HYPQ

C. Krattenthaler

Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. e-mail: KRATT@AP.UNIVIE.AC.AT

                         WWW: http://www.mat.univie.ac.at/People/kratt

This is a MATHEMATICA package for handling basic hypergeometric series. It provides quite a few tools for (A) manipulating q-factorial expressions (B) transforming q-binomial sums into basic hypergeometric notation (C) summing basic hypergeometric series (D) transforming basic hypergeometric series (E) applying contiguous relations (F) doing formal limits of basic hypergeometric expressions (G) transforming basic hypergeometric MATHEMATICA expressions into TEX-code.

The tools for items (A), (B), (F), (G) are contained in the file hyp.q, the basic package. This file must be loaded at the very beginning of your MATHEMATICA session. (Ignore error messages occuring when loading hyp.q.) The file hyp.q defines the basic objects, the rules and functions for items (A), (B), (F), and (G), and predefines all the remaining ones. The tools for (C) are the contents of the file summatio.q, those for (D) are the contents of the files transfor.q and transfor.qli, and those for (E) are the contents of the file contig.q. You also have access to summation and transformation formulas in form of equations. This is the contents of the files summatio.qgl and transfor.qgl, respectively. The file output defines some nice MATHEMATICA output features for SUM, Product,

Integrate, Abs, Floor, Ceiling, Pi, and Infinity.

However, the philosophy of this package is:

Do it by yourself!

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

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

For a brief summary of the main features of this package the user is referred to [2] which is the contents of the AMS-TEX file hyp_{–}hypq.tex.

Basic hypergeometric notation

All the notation and terminology is adopted from [1, pp. 1–6]. Given a (fixed) complex number q with |q| < 1, the basic hypergeometric series is defined by

rϕ_{s}[ \begin{matrix}a_{1}, …, a_{r} \\ b_{1}, …, b_{s} \end{matrix}; q, z] = ∑_{n=0}^{∞} \frac{(a_{1}; q)_{n} ⋯ (a_{r}; q)_{n}}{(q; q)_{n}(b_{1}; q)_{n} ⋯ (b_{s}; q)_{n}} ((−1)^{n}q^{\binom{n}{2}})^{s−r+1} z^{n} ,

where the rising q-factorial (a; q)_{n} is given by (a; q)_{n} := (1 − a)(1 − aq) ⋯ (1 − aq^{n−1}), n ≥ 1, (a)_{0} := 1. Also the infinite q-factorial (a; q)_{∞} := ∏_{i=0}^{∞}(1 − aq^{i}) is used. A basic hypergeometric series _{r+1}ϕ_{r} is called very well-poised if a_{i}b_{i} = qa_{0} for i = 1, 2, …, r, and among the parameters a_{i} occur both q\sqrt{a_{0}} and −q\sqrt{a_{0}}. We use the standard abbreviation for very well-poised basic hypergeometric series,

r+1W_{r}(a_{0}; a_{3}, a_{4}, …, a_{r}; z) := _{r+1}ϕ_{r}[ \begin{matrix}a_{0}, q\sqrt{a_{0}}, −q\sqrt{a_{0}}, a_{3}, a_{4}, …, a_{r} \\ \sqrt{a_{0}}, −\sqrt{a_{0}}, qa_{0}/a_{2}, qa_{0}/a_{3}, …, qa_{0}/a_{r} \end{matrix}; z] .

A convenient notation for basic hypergeometric series with several bases is

rΦ_{s}[ \begin{matrix}a_{1}^{(1)}, …, a_{r_{1}}^{(1)}; …; a_{1}^{(p)}, …, a_{r_{p}}^{(p)} \\ b_{1}^{(1)}, …, b_{s_{1}}^{(1)}; …; b_{1}^{(p)}, …, b_{s_{p}}^{(p)} \end{matrix} ; q_{1}, …, q_{p}; z]

= ∑_{n=0}^{∞} \frac{z^{n}}{(q_{1}; q_{1})_{n}} (−1)^{n}q\binom{n}{2}_{1} ∏_{i=1}^{p} \frac{(a_{1}^{(i)}; q_{i})_{n} ⋯ (a_{r_{i}}^{(i)}; q_{i})_{n}}{(b_{1}^{(i)}; q_{i})_{n} ⋯ (b_{s_{i}}^{(i)}; q_{i})_{n}} ((−1)^{n}q^{\binom{n}{2}})^{s_{i}−r_{i}} .

The bilateral basic hypergeometric series is defined by

rψ_{s}[ \begin{matrix}a_{1}, …, a_{r} \\ b_{1}, …, b_{s} \end{matrix}; q, z] = ∑_{n=−∞}^{∞} \frac{(a_{1}; q)_{n} ⋯ (a_{r}; q)_{n}}{(b_{1}; q)_{n} ⋯ (b_{s}; q)_{n}} ((−1)^{n}q^{\binom{n}{2}})^{s−r+1} z^{n} ,

We also use the compact Gasper-Rahman notation

(a_{1}, a_{2}, …, a_{r}; q)_{n} := (a_{1}; q)_{n} (a_{2}; q)_{n} ⋯ (a_{r}; q)_{n}.

