrace 3%
\MATHrbrace  (%
\MATHlbrace \MATHbackslash displaystyle a; b, c%
\MATHrbrace ; %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace )
\goodbreakpoint%
In[4]:= TeXFV
\goodbreakpoint%
In[5]:= hypAttributes
\goodbreakpoint%
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses F[] for very well-poised basic hypergeometric series.
\goodbreakpoint%
In[6]:= TeXForm[\%2]
\goodbreakpoint%
Out[6]//TeXForm= 
  %
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 4%
\MATHrbrace  F \MATHtief %
\MATHlbrace 3%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, 1 + %
\MATHlbrace a\MATHbackslash over 2%
\MATHrbrace , b, c%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  %
\MATHlbrace a\MATHbackslash over 2%
\MATHrbrace ,
   1 + a - b, 1 + a - c%
\MATHrbrace \MATHbackslash endmatrix ; %
\MATHlbrace \MATHbackslash displaystyle z%
\MATHrbrace \MATHbackslash right ] 
\goodbreakpoint%
In[7]:= TeXFV
\goodbreakpoint%
In[8]:= hypAttributes

Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\endMATH 
\Seealso F, V, hypAttributes.

\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[p[a/3,2*n],filename]
\goodbreakpoint%
In[2]:= TeXMat[F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z],filename,"A hypergeometric \MATHtief 2F\MATHtief 1 series"]
\goodbreakpoint%
In[3]:= !type filename.m
A[1]:=
p[a/3, 2*n]
"A hypergeometric \MATHtief 2F\MATHtief 1 series"
A[2]:=
F[%
\MATHlbrace a, b%
\MATHrbrace , %
\MATHlbrace c%
\MATHrbrace , z]
\goodbreakpoint%
In[3]:= !type filename.tex
A[1]:=
(%
\MATHlbrace  \MATHbackslash textstyle %
\MATHlbrace a\MATHbackslash over 3%
\MATHrbrace %
\MATHrbrace ) \MATHtief %
\MATHlbrace 2\MATHbackslash ,n%
\MATHrbrace  
"A hypergeometric \MATHtief 2F\MATHtief 1 series"
A[2]:=
%
\MATHlbrace %
\MATHrbrace  \MATHtief %
\MATHlbrace 2%
\MATHrbrace  F \MATHtief %
\MATHlbrace 1%
\MATHrbrace  \MATHbackslash !\MATHbackslash left [ \MATHbackslash matrix %
\MATHlbrace  a, b%
\MATHrbrace \MATHbackslash \MATHbackslash  %
\MATHlbrace  c%
\MATHrbrace \MATHbackslash endmatrix ; %
\MATHlbrace \MATHbackslash displaystyle
  z%
\MATHrbrace \MATHbackslash right ] 
\endMATH
\Seealso AmSTeX, AmSLaTeX, LaTeX, TeX, TeXFV, SchreibeZahl.