The file hyp.q

The objects which are defined in the file hyp.q are

AbsGreater, AbsSmaller, AbsUndetermined, Add, AmSLaTeX, AmSTeX, baszerl1, baszerl2, baszus1,
baszus2, Binomialpq, Binomialq, Div, Drucke, Ers, erw1, erw2, Expandq, Factorialpq, Factorialq,
Gleichung, GlTausche, Hoch, hypqAttributes, inv1, inv2, LaTeX, Limes, lina1, lina2, linz, LS, Mal,
ManipulationsListe, MinusOne, Multinomialpq, Multinomialq, neg1, neg2, ph, Ph, phCancel, PhEinf,
phEinf, phFormat, Phinv, phinv, PhOrdne, phOrdne, PhPerm, phPerm, Phph, phPh, phps, PhSUM, phSUM,
phTausche, PosListe, pq, PQ, pqaufl, pqinf, pqinfzerl, pqinfzus, PQSort, pqzerl, pqzus, ps, psEinf,
psinv, psOrdne, psPerm, psph, psShift, psSUM, RS, SchreibeZahl, SimplifyPQ, Sub, Subst, SUM, SUMErw1,
SUMErw2, SUMExpand, SUMInfinity, SUMph, SUMPh, SUMps, SUMRegeln, SUMSammle, SUMShift, SUMTausche,
SUMUmkehr, SUMZerl, TeX, TeXMat, TeXphW, trans, W, zerl, zus1, zus2, zus3.

These objects can be divided into 9 groups: There are the basic objects,

Binomialpq, Binomialq, Factorialpq, Factorialq, Multinomialpq, Multinomialq, ph, Ph, pq, pqinf, ps,
SUM, W,

the rules for manipulating q-factorial expressions

baszerl1, baszerl2, baszus1, baszus2, erw1, erw2, Expandq, inv1, inv2, lina1, lina2, linz, MinusOne,
neg1, neg2, pqaufl, pqinfzerl, pqinfzus, pqzerl, pqzus, trans, zerl, zus1, zus2, zus3,

the rules for manipulating sums and basic hypergeometric series,

The MATHEMATICA package HYPQ 3

    phCancel, phEinf, PhEinf, phFormat, Phinv, phinv, PhOrdne, phOrdne, PhPerm, phPerm, Phph, phPh, phps,
    PhSUM, phSUM, phTausche, ps, psEinf, psinv, psOrdne, psPerm, psph, psShift, psSUM, SUMErw1, SUMErw2,
    SUMExpand, SUMInfinity, SUMph, SUMPh, SUMps, SUMRegeln, SUMSammle, SUMShift, SUMTausche, SUMUmkehr,
    SUMZerl,

two functions for controlled use of rules,

    Ers, Posliste,

one function for substitution of an expression instead of another expression,

    Subst,

some objects for doing limits of basic hypergeometric expressions,

    AbsGreater, AbsSmaller, AbsUndetermined, Limes,

one object for simplifying arguments in basic hypergeometric expressions,

    SimplifyPQ,

some objects for converting expressions into TEX-code,

    AmSLaTeX, AmSTeX, LaTeX, TeX, TeXMat, TeXphW,

two objects for on-line help,

    hypqAttributes, ManipulationsListe,

and the function

    Drucke

which enables you to directly send an expression to the printer. Finally there are

    Add, Div, Gleichung, GlTausche, Hoch, LS, Mal, PQSort, RS, Sub

for manipulating equations and writing expressions in a “normalized” form (PQSort) in order to be able to quickly check if two expressions agree. These objects are particularly important when using objects from summatio.qgl and

transfor.qgl.

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 Ers and PosListe.

                                The file summatio.q

This file contains most of the summation formulas of the book [1], including the (almost) complete Appendix II of [1], 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 summatio.q are

    S0110, S1001, S1101, S1102, S1110, S2101, S2102, S2103, S2104, S2105, S2106, S2107, S2161, S2201,
    S2202, S2210, S3201, S3202, S3203, S3204, S3261, S3310, S4301, S4302, S4303, S4304, S4305, S4306,
    S4307, S4308, S4361, S4410, S5401, S5402, S5501, S6501, S6502, S6610, S8701, S8702, S8703, S8704,
    S8761, S10901, SListe, SumListe.

The numbering of each rule is S⟨d1⟩⟨d2⟩⟨n1⟩ following the following system: The number ⟨d1⟩ is the number of the upper parameters, the number ⟨d2⟩ is the number of the lower parameters of the basic hypergeometric series to which the rule applies. The number ⟨n1⟩ allows to distinguish the rules applying to basic hypergeometric series with equal numbers of upper and lower parameters. ⟨n1⟩ is within the range 01–30 if the summation is a one-term summation, it is within the range 61–90 if the summation is a two- or more-term summation.