\Name Tgl2103
\Description Transformation formula (\cite{\SlatAC}, (1.3.15)) 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}, Appendix (III.4), $q\uparrow1$) in form of an equation.
It is the same transformation as that in \hbox{\tt T2104}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2106
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.6), $q\uparrow1$) 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}, Appendix (III.7), $q\uparrow1$) in form of an equation.
It is the same transformation as that in \hbox{\tt T2107}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2110  
\Description Transformation formula (\cite{\RaVeAA}, (3.2)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2110}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2112 
\Description Transformation formula (\cite{\RaVeAA}, (5.10)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2112}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2131
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.6), reversed, $q\uparrow1$) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T2131}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2132
\Description Transformation formula (\cite{\SlatAC}, (2.5.7)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2132}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2133 
\Description Transformation formula (\cite{\RaVeAA}, (5.12)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2133}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2134 
\Description Transformation formula (\cite{\BailAA}, Ex. 4.(iii), p. 97, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2134}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2135 
\Description Transformation formula (\cite{\BailAA}, Ex. 4.(iii), p. 97) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2135}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2136 
\Description Transformation formula (\cite{\RaVeAA}, (5.10), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2136}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2137 
\Description Transformation formula (\cite{\RaVeAA}, (5.12), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2137}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2138 
\Description Transformation formula (\cite{\RaVeAA}, (3.31), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2138}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2139 
\Description Transformation formula (\cite{\RaVeAA}, (3.31)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2139}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2140 
\Description Transformation formula (\cite{\RaVeAA}, (3.2), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2140}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2141 
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2141}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2163 
\Description Transformation formula (\cite{\GaRaAA}, Ex 3.8, $q\to 1$; 
\cite{\SlatAC}, pp. 36/37) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2163}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2191
\Description Transformation formula (\cite{\SlatAC}, (1.8.10)) in form of an equation.
It is the same transformation as that in \hbox{\tt T2191}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl2192 
\Description Transformation formula (\cite{\GaRaAA}, Ex 3.8, $q\to 1$, 
reversed; \cite{\SlatAC}, pp. 36/37) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T2192}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3204
\Description Transformation formula (\cite{\BailAA}, Ex.~7, p.~98) 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{\SlatAC}, (2.3.3.7)) 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{\BailAA}, Ex.~7, p.~98, terminating form) 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} (3.1.1)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3207}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3217
\Description Transformation formula (\cite{\BailAA} 4.4(2), reversed) in form of an equation.
It is the same transformation as that in \hbox{\tt T3217}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3231
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.21), $q\uparrow1$) in form of an equation.
It is the same transformation as that in \hbox{\tt T3231}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3232
\Description Transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.97,
reversed, first form) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3232}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3233
\Description Transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.97,
reversed, second form) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3233}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3234
\Description Transformation formula (\cite{\SlatAC}, (2.5.7), reversed) 
in form of an equation.
It is the same transformation as that in \hbox{\tt T3234}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3235 
\Description Transformation formula (\cite{\GaRaAA}, Ex 3.4, $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3235}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3236 
\Description Transformation formula (\cite{\GaRaAA}, (3.4.8), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3236}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3237 
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3237}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3238 
\Description Transformation formula (\cite{\BailAA}, Ex. 6, p. 97) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3238}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3239
\Description Transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.~97) in form of an equation.
It is the same transformation as that in \hbox{\tt T3239}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3240 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3240}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3261
\Description Transformation formula (\cite{\GaRaAA}, (III.33), $q\uparrow1$) 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{\SlatAC}, (4.3.4.2)) in form of an equation.
It is the same transformation as that in \hbox{\tt T3262}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3263 
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.33), 
$q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3263}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3264 
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.34), 
$q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3264}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl3267 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.6, $q\to 1$) 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, 
$q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T3268}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4301
\Description Transformation formula (\cite{\SlatAC}, (4.3.5.1)) 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}, Appendix (III.16), $q\uparrow1$) 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{\SlatAC}, (2.4.1.1)) 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{\SlatAC}, (4.3.6.4)) in form of an equation.
It is the same transformation as that in \hbox{\tt T4304}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4306
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.21), $q\uparrow1$, reversed) in form of an equation.
It is the same transformation as that in \hbox{\tt T3231}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4309 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(i), $q\to 1$) 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), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T4310}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4312 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.4, $q\to 1$) 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, $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T4313}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4331 
\Description Transformation formula (\cite{\BailAA}, Ex. 6, p. 97, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T4331}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4332 
\Description Transformation formula (\cite{\BailAA}, 4.6(1)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T4332}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4362 
\Description Transformation formula (\cite{\SlatAC}, (2.4.4.3), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T4362}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl4391 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T4391}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5401
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.4)) 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}, Appendix (III.26), $q\uparrow1$) 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{\BailAA}, 4.6(1), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T5403}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl5468 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.25, $q\to 1$) 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{\SlatAC}, (2.4.3.3)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T6501}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl6531
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.5)) in form of an equation.
It is the same transformation as that in \hbox{\tt T6531}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl6532 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(ii), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T6532}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl6533 
\Description Transformation formula (\cite{\GaRaAA}, (3.10.4), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T6533}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl6534 
\Description Transformation formula (\cite{\BailAA}, 4.4(2)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T6534}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7631
\Description Transformation formula (\cite{\SlatAC}, (2.4.1.1), reversed) in form of an equation.
It is the same transformation as that in \hbox{\tt T7631}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7632
\Description Transformation formula (\cite{\SlatAC}, (2.4.1.1), reversed) in form of an equation.
It is the same transformation as that in \hbox{\tt T7632}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7633
\Description Transformation formula (\cite{\SlatAC}, (4.3.6.4), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T7633}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7634
\Description Transformation formula (\cite{\BailAA}, 7.5.(1)) in form of an equation.
It is the same transformation as that in \hbox{\tt T7634}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7635
\Description Transformation formula (\cite{\BailAA}, 7.5.(2)) in form of an equation.
It is the same transformation as that in \hbox{\tt T7635}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7636 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.10), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T7636}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7637 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.13(i), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T7637}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7691
\Description Transformation formula (\cite{\SlatAC}, (2.4.4.3)) in form of an equation.
It is the same transformation as that in \hbox{\tt T7691}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7692
\Description Transformation formula (\cite{\SlatAC}, (4.3.7.8)) in form of an equation.
It is the same transformation as that in \hbox{\tt T7692}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7693 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.15, $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T7693}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl7694 
\Description Transformation formula (\cite{\SlatAC}, (4.3.7.8), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T7694}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8731 
\Description Transformation formula (\cite{\RaVeAA}, (7.7), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T8731}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl8732 
\Description Transformation formula (\cite{\RaVeAA}, (7.8), $q\to 1$, reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T8732}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9831
\Description Transformation formula (\cite{\SlatAC}, (2.4.4.1)) in form of an equation.
It is the same transformation as that in \hbox{\tt T9831}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9832
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.4), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T9832}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9833
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.26), $q\uparrow1$, reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T9833}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9834
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.5), reversed) in form 
  of an equation.
It is the same transformation as that in \hbox{\tt T9834}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9835 
\Description Transformation formula (\cite{\BailAA}, 7.6(1)) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9835}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9836 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 3.21(iii), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9836}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9837 
\Description Transformation formula (\cite{\SlatAC}, (2.4.3.3), reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9837}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9838 
\Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15, $q\to1$, 
reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9838}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9891
\Description Transformation formula (\cite{\GaRaAA}, Appendix (III.39), $q\uparrow1$) in form of an equation.
It is the same transformation as that in \hbox{\tt T9891}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9892 
\Description Transformation formula (\cite{\GaRaAA}, (3.5.7), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9892}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9893 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.30, $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9893}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl9894 
\Description Transformation formula (\cite{\GaRaAA}, Ex. 2.25, $q\to 1$, 
reversed) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T9894}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl111031 
\Description Transformation formula (\cite{\RaVeAA}, (7.8), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T111031}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tgl111032 
\Description Transformation formula (\cite{\RaVeAA}, (7.7), $q\to 1$) in form of 
an equation.
It is the same transformation as that in \hbox{\tt T111032}.
\Seealso Sgl2101, TransListe\$gl, Gleichung.

\Name Tli2103
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (1.3.15)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2103}.
\Seealso Tli2107, Tli2104, TransListe\$gl, Gleichung.

\Name Tli2104
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.4), $q\uparrow1$) in form of a rule.
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 Tli2107}-example. For information of how to
read the resulting listing confer {\tt Tli2107}.
\MATH
In[4]:= \%2/.Tli2104
\goodbreakpoint%
               %
\MATHluEck            %
\MATHruEck 
               %
\MATHvStrich  -m, -n    %
\MATHvStrich 
Out[4]= %
\MATHlbrace 2,  F %
\MATHvStrich        ; z %
\MATHvStrich , %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace 
            2 1%
\MATHvStrich    a       %
\MATHvStrich 
               %
\MATHloEck            %
\MATHroEck 
\goodbreakpoint%
                %
\MATHluEck                          %
\MATHruEck 
           m    %
\MATHvStrich  -m, 1 - a - m    1 - z  %
\MATHvStrich 
          z   F %
\MATHvStrich               ; -(-----) %
\MATHvStrich  (a + n)
             2 1%
\MATHvStrich  1 - a - m - n      z    %
\MATHvStrich         m
                %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser          ------------------------------------------%
\MATHrbrace %
\MATHrbrace , T2107%
\MATHrbrace , T2104%
\MATHrbrace , 
                             (a)
                                m
\goodbreakpoint% 
               %
\MATHluEck                          %
\MATHruEck 
          n    %
\MATHvStrich  -n, 1 - a - n    1 - z  %
\MATHvStrich 
         z   F %
\MATHvStrich               ; -(-----) %
\MATHvStrich  (a + n)
            2 1%
\MATHvStrich  1 - a - m - n      z    %
\MATHvStrich         m
               %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser     %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace ------------------------------------------%
\MATHrbrace %
\MATHrbrace , T2107%
\MATHrbrace , 
                            (a)
                               m
\goodbreakpoint%
\MATHgroesser      %
\MATHlbrace FPerm[2, 1, u], T2104%
\MATHrbrace %
\MATHrbrace , 
\goodbreakpoint% 
              %
\MATHluEck                          %
\MATHruEck 
         m    %
\MATHvStrich  -m, 1 - a - m    1 - z  %
\MATHvStrich 
        z   F %
\MATHvStrich               ; -(-----) %
\MATHvStrich  (a + m)
           2 1%
\MATHvStrich  1 - a - m - n      z    %
\MATHvStrich         n
              %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser     %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace ------------------------------------------%
\MATHrbrace , 
                           (a)
                              n
\goodbreakpoint% 
\MATHgroesser       %
\MATHlbrace FPerm[2, 1, u], T2107%
\MATHrbrace %
\MATHrbrace , %
\MATHlbrace FPerm[2, 1, u], T2104%
\MATHrbrace %
\MATHrbrace , 
\goodbreakpoint% 
              %
\MATHluEck                          %
\MATHruEck 
         n    %
\MATHvStrich  -n, 1 - a - n    1 - z  %
\MATHvStrich 
        z   F %
\MATHvStrich               ; -(-----) %
\MATHvStrich  (a + m)
           2 1%
\MATHvStrich  1 - a - m - n      z    %
\MATHvStrich         n
              %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser     %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace ------------------------------------------%
\MATHrbrace , 
                           (a)
                              n
\goodbreakpoint% 
\MATHgroesser       %
\MATHlbrace FPerm[2, 1, u], T2107%
\MATHrbrace %
\MATHrbrace , T2104%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[5]:= \%1/.FPerm[2,1,u]/.T2107/.FPerm[2,1,u]/.T2104
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
              %
\MATHluEck                          %
\MATHruEck 
         m    %
\MATHvStrich  -m, 1 - a - m    1 - z  %
\MATHvStrich 
        z   F %
\MATHvStrich               ; -(-----) %
\MATHvStrich  (a + m)
           2 1%
\MATHvStrich  1 - a - m - n      z    %
\MATHvStrich         n
              %
\MATHloEck                          %
\MATHroEck 
Out[5]= ------------------------------------------
                           (a)
                              n
\endMATH
\Seealso Tli2107, TransListe, Gleichung.