For terminating series there is a check if one of the parameters is of the form q^{−n} where n is a nonnegative integer. Depending on your input you might be asked if some expression is a nonnegative integer (see the examples for S3201). Be sure to give an affirmative answer only for 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. SUMUmkehr, phSUM, or in case that automatic evaluating is active (cf. PQ). The rule SListe enables you to quickly check if one of the summation rules S0110–S8761 can be directly applied.

                            The file transfor.q

This file contains most of the transformation formulas of the book [1], including the (almost) complete Appendix III of [1], 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 transfor.q are

T2101, T2102, T2103, T2104, T2105, T2106, T2107, T2108, T2109, T2110, T2111, T2112, T2161, T2162, T2163,
T2201, T2202, T3101, T3201, T3202, T3203, T3204, T3205, T3206, T3207, T3208, T3209, T3210, T3211, T3212,
T3213, T3214, T3215, T3216, T3217, T3261, T3262, T3263, T3264, T3265, T3266, T3267, T3268, T3269, T4301,
T4302, T4303, T4304, T4305, T4306, T4307, T4308, T4309, T4310, T4311, T4312, T4313, T4361, T4362, T5401,
T5402, T5403, T5404, T5405, T5461, T5462, T5463, T5464, T5465, T5466, T5467, T5468, T5469, T6501, T7601,
T7701, T8701, T8702, T8703, T8704, T8705, T8706, T8707, T8708, T8709, T8710, T8711, T8761, T8762, T8763,
T8764, T8810, T10901, T10902, T10903, T10904, T10905, T10906, T10907, T10961, T10962, T10963, T101010,
T121101, T121102, T121103, T121104, T121105, T121106, T121107, T121161, TListe, TransListe.

The comments for the file summatio.q regarding the numbering of the rules and optional questions for input also apply here. The rule TListe enables you to quickly check if one of the transformation rules T2101–T10961 can be directly applied.

                           The file transfor.qli

Each of the objects of this file corresponds to a transformation rule of the file transfor.q. Each object gives a list of all the outcomes under application of a particular transformation after before having permuted the upper and lower parameters of the involved basic hypergeometric series. All the objects in this file are rules. These rules help to prove conjectured transformation formulas quickly. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by transfor.qli are

Tli2101, Tli2102, Tli2103, Tli2104, Tli2105, Tli2106, Tli2107, Tli2108, Tli2109, Tli2110, Tli2111,
Tli2112, Tli2161, Tli2162, Tli2163, Tli2201, Tli2202, Tli3101, Tli3201, Tli3202, Tli3203, Tli3204,
Tli3205, Tli3206, Tli3207, Tli3208, Tli3209, Tli3210, Tli3211, Tli3212, Tli3213, Tli3214, Tli3215,
Tli3216, Tli3261, Tli3262, Tli3263, Tli3264, Tli3265, Tli3266, Tli3267, Tli3268, Tli3269, Tli4301,
Tli4302, Tli4303, Tli4304, Tli4305, Tli4306, Tli4307, Tli4308, Tli4309, Tli4310, Tli4311, Tli4312,
Tli4313, Tli4361, Tli4362, Tli5401, Tli5402, Tli5403, Tli5404, Tli5405, Tli5461, Tli5462, Tli5463,
Tli5464, Tli5465, Tli5466, Tli5467, Tli5468, Tli5469, Tli6501, Tli7601, Tli8701, Tli8702, Tli8703,
Tli8704, Tli8705, Tli8706, Tli8707, Tli8708, Tli8709, Tli8710, Tli8711, Tli8761, Tli8762, Tli8763,
Tli8764, Tli8810, Tli10901, Tli10902, Tli10903, Tli10904, Tli10905, Tli10906, Tli10907, Tli10961,
Tli10962, Tli10963, Tli101010, Tli121101, Tli121102, Tli121103, Tli121104, Tli121105, Tli121106,
Tli121107, Tli121161.

                   The files summatio.qgl and transfor.qgl

These files contain the same summations, respectively transformations, as summatio.q, respectively transfor.q, but in form of equations. You do not have to load these files by hand since they are loaded automatically once an object of this file is called. The respective objects are

Sgl0110, Sgl1001, Sgl1101, Sgl1102, Sgl1110, Sgl2101, Sgl2102, Sgl2103, Sgl2104, Sgl2105, Sgl2106,
Sgl2107, Sgl2161, Sgl2201, Sgl2202, Sgl2210, Sgl3201, Sgl3202, Sgl3203, Sgl3204, Sgl3261, Sgl3310,