\Name Tli2106
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.6), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T2106}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2107
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.7), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T2107}.
\Example
\MATH
In[1]:= F[%
\MATHlbrace -m,-n%
\MATHrbrace ,%
\MATHlbrace a%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck            %
\MATHruEck 
           %
\MATHvStrich  -m, -n    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich        ; z %
\MATHvStrich 
        2 1%
\MATHvStrich    a       %
\MATHvStrich 
           %
\MATHloEck            %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.Tli2107
Is m a nonnegative integer?
[y|n]:  y
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
                                    %
\MATHluEck                       %
\MATHruEck 
                                    %
\MATHvStrich     -m, -n            %
\MATHvStrich 
                                  F %
\MATHvStrich               ; 1 - z %
\MATHvStrich  (a + n)
               %
\MATHluEck            %
\MATHruEck      2 1%
\MATHvStrich  1 - a - m - n        %
\MATHvStrich         m
               %
\MATHvStrich  -m, -n    %
\MATHvStrich         %
\MATHloEck                       %
\MATHroEck 
Out[2]= %
\MATHlbrace 1,  F %
\MATHvStrich        ; z %
\MATHvStrich , %
\MATHlbrace %
\MATHlbrace %
\MATHlbrace ------------------------------------%
\MATHrbrace %
\MATHrbrace , 
            2 1%
\MATHvStrich    a       %
\MATHvStrich                      (a)
               %
\MATHloEck            %
\MATHroEck                         m
\goodbreakpoint% 
                   %
\MATHluEck                       %
\MATHruEck 
                   %
\MATHvStrich     -n, -m            %
\MATHvStrich 
                 F %
\MATHvStrich               ; 1 - z %
\MATHvStrich  (a + m)
                2 1%
\MATHvStrich  1 - a - m - n        %
\MATHvStrich         n
                   %
\MATHloEck                       %
\MATHroEck 
\MATHgroesser      T2107%
\MATHrbrace , %
\MATHlbrace %
\MATHlbrace ------------------------------------%
\MATHrbrace , 
                                (a)
                                   n
\goodbreakpoint% 
\MATHgroesser      %
\MATHlbrace FPerm[2, 1, u], T2107%
\MATHrbrace %
\MATHrbrace %
\MATHrbrace 

\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 ${}_2F_1\!\[\matrix -m,-n\\a\endmatrix; z\]$, 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 second of them is obtained by
first permuting the upper parameters by {\tt FPerm[2,1,u]} and then
applying {\tt T2101}.
\MATH
\goodbreakpoint%
In[3]:= \%1/.FPerm[2,1,u]/.T2107
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
           %
\MATHluEck                       %
\MATHruEck 
           %
\MATHvStrich     -n, -m            %
\MATHvStrich 
         F %
\MATHvStrich               ; 1 - z %
\MATHvStrich  (a + m)
        2 1%
\MATHvStrich  1 - a - m - n        %
\MATHvStrich         n
           %
\MATHloEck                       %
\MATHroEck 
Out[3]= ------------------------------------
                        (a)
                           n
\endMATH
For an example of how to iterate {\tt Tli}-rules see {\tt Tli2104}.
\Seealso Tli2104, TransListe, Gleichung.

\Name Tli2110  
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (3.2)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2110}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2112 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (5.10)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2112}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2131
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.6), reversed, $q\uparrow1$) in form 
  of a rule.
The transformation is the same as that in \hbox{\tt T2131}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2132
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.5.7)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2132}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2133 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (5.12)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2133}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.


\Name Tli2134 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex. 4.(iii), p. 97, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T2134}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2135 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex. 4.(iii), p. 97) in form of a rule.
The transformation is the same as that in \hbox{\tt T2135}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2136 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (5.10), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T2136}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2137 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (5.12), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T2137}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2138 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (3.31), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T2138}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2139 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (3.31)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2139}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2140 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (3.2), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T2140}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2141 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.4.8), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T2141}.
\Seealso Tli2107, 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, $q\to 1$; 
\cite{\SlatAC}, pp. 36/37) in form of a rule.
The transformation is the same as that in \hbox{\tt T2163}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2191
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (1.8.10)) in form of a rule.
The transformation is the same as that in \hbox{\tt T2191}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli2192 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex 3.8, $q\to 1$, 
reversed; \cite{\SlatAC}, pp. 36/37) in form of a rule.
The transformation is the same as that in \hbox{\tt T2192}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3204
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex.~7, p.~98) in form of a rule.
The transformation is the same as that in \hbox{\tt T3204}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3205
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.3.3.7)) in form of a rule.
The transformation is the same as that in \hbox{\tt T3205}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3206
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex.~7, p.~98, terminating form) in form of a rule.
The transformation is the same as that in \hbox{\tt T3206}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3207
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA} (3.1.1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T3207}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3217
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA} 4.4(2), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3217}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3231
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.21), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T3231}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3232
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.97,
reversed, first form) 
in form of a rule.
The transformation is the same as that in \hbox{\tt T3232}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3233
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.97,
reversed, second form) 
in form of a rule.
The transformation is the same as that in \hbox{\tt T3233}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3234
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.5.7), reversed) 
in form of a rule.
The transformation is the same as that in \hbox{\tt T3234}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3235 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex 3.4, $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3235}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3236 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.4.8), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T3236}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3237 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.10.4), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3237}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3238 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex. 6, p. 97) in form of a rule.
The transformation is the same as that in \hbox{\tt T3238}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3239
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex.~4.(iv), p.~97) in form of a rule.
The transformation is the same as that in \hbox{\tt T3239}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3240 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.5.10), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3240}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3261
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (III.33), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T3261}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3262
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (4.3.4.2)) in form of a rule.
The transformation is the same as that in \hbox{\tt T3262}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3263 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.33), 
$q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3263}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli3264 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.34), 
$q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3264}.
\Seealso Tli2107, 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, $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T3267}.
\Seealso Tli2107, 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, 
$q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3268}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4301
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (4.3.5.1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T4301}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4302
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.16), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T4302}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4303
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.1.1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T4303}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4304
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (4.3.6.4)) in form of a rule.
The transformation is the same as that in \hbox{\tt T4304}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4306
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.21), $q\uparrow1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T3231}.
\Seealso Tli2107, 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), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T4309}.
\Seealso Tli2107, 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), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T4310}.
\Seealso Tli2107, 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, $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T4312}.
\Seealso Tli2107, 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, $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T4313}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4331 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, Ex. 6, p. 97, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T4331}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.



\Name Tli4332 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, 4.6(1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T4332}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4362 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.4.3), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T4362}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli4391 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.5.7), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T4391}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli5401
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.3.4)) in form of a rule.
The transformation is the same as that in \hbox{\tt T5401}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli5402
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.26), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T5402}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli5403 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, 4.6(1), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T5403}.
\Seealso Tli2107, 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, $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T5468}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli6501 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.3.3)) in form of a rule.
The transformation is the same as that in \hbox{\tt T6501}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli6531
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.3.5)) in form of a rule.
The transformation is the same as that in \hbox{\tt T6531}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli6532 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex. 2.13(ii), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T6532}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli6533 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.10.4), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T6533}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli6534
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, 4.4(2)) in form of a rule.
The transformation is the same as that in \hbox{\tt T6534}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7631
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.1.1), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T7631}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7632
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.1.1), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T7632}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7633
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (4.3.6.4), reversed) in form 
  of a rule.
The transformation is the same as that in \hbox{\tt T7633}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7634
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, 7.5.(1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T7634}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7635
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, 7.5.(2)) in form of a rule.
The transformation is the same as that in \hbox{\tt T7635}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7636 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.5.10), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T7636}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7637 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex. 2.13(i), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T7637}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7691
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.4.3)) in form of a rule.
The transformation is the same as that in \hbox{\tt T7691}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7692
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (4.3.7.8)) in form of a rule.
The transformation is the same as that in \hbox{\tt T7692}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7693 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex. 2.15, $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T7693}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli7694 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (4.3.7.8), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T7694}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli8731 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (7.7), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T8731}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli8732 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (7.8), $q\to 1$, reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T8732}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9831
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.4.1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T9831}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9832
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.3.4), reversed) in form 
  of a rule.
The transformation is the same as that in \hbox{\tt T9832}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9833
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.26), $q\uparrow1$, reversed) in form 
  of a rule.
The transformation is the same as that in \hbox{\tt T9833}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9834
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.3.5), reversed) in form 
  of a rule.
The transformation is the same as that in \hbox{\tt T9834}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9835 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\BailAA}, 7.6(1)) in form of a rule.
The transformation is the same as that in \hbox{\tt T9835}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9836 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex. 3.21(iii), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T9836}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9837 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\SlatAC}, (2.4.3.3), reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T9837}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9837 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex.~8.15, $q\to1$, 
reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T9838}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9891
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Appendix (III.39), $q\uparrow1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T9891}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9892 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, (3.5.7), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T9892}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9893 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex. 2.30, $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T9893}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli9894 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\GaRaAA}, Ex. 2.25, $q\to 1$, 
reversed) in form of a rule.
The transformation is the same as that in \hbox{\tt T9894}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli111031 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (7.8), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T111031}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name Tli111032 
\Description Rule that gives a list of all possible outcomes under
application of the
transformation formula (\cite{\RaVeAA}, (7.7), $q\to 1$) in form of a rule.
The transformation is the same as that in \hbox{\tt T111032}.
\Seealso Tli2107, Tli2104, TransListe, Gleichung.

\Name TListe
\Description Rule that gives for a hypergeometric series a list of 
   applicable transformation formulas. Each entry of this list has the 
   format
\hbox{\tt $\{$ArgumentPermutations,T$\langle$number$\rangle$$\}$}, 
where \hbox{\tt ArgumentPermutations} 
   is a sequence of reorderings of the parameters of the hypergeometric
   series (given in terms of \hbox{\tt FPerm} and \hbox{\tt
FTausche}) and \hbox{\tt T$\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 FOrdne} 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]:= F[%
\MATHlbrace a,b%
\MATHrbrace ,%
\MATHlbrace c%
\MATHrbrace ,z]
\goodbreakpoint%
           %
\MATHluEck          %
\MATHruEck 
           %
\MATHvStrich  a, b    %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich      ; z %
\MATHvStrich 
        2 1%
\MATHvStrich   c      %
\MATHvStrich 
           %
\MATHloEck          %
\MATHroEck 
\goodbreakpoint%
In[2]:= \%/.TListe
Is -a a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -b a nonnegative integer?
[y|n]:  n%
\goodbreakpoint%
Be sure to apply "FOrdne" before using the following information!
\goodbreakpoint%
Out[2]= %
\MATHlbrace %
\MATHlbrace T2103%
\MATHrbrace , %
\MATHlbrace T2104%
\MATHrbrace , %
\MATHlbrace T2191%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[3]:= F[%
\MATHlbrace -n,2*a,2*b%
\MATHrbrace ,%
\MATHlbrace -2*n,1/2+a+b%
\MATHrbrace ,1]
\goodbreakpoint%
           %
\MATHluEck   -n, 2 a, 2 b      %
\MATHruEck 
           %
\MATHvStrich                     %
\MATHvStrich 
Out[3]=  F %
\MATHvStrich        1        ; 1 %
\MATHvStrich 
        3 2%
\MATHvStrich  -2 n, - + a + b    %
\MATHvStrich 
           %
\MATHloEck        2            %
\MATHroEck 
\goodbreakpoint%
In[4]:= \%/.TListe%
\par\penalty-5000\leavevmode%
Is -2*b a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -2*a a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is n a nonnegative integer?
[y|n]:  y%
\goodbreakpoint%
Be sure to apply "FOrdne" before using the following information!
\goodbreakpoint%
Out[4]= %
\MATHlbrace %
\MATHlbrace T3204%
\MATHrbrace , %
\MATHlbrace T3205%
\MATHrbrace , %
\MATHlbrace T3206%
\MATHrbrace , %
\MATHlbrace T3207%
\MATHrbrace , %
\MATHlbrace T3217%
\MATHrbrace , %
\MATHlbrace T3261%
\MATHrbrace , %
\MATHlbrace T3262%
\MATHrbrace , 
 