    Sgl4301, Sgl4302, Sgl4303, Sgl4304, Sgl4305, Sgl4306, Sgl4307, Sgl4308, Sgl4361, Sgl4410, Sgl5401,
    Sgl5402, Sgl5501, Sgl6501, Sgl6502, Sgl6610, Sgl8701, Sgl8702, Sgl8703, Sgl8704, Sgl8761, Sgl10901,
    SumListe$gl,

and

    Tgl2101, Tgl2102, Tgl2103, Tgl2104, Tgl2105, Tgl2106, Tgl2107, Tgl2108, Tgl2109, Tgl2110, Tgl2111,
    Tgl2112, Tgl2161, Tgl2162, Tgl2163, Tgl2201, Tgl2202, Tgl3101, Tgl3201, Tgl3202, Tgl3203, Tgl3204,
    Tgl3205, Tgl3206, Tgl3207, Tgl3208, Tgl3209, Tgl3210, Tgl3211, Tgl3212, Tgl3213, Tgl3214, Tgl3215,
    Tgl3216, Tgl3261, Tgl3262, Tgl3263, Tgl3264, Tgl3265, Tgl3266, Tgl3267, Tgl3268, Tgl3269, Tgl4301,
    Tgl4302, Tgl4303, Tgl4304, Tgl4305, Tgl4306, Tgl4307, Tgl4308, Tgl4309, Tgl4310, Tgl4311, Tgl4312,
    Tgl4313, Tgl4361, Tgl4362, Tgl5401, Tgl5402, Tgl5403, Tgl5404, Tgl5405, Tgl5461, Tgl5462, Tgl5463,
    Tgl5464, Tgl5465, Tgl5466, Tgl5467, Tgl5468, Tgl5469, Tgl6501, Tgl7601, Tgl8701, Tgl8702, Tgl8703,
    Tgl8704, Tgl8705, Tgl8706, Tgl8707, Tgl8708, Tgl8709, Tgl8710, Tgl8711, Tgl8761, Tgl8762, Tgl8763,
    Tgl8764, Tgl8810, Tgl10901, Tgl10902, Tgl10903, Tgl10904, Tgl10905, Tgl10906, Tgl10907, Tgl10961,
    Tgl10962, Tgl10963, Tgl101010, Tgl121101, Tgl121102, Tgl121103, Tgl121104, Tgl121105, Tgl121106,
    Tgl121107, Tgl121161, TransListe$gl.

When calling one of these objects you will be put two questions. If the variables of the called summation or transformation are undefined, the first question is

Do you want to set values for the equation?  [y|n]:

Enter y if you want to set values, even only for some of them, if you do not need to set values enter n. If some of the variables of the called summation or transformation are already defined, you will be asked as first question

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]:

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 nv.

The second question concerns the base q. Either it is

Do you want to set a value for q in the equation?  [y|n]:

or

q has a value.  Should q be cleared?
Do you want to set a value for q in the equation (v)?  [y|n|yv|nv]:

The explanations for the first question also apply here. (Cf. the examples in Sgl2101).

In addition there are the functions and variables

    Add, Div, Gleichung, GlTausche, Hoch, LS, Mal, RS, Sub,

for manipulating equations. In fact, once you have called one of the objects Sgl∗ or Tgl∗, the right-hand side of the displayed equation will have been assigned to RS, the left-hand side to LS, and thus the equation itself to Gleichung (cf. the example in Gleichung). The functions Add, Div, GlTausche, Hoch, Mal, Sub, and also Ers, allow you to manipulate the equation.

                                 The file contig.q

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 contig.q are

    C01, C02, C11, C12, C13, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30,
    C31, C32, C33, C34, C35, C36, C37, C38, C39, C40, C41, C42, C43, C44, C45, C46, C47, C48, C49, C50, C51, C52,
    C53, C54, C55, C56, C57, C58, C59, C60, C61, C62, C63, C64, C65, C66, C67, C68, C69, C70, C71, C72, C73, C74,
    C75, C76, C77, C78, C79, C80, C81, C82, C83, C84, C85, C86, C87, C88, C89, C90, C91, C92, C93, C94, C95, C96,
    C97, C98, C99, C100, C101, C102, C103, C104, C105, C106, C107, C108, C109, C110, C111, C112, C113, C114,
    C115, C116, C117, C118, C119, C120, C121, ContigListe.

Simultaneous use of HYP and HYPQ

It is possible to load both packages, HYP and HYPQ. In this case, the objects of the package that is loaded last will override the respective objects of the other package which have identical names. However, you can use the overrided objects by calling them by their full names. To determine the full name of an object the following rule applies:

If the object Object is defined in the file File.ext, then the full name of Object is File`ext`Object.

For instance, if you load hyp.q first and then hyp.m and want to use Limes with a basic hypergeometric expression, then you have to type Hyp`q`Limes instead of Limes. (Calling Limes would invoke the ordinary hypergeometric Limes.) For information on contexts in MATHEMATICA confer [4].

On-line help

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

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

The screen output

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

[            ]
| a, b       |
Out[1]=  ph |     ; q, z |
2  1|  c         |
[            ]
In[2]:= ph[{a,b,c},{d,e,0},q,z]
[               ]
| a, b, c       |
Out[2]=  ph |        ; q, z |
3  3| d, e, 0       |
[               ]

8 The MATHEMATICA package HYPQ

A brief dictionary

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.