\MATHgroesser     %
\MATHlbrace FPerm[2,3,1,u], FTausche[1,2,l], T3231%
\MATHrbrace %
\MATHrbrace 
\goodbreakpoint%
In[5]:= \%3/.FOrdne/.FPerm[2,3,1,u]/.FTausche[1,2,l]/.T3231%
\par\penalty-5000\leavevmode%
Is -2*a a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -2*b a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is n a nonnegative integer?
[y|n]:  y
\goodbreakpoint%
           %
\MATHluEck       a, b, -n, -n         %
\MATHruEck 
           %
\MATHvStrich                            %
\MATHvStrich 
Out[5]=  F %
\MATHvStrich  1              1 - 2 n; 1 %
\MATHvStrich 
        4 3%
\MATHvStrich  - + a + b, -n, -------    %
\MATHvStrich 
           %
\MATHloEck  2                 2       %
\MATHroEck 
\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 Whipple's transformation, the second by a number
of specialized very well-poised $_7\phi_6$ transformations.
These facts are also observed by using this package.
\vskip10pt
\MATH
In[6]:= F[%
\MATHlbrace a,1+a/2,b,c,d,e%
\MATHrbrace ,%
\MATHlbrace a/2,1+a-b,1+a-c,1+a-d,1+a-e%
\MATHrbrace ,-1]
\goodbreakpoint%
                                a
           %
\MATHluEck              a, 1 + -, b, c, d, e                  %
\MATHruEck 
           %
\MATHvStrich                     2                              %
\MATHvStrich 
Out[6]=  F %
\MATHvStrich                                               ; -1 %
\MATHvStrich 
        6 5%
\MATHvStrich  a                                                 %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e     %
\MATHroEck 
             2
\goodbreakpoint%
In[7]:= \%/.TListe
\goodbreakpoint%
Out[7]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[8]:= Tgl7632
Do you want to set values for the equation? [y|n]: n
\goodbreakpoint%
                                    a
           %
\MATHluEck                  a, 1 + -, b, c, d, e, -n                    %
\MATHruEck 
           %
\MATHvStrich                         2                                    %
\MATHvStrich 
Out[8]=  F %
\MATHvStrich                                                          ; 1 %
\MATHvStrich  == 
        7 6%
\MATHvStrich  a                                                           %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n    %
\MATHroEck 
             2
\goodbreakpoint% 
        %
\MATHluEck                                          %
\MATHruEck  (1 + a, 1 + a - d - e)
        %
\MATHvStrich        1 + a - b - c, d, e, -n           %
\MATHvStrich                        n
\MATHgroesser      F %
\MATHvStrich                                      ; 1 %
\MATHvStrich  -----------------------
     4 3%
\MATHvStrich  1 + a - b, 1 + a - c, -a + d + e - n    %
\MATHvStrich  (1 + a - d, 1 + a - e)
        %
\MATHloEck                                          %
\MATHroEck                        n
\goodbreakpoint%
In[9]:= Limes[\%,n-\MATHgroesser Infinity]
\goodbreakpoint%
                                a
           %
\MATHluEck              a, 1 + -, b, c, d, e                  %
\MATHruEck 
           %
\MATHvStrich                     2                              %
\MATHvStrich 
Out[9]=  F %
\MATHvStrich                                               ; -1 %
\MATHvStrich  == 
        6 5%
\MATHvStrich  a                                                 %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e     %
\MATHroEck 
             2
\goodbreakpoint% 
        %
\MATHluEck                          %
\MATHruEck 
        %
\MATHvStrich  1 + a - b - c, d, e     %
\MATHvStrich 
      F %
\MATHvStrich                      ; 1 %
\MATHvStrich  \MATHGamma (1 + a - d) \MATHGamma (1 + a - e)
     3 2%
\MATHvStrich  1 + a - b, 1 + a - c    %
\MATHvStrich 
        %
\MATHloEck                          %
\MATHroEck 
\MATHgroesser     --------------------------------------------------------
                    \MATHGamma (1 + a) \MATHGamma (1 + a - d - e)
\goodbreakpoint%
In[10]:= F[%
\MATHlbrace 1+a/2,1,c,d,e,f%
\MATHrbrace ,%
\MATHlbrace a/2,1+a-c,1+a-d,1+a-e,1+a-f%
\MATHrbrace ,1]
\goodbreakpoint%
                              a
            %
\MATHluEck              1 + -, 1, c, d, e, f                 %
\MATHruEck 
            %
\MATHvStrich                  2                                %
\MATHvStrich 
Out[10]=  F %
\MATHvStrich                                               ; 1 %
\MATHvStrich 
         6 5%
\MATHvStrich  a                                                %
\MATHvStrich 
            %
\MATHloEck  -, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a - f    %
\MATHroEck 
              2
\goodbreakpoint%
In[11]:= \%/.TListe
\goodbreakpoint%
Out[11]= %
\MATHlbrace %
\MATHrbrace 
\goodbreakpoint%
In[12]:= \%\%/.FEinf
Add the parameter: a
\goodbreakpoint%
                                 a
            %
\MATHluEck              a, 1 + -, 1, c, d, e, f                 %
\MATHruEck 
            %
\MATHvStrich                     2                                %
\MATHvStrich 
Out[12]=  F %
\MATHvStrich                                                  ; 1 %
\MATHvStrich 
         7 6%
\MATHvStrich     a                                                %
\MATHvStrich 
            %
\MATHloEck  a, -, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a - f    %
\MATHroEck 
                 2
\goodbreakpoint%
In[13]:= \%/.TListe%
\par\penalty-5000\leavevmode%
Is -f a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -1 - a + e + f a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -1 - a + d + f a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -1 - a + d + e a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -1 - a + c + f a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -1 - a + c + e a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -1 - a + c + d a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -a + f a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -a + e a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -a + d a nonnegative integer?
[y|n]:  n%
\par\penalty-5000\leavevmode%
Is -a + c a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
Be sure to apply "FOrdne" before using the following information!
\goodbreakpoint%
Out[13]= %
\MATHlbrace %
\MATHlbrace T7631%
\MATHrbrace , %
\MATHlbrace T7634%
\MATHrbrace , %
\MATHlbrace T7635%
\MATHrbrace , %
\MATHlbrace T7691%
\MATHrbrace , %
\MATHlbrace T7692%
\MATHrbrace %
\MATHrbrace 
\endMATH
\Seealso SListe, FPerm, FTausche, TransListe.