A German–English vocabulary

German  English  Mathematica objects in HYPQ

containing the word

abspalten  split  lina1, lina2
auflösen  dissolve  pqaufl
drucken  print  Drucke
einfügen  insert  phEinf
ersetzen  replace  Ers

erweitern  extend  erw1, erw2, SUMErw1, SUMErw2

Gleichung  equation  Gleichung
”2 hoch 3”  ”2 to the 3”  Hoch
”2 mal 3”  ”2 times 3”  Mal
ordnen  order  phOrdne

Regel  rule  SUMRegeln

sammeln  collect  SUMSammle
schreiben  write  SchreibeZahl
tauschen  exchange, interchange SUMTausche, GlTausche
umkehren  reverse  SUMUmkehr

Zahl  number  SchreibeZahl
zerlegen  split  zerl1, pqzerl, pqinfzerl, SUMZerl

zusammenfassen put together  zus1, pqzus, pqinfzus

The MATHEMATICA package HYPQ 9

An English–German vocabulary

English  German  Mathematica objects in HYPQ

containing the word

collect  sammeln  SUMSammle
dissolve  auflösen  pqaufl
equation  Gleichung  Gleichung

exchange  tauschen  SUMTausche, GlTausche
extend  erweitern  erw1, erw2, SUMErw1, SUMErw2

insert  einfügen  phEinf

interchange tauschen  SUMTausche, GlTausche
number  Zahl  SchreibeZahl

order  ordnen  phOrdne
print  drucken  Drucke
replace  ersetzen  Ers
reverse  umkehren  SUMUmkehr

rule  Regel  SUMRegeln

split  abspalten  lina1, lina2

split  zerlegen  zerl1, pqzerl, pqinfzerl, SUMZerl

”2 times 3” ”2 mal 3”  Mal
”2 to the 3” ”2 hoch 3”  Hoch
put together zusammenfassen zus1, pqzus, pqinfzus
write  schreiben  SchreibeZahl

Alphabetic List of the objects with desriptions

AbsGreater

\underline{Description}: Function for declaring the absolute value of a variable or expression to be greater than 1. This declaration is used by Limes. By default the absolute value of q is defined to be smaller than 1.

\underline{Usage}: AbsGreater[Expr].

\underline{Example(s)}:

In[1]:= Limes[pq[a,n,p],n-⟩Infinity]
Is  p  smaller than 1?
[y|n|u]:  y
Out[1]= (a;p)

∞

In[2]:= Limes[pq[a*p^{∧}-n,n,p],n-⟩Infinity]
Is n even, odd, or neither of both?
[e|o|n]:  n
Is  a  smaller than 1?
[y|n|u]:  u
Is  a  smaller than 1?
[y|n|u]:  u
The expression
                  2
    n  n  -n/2 - n /2  p
(-1)  a  p            (-;p)
                       a ∞
was obtained.
Therefore the limit n -⟩ ∞ could not be determined.
Here is your expression:
         a
Out[2]= (--; p)
          n    n
         p
In[3]:= AbsGreater[p]
In[4]:= Limes[pq[a*p^{∧}-n,n,p],n-⟩Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]:  e

         a 1
Out[4]= (-;-)
         p p ∞
\underline{See also}: AbsSmaller, AbsUndetermined, Limes.

AbsSmaller

\underline{Description}: Function for declaring the absolute value of a variable or expression to be smaller than 1. This declaration is used by Limes. By default the absolute value of q is defined to be smaller than 1.

\underline{Usage}: AbsSmaller[Expr].

\underline{Example(s)}:

In[1]:= Limes[pq[a,n,1/p],n-⟩Infinity]
Is  p  smaller than 1?
[y|n|u]:  n
           1
Out[1]= (a;-)
           p ∞
In[2]:= Limes[pq[a*p^{∧}n,n,1/p],n-⟩Infinity]
Is n even, odd, or neither of both?
[e|o|n]:  n
Is  a  smaller than 1?
[y|n|u]:  u
Is  a  smaller than 1?
[y|n|u]:  u
The expression
                 2
    n  n  n/2 + n /2   1  1
(-1)  a  p           (---;-)
                      a p p ∞
was obtained.
Therefore the limit n -⟩ ∞ could not be determined.
Here is your expression:
            n  1
Out[2]= (a p ; -)
               p n
In[3]:= AbsSmaller[p]

12 Alphabetic List: AbsUndetermined

In[4]:= Limes[pq[a*p^{∧}n,n,1/p],n-⟩Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]:  e
Out[4]= (a p;p)

∞

\underline{See also}: AbsGreater, AbsUndetermined, Limes.

AbsUndetermined \underline{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 Limes. By default the absolute value of q is defined to be smaller than 1.

\underline{Usage}: AbsUndetermined[Expr].

\underline{Example(s)}:

In[1]:= AbsSmaller[p]
In[2]:= Limes[pq[a,n,p],n-⟩Infinity]
Out[2]= (a;p)

∞

In[3]:= AbsGreater[p]
In[4]:= Limes[pq[a*p^{∧}-n,n,p],n-⟩Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]:  e
         a 1
Out[4]= (-;-)
         p p ∞
In[5]:= AbsUndetermined[p]
In[6]:= Limes[pq[a,n,p],n-⟩Infinity]
Is  p  smaller than 1?
[y|n|u]:  u
The expression
Indeterminate
was obtained.
Therefore the limit n -⟩ ∞ could not be determined.
Here is your expression:

Out[6]= (a; p)
              n
In[7]:= Limes[pq[a*p^{∧}-n,n,p],n-⟩Infinity]
Is  p  smaller than 1?
[y|n|u]:  u
The expression
Indeterminate
was obtained.
Therefore the limit n -⟩ ∞ could not be determined.
Here is your expression:
         a
Out[7]= (--; p)
          n    n
         p
\underline{See also}: AbsGreater, AbsSmaller, Limes.

Add

\underline{Description}: Function that adds Expr to Gleichung.

\underline{Usage}: Add[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= Add[pq[c/a,n]]
                                                    n  c
                                                   a  (-; q)
                 -n            c          c            a    n
Out[2]= ϕ a, q  ; q, q   + (-; q)  == (-; q)  + ----------
        2 1                    a    n     a    n    (c; q)
               c                                          n

In[3]:= Gleichung
                                                    n  c
                                                   a  (-; q)
                 -n            c          c            a    n
Out[3]= ϕ a, q  ; q, q   + (-; q)  == (-; q)  + ----------
        2 1                    a    n     a    n    (c; q)
               c                                          n
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Div, Sub, Hoch, GlTausche, Ers.

AmSLaTeX

\underline{Description}: Switch that changes the output of TeXForm to be usable with AMS-LaTEX. By default the output of TeXForm is usable with AMS-TEX.

\underline{Usage}: AmSLaTeX.

\underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[2]:= TeXForm[ph[{a,b},{c},q,z]]
Out[2]//TeXForm=
  {} {2} \phi {1} \! \left [ \matrix \let \over / a, b\\ \let \over /
   c\endmatrix ;q, {\displaystyle z} \right ]
In[3]:= AmSLaTeX
In[4]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[5]:= TeXForm[ph[{a,b},{c},q,z]]
Out[5]//TeXForm=
  {} {2} \phi {1} \! \left [ \begin{matrix} \let \over / a, b\\ \let \over /
   c\end{matrix} ;q, {\displaystyle z} \right ]
\underline{See also}: AmSTeX, LaTeX, TeX, TeXMat, TeXphW.

Alphabetic List: baszerl1 15

AmSTeX \underline{Description}: Switch that changes the output of TeXForm to be usable with AMS-TEX. By default the output of

TeXForm is usable with AMS-TEX.

\underline{Usage}: AmSTeX.

\underline{Example(s)}:

In[1]:= TeX
In[2]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[3]:= TeXForm[ph[{a,b},{c},q,z]]
Out[3]//TeXForm=
  {} {2} \phi {1} \! \left [ \matrix { \let \over / a, b\cr \let \over / c}
   ;q, {\displaystyle z} \right ]
In[4]:= AmSTeX
In[5]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[6]:= TeXForm[ph[{a,b},{c},q,z]]
Out[6]//TeXForm=
  {} {2} \phi {1} \! \left [ \matrix \let \over / a, b\\ \let \over /
   c\endmatrix ;q, {\displaystyle z} \right ]
\underline{See also}: AmSLaTeX, LaTeX, TeX, TeXMat, TeXphW.

baszerl1

\underline{Description}: \begin{matrix}(a; q)_{n} → ∏_{k=0}^{m−1}(aq^{k}; q^{m})_{n/m}, \\ (a; q)_{∞} → ∏_{k=0}^{m−1}(aq^{k}; q^{m})_{∞}. \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.baszerl1.

\underline{Example(s)}:

In[1]:= pq[a,n]

16 Alphabetic List: baszerl2

Out[1]= (a; q)
              n
In[2]:= %/.baszerl1
split into ?  terms:  2
             2            2
Out[2]= (a; q )    (a q; q )
               n/2          n/2
In[3]:= pq[b,4*m]
Out[3]= (b; q)
              4 m
In[4]:= %/.baszerl1
split into ?  terms:  4
             4          4       2   4       3   4
Out[4]= (b; q )  (b q; q )  (b q ; q )  (b q ; q )
               m          m           m           m
In[5]:= pqinf[c,q^{∧}(1/2)]
Out[5]= (c;Sqrt[q])

∞

In[6]:= %/.baszerl1
split into ?  terms:  4
            2               2         2       3/2  2
Out[6]= (c;q )  (c Sqrt[q];q )  (c q;q )  (c q   ;q )

∞  ∞  ∞  ∞

\underline{See also}: baszerl2, baszus1, baszus2, Ers, PosListe, ManipulationsListe.

baszerl2

\underline{Description}: \begin{matrix}(a; q)_{n} → ∏_{k=0}^{m−1}(e^{2πik/m}a^{1/m}; q^{1/m})_{n}, \\ (a; q)_{∞} → ∏_{k=0}^{m−1}(e^{2πik/m}a^{1/m}; q^{1/m})_{∞}. \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.baszerl2.