\Name trans
\Description $(a)_n \to (-1)^n\,(1-n-a)_n$.
\Usage Expr/.trans.
\Example 
\MATH
In[1]:= p[a,n]
\goodbreakpoint%
Out[1]= (a)
           n
\goodbreakpoint%
In[2]:= \%/.trans
\goodbreakpoint%
            n
Out[2]= (-1)  (1 - a - n)
                         n
\endMATH 
\Seealso Ers, PosListe, ManipulationsListe.

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

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


\Name V
\Description \hbox{\tt V[a,List,z]} is the very well-poised  hypergeometric
series.
\Usage V[a,List,z].
\Example 
\MATH
In[1]:= V[a,%
\MATHlbrace b,c%
\MATHrbrace ,z]
\goodbreakpoint%
                        a
           %
\MATHluEck      a, 1 + -, b, c         %
\MATHruEck 
           %
\MATHvStrich             2               %
\MATHvStrich 
Out[1]=  F %
\MATHvStrich                         ; z %
\MATHvStrich 
        4 3%
\MATHvStrich  a                          %
\MATHvStrich 
           %
\MATHloEck  -, 1 + a - b, 1 + a - c    %
\MATHroEck 
             2
\endMATH 
\Seealso F, TeXFV, P, FFormat.

\Name ZB
\Description Rule that looks for a recurrence relation for a sum or 
  hypergeometric series using Zeilberger's algorithm \cite{\ZeilAM,
\ZeilAN, \ZeilAV}. 

Here a call is made to the function {\tt Zb} of the {\sl Mathematica} 
implementation of Gosper's and Zeilberger's algorithms written by Peter Paule and
Markus Schorn. The current version~1.1 or updates can be received via
e-mail request to \hbox{\tt peter.paule\@risc.uni-linz.ac.at}. This
implemenation provides the user with the objects
\medskip\noindent
{\leftskip20pt\rightskip20pt
\hbox{\tt Zb}, \hbox{\tt Gosper}, \hbox{\tt RunMode}, \hbox{\tt FileName}, 
\hbox{\tt SolAmount},
\hbox{\tt Fnk}, \hbox{\tt GoRat}, \hbox{\tt GoSol}, \hbox{\tt Cert}, 
  \hbox{\tt DegBound}, \hbox{\tt System}, \hbox{\tt SystemDimension}.
\par}
\medskip\noindent
Also within the package HYP,
all these objects work as described in the documentation of this
implementation. Therefore the user is referred to this documentation
and the description
\cite{\PaScAA} in order to learn about the
various features of these objects. 

The package HYP provides two additional objects, {\tt ZB} and
\hbox{\tt GOSPER}. The rule {\tt ZB} allows to apply Zeilberger's algorithm
directly to an expression containing a {\tt SUM} or a hypergeometric
series.
\Usage Expr/.ZB[recvar, order].
\Example
\MATH
In[1]:= SUM[Binomial[N,k]*Binomial[M,L-k],%
\MATHlbrace k,0,Infinity%
\MATHrbrace ]
\goodbreakpoint%
          \MATHinfty 
        %
\MATHluEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHruEck (        ) (   )
         \MATHbackslash    (   M    ) ( N )
Out[1]=   \MATHgroesser   (        ) (   )
         /   ( -k + L ) ( k )
        %
\MATHloEck %
\MATHhStrich %
\MATHhStrich %
\MATHhStrich %
\MATHroEck (        ) (   )
         k=0
\goodbreakpoint%
In[2]:= \%/.ZB[L,1]
 
    Peter Paule and Markus Schorn's implementation of the
             Zeilberger algorithm. (Version 1.1)
\goodbreakpoint%
Out[2]= %
\MATHlbrace (-L + M + N) SUM[L] + (-1 - L) SUM[1 + L] == 0%
\MATHrbrace 
\goodbreakpoint%
In[3]:= \%1/.SUMF
\goodbreakpoint%
           %
\MATHluEck               %
\MATHruEck 
           %
\MATHvStrich   -L, -N      %
\MATHvStrich 
         F %
\MATHvStrich           ; 1 %
\MATHvStrich  (1 - L + M)
        2 1%
\MATHvStrich  1 - L + M    %
\MATHvStrich             L
           %
\MATHloEck               %
\MATHroEck 
Out[3]= --------------------------------
                      (1)
                         L
\goodbreakpoint%
In[4]:= \%/.ZB[L,1]
 
    Peter Paule and Markus Schorn's implementation of the
             Zeilberger algorithm. (Version 1.1)
Is N a nonnegative integer?
[y|n]:  n
\goodbreakpoint%
Out[4]= %
\MATHlbrace (-L + M + N) SUM[L] + (-1 - L) SUM[1 + L] == 0%
\MATHrbrace 
\endMATH
\Seealso GOSPER.

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

\Name zus1
\Description \vtab $(a)_n(a+n)_m \to (a)_{n+m}$,\\
           $(a)_n/\Gamma(a+n) \to 1/\Gamma(a)$.
\endvtab
\Usage Expr/.zus1.
\Example 
\MATH
In[1]:= p[a,2*n]*p[a+2*n,m-n]
\goodbreakpoint%
Out[1]= (a)    (a + 2 n)
           2 n          m - n
\goodbreakpoint%
In[2]:= \%/.zus1
\goodbreakpoint%
Out[2]= (a)
           m + n
\goodbreakpoint%
In[3]:= p[a,2*n]/GAMMA[a+2*n]
\goodbreakpoint%
          (a)
             2 n
Out[3]= ----------
        \MATHGamma (a + 2 n)
\goodbreakpoint%
In[4]:= \%/.zus1
\goodbreakpoint%
         1
Out[4]= ----
        \MATHGamma (a)
\endMATH 
\Seealso zus2, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

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

\Name zus3
\Description $(a)_n/(b)_m \to (a)_{n-m}$,
\vskip6pt
\leavevmode\hskip10pt  provided $a+n=b+m$, and
\vskip6pt
\leavevmode\hphantom{Description: }$\Gamma(a+n)/\Gamma(a) \to
(a)_{n}$.
\Usage Expr/.zus3.
\Example 
\MATH
In[1]:= p[a+m,n]/p[a+n,m]
\goodbreakpoint%
        (a + m)
               n
Out[1]= --------
        (a + n)
               m
\goodbreakpoint%
In[2]:= \%/.zus3
\goodbreakpoint%
Out[2]= (a + m)
               -m + n
\goodbreakpoint%
In[3]:= GAMMA[a]/GAMMA[a+n]
\goodbreakpoint%
          \MATHGamma (a)
Out[3]= --------
        \MATHGamma (a + n)
\goodbreakpoint%
In[4]:= \%/.zus3
\goodbreakpoint%
Out[4]= (a + n)
               -n
\goodbreakpoint%
In[5]:= \%/.neg1
\goodbreakpoint%
         1
Out[5]= ----
        (a)
           n
\endMATH 
\Seealso zus1, zus2, erw1, erw2, Ers, PosListe, ManipulationsListe.

\Refs
\ref\no \BailAA\by W. N. Bailey \yr 1935 
\book Generalized hypergeometric series
\publ Cambridge University Press
\publaddr Cambridge\endref

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

\ref\no \GospAB\by R. W. Gosper \yr 1978 \paper Decision procedure
for indefinite hypergeometric summation\jour Proc\. Nat\. Acad\.
Sci\. USA\vol 75\pages 40--42\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 \PaScAA\by P. Paule and M. Schorn\paper A {\sl Mathematica\/}
version of Zeilberger's algorithm for proving binomial coefficient
identities\jour J. Symbolic Comput\.
\toappear \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 \SlatAC\by L. J. Slater \yr 1966 
\book Generalized hypergeometric functions
\publ Cambridge University Press
\publaddr Cambridge\endref

\ref\no \VeJaAH\by A.    Verma and V. K. Jain \yr 1985 \paper Some 
summation formulae for nonterminating basic hypergeometric series\jour 
SIAM J. Math\. Anal\.\vol 16\pages 647--655\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

\ref\no \ZeilAM\by D.    Zeilberger \yr 1990 \paper A fast algorithm for 
proving terminating hypergeometric identities\jour 
Discrete Math\.\vol 80\pages 207--211\endref

\ref\no \ZeilAN\by D.    Zeilberger \yr 1990 \paper A holonomic systems 
approach to special functiions identities\jour J. 
Comput\. Appl\. Math\.\vol 32\pages 321--368\endref

\ref\no \ZeilAV\by D.    Zeilberger \yr 1991 \paper The method of 
creative telescoping\jour J. Symbolic Comput\.\vol 11\pages 195--204\endref

\endRefs
\enddocument