\underline{Example(s)}:

In[1]:= pq[a,n]
Out[1]= (a; q)
              n

In[2]:= %/.baszerl2
split into ?  terms:  2
Out[2]= (-Sqrt[a]; Sqrt[q])  (Sqrt[a]; Sqrt[q])
                           n                   n
In[3]:= pq[a^{∧}2,n,q^{∧}4]
          2   4
Out[3]= (a ; q )
                n
In[4]:= %/.baszerl2
split into ?  terms:  4
Out[4]= (-Sqrt[a]; q)  (-I Sqrt[a]; q)  (I Sqrt[a]; q)  (Sqrt[a]; q)
                     n                n               n             n
In[5]:= pqinf[c,q^{∧}3]
            3
Out[5]= (c;q )

∞

In[6]:= %/.baszerl2
split into ?  terms:  3

          1/3       (2 I)/3 π 1/3       (4 I)/3 π 1/3

Out[6]= (c   ;q)  (E          c   ;q)  (E          c   ;q)

∞  ∞  ∞

\underline{See also}: Ers, PosListe, ManipulationsListe.

baszus1

\underline{Description}: \begin{matrix}(a; q)_{n} → (a; q^{1/m})_{mn}/ ∏_{k=1}^{m−1}(aq^{k/m}; q)_{n}, \\ (a; q)_{∞} → (a; q^{1/m})_{∞}/ ∏_{k=1}^{m−1}(aq^{k/m}; q)_{∞}. \end{matrix}

The parameter m has to be entered on request. This operation is basically the inverse of baszerl1.

\underline{Usage}: Expr/.baszus1.

\underline{Example(s)}:

In[1]:= pq[a,n,q^{∧}2]*pq[a*q,n,q^{∧}2]
             2          2
Out[1]= (a; q )  (a q; q )
               n          n

In[2]:= Ers[%,baszus1,{1}]
put together ?  terms:  2
put together ?  terms:  2
Out[2]= (a; q)
              2 n
In[3]:= pqinf[a/q,q^{∧}2]*pqinf[a,q^{∧}2]*pq[b,m]*pq[b*q^{∧}(1/3),m]*pq[b*q^{∧}(2/3),m]
            2    a  2               1/3          2/3
Out[3]= (a;q )  (-;q )  (b; q)  (b q   ; q)  (b q   ; q)
              ∞ q ∞  m            m            m
In[4]:= Ers[%,baszus1,{3}]
put together ?  terms:  3
put together ?  terms:  3
            2    a  2        1/3
Out[4]= (a;q )  (-;q )  (b; q   )
              ∞ q ∞  3 m
In[5]:= Ers[%,baszus1,{2}]
put together ?  terms:  2
put together ?  terms:  2
         a          1/3
Out[5]= (-;q)  (b; q   )
         q ∞  3 m
\underline{See also}: Ers, PosListe, ManipulationsListe.

baszus2

\underline{Description}: \begin{matrix}(a; q)_{n} → (a^{m}; q^{m})_{n}/ ∏_{k=1}^{m−1}(e^{2πik/m}a; q)_{n}, \\ (a; q)_{∞} → (a^{m}; q^{m})_{∞}/ ∏_{k=1}^{m−1}(e^{2πik/m}a; q)_{∞}. \end{matrix}

The parameter m has to be entered on request. This operation is basically the inverse of baszerl2.

\underline{Usage}: Expr/.baszus2.

\underline{Example(s)}:

In[1]:= pq[a,m,q^{∧}2]*pq[-a,m,q^{∧}2]
              2        2
Out[1]= (-a; q )  (a; q )
                m        m
In[2]:= Ers[%,baszus2,{1}]
put together ?  terms:  2
put together ?  terms:  2

          2   4
Out[2]= (a ; q )
                m
In[3]:= pqinf[-A,q^{∧}(1/2)]*pqinf[A,q^{∧}(1/2)]*pq[-a,n]*pq[-I*a,n]*pq[I*a,n]*
        pq[a,n]
Out[3]= (-A;Sqrt[q])  (A;Sqrt[q])  (-a; q)  (-I a; q)  (I a; q)  (a; q)
                    ∞  ∞  n          n         n       n
In[4]:= Ers[%,baszus2,{3}]
put together ?  terms:  4
put together ?  terms:  4
                                     4   4
Out[4]= (-A;Sqrt[q])  (A;Sqrt[q])  (a ; q )

∞  ∞  n

In[5]:= Ers[%,baszus2,{1}]
put together ?  terms:  2
put together ?  terms:  2
          2       4   4
Out[5]= (A ;q)  (a ; q )
              ∞  n

Whenever you use a rule, that asks you for an input, within Ers, you will get the reqest for input twice. The value which is entered last is used.

\underline{See also}: Ers, PosListe, ManipulationsListe.

Binomialpq

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

\underline{Usage}: Binomialpq[n,k,q]
   or: Binomialpq[n,k].

\underline{Example(s)}:

In[1]:= Binomialpq[n,k]
          1 - k + n
        (q         ; q)
                       k
Out[1]= ----------------
            (q; q)
                  k
In[2]:= Binomialpq[6,3]

          4
        (q ; q)
               3
Out[2]= --------
        (q; q)
              3
\underline{See also}: Binomialq, Multinomialpq, Multinomialq, Factorialq, Factorialpq.

Binomialq

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

\underline{Usage}: Binomialq[n,k,q]
   or: Binomialq[n,k].

\underline{Example(s)}:

In[1]:= Binomialq[n+k,n-k]

          k + n
Out[1]=
          -k + n
                  q
In[2]:= Binomialq[6,3]
                   2      3      4      5      6      7    8    9
Out[2]= 1 + q + 2 q  + 3 q  + 3 q  + 3 q  + 3 q  + 2 q  + q  + q
\underline{See also}: Binomialpq, Multinomialpq, Multinomialq, Factorialq, Factorialpq.

C01

\underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] ⟶ 1 + \frac{(−1)^{1−r+s}z}{(1 − q)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}q, (qA) \\ q^{2}, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C01.
\underline{See also}: C64, ContigListe, Ers, PosListe.

C02 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A), q \\ (B) \end{matrix} ; q, z] ⟶ −\frac{(−1)^{1−r+s}q^{1−r+s}}{z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})} + \frac{(−1)^{1−r+s}q^{1−r+s}}{z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q), q \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C02.
\underline{See also}: C64, ContigListe, Ers, PosListe.

C11 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, qz] + (−1)^{1−r+s}z \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C11.
\underline{See also}: C64, ContigListe, Ers, PosListe.

C12 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, \frac{z}{q} ] − \frac{(−1)^{1−r+s}z}{q} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ (qB) \end{matrix} ; q, q^{−r+s}z]

\underline{Usage}: Expr/.C12.
\underline{See also}: C64, ContigListe, Ers, PosListe.

C13 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s}q^{1−r+s}}{z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s}q^{1−r+s}}{z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ (B/q) \end{matrix} ; q, q^{r−s}z]

\underline{Usage}: Expr/.C13.
\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s}az}{q} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}a, (qA) \\ (qB) \end{matrix} ; q, q^{1−r+s}z] +_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C14[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C15 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ (B) \end{matrix} ; q, z] − (−1)^{1−r+s}az \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, (qA) \\ (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C15[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C16 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s}q^{2−r+s}}{az} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s}q^{2−r+s}}{az} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C16[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C17 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z] ⟶ (−1)^{1−r+s}z \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}a, (qA) \\ (qB) \end{matrix} ; q, q^{1−r+s}z] +_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ (B) \end{matrix} ; q, qz]

\underline{Usage}: Expr/.C17[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ (B) \end{matrix} ; q, \frac{z}{q} ] − \frac{(−1)^{1−r+s}z}{q} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, (qA) \\ (qB) \end{matrix} ; q, q^{−r+s}z]

\underline{Usage}: Expr/.C18[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C19 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s}q^{1−r+s}}{z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s}q^{1−r+s}}{z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, (A/q) \\ (B/q) \end{matrix} ; q, q^{r−s}z]

\underline{Usage}: Expr/.C19[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C20 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ bq, (B) \end{matrix} ; q, z] + \frac{(−1)^{1−r+s}bz}{(1 − b) (1 − bq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ bq^{2}, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C20[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C21 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ b/q, (B) \end{matrix} ; q, z] − \frac{(−1)^{1−r+s}bz}{(1 − b) (1 − \frac{b}{q} ) q} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ bq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C21[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{3−r+s}}{bz} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ b/q^{2}, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{3−r+s}}{bz} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ b/q, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C22[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C23 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ bq, (B) \end{matrix} ; q, qz] + \frac{(−1)^{1−r+s}z}{(1 − b) (1 − bq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ bq^{2}, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C23[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C24 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ b/q, (B) \end{matrix} ; q, \frac{z}{q} ] − \frac{(−1)^{1−r+s}z}{(1 − b) (1 − \frac{b}{q} ) q} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ bq, (qB) \end{matrix} ; q, q^{−r+s}z]

\underline{Usage}: Expr/.C24[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{1−r+s}}{z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ b/q^{2}, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{1−r+s}}{z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ b/q, (B/q) \end{matrix} ; q, q^{r−s}z]

\underline{Usage}: Expr/.C25[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C26 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(1 − b)}{a − b}_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ (B) \end{matrix} ; q, \frac{z}{q} ] + \frac{(1 − a)}{−a + b}_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C26[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C27 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(b − \frac{a}{q})}{1 − \frac{a}{q}}_{r}ϕ_{s}[ \begin{matrix}a/q, b, (A) \\ (B) \end{matrix} ; q, qz] + \frac{(1 − b)}{1 − \frac{a}{q}}_{r}ϕ_{s}[ \begin{matrix}a/q, bq, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C27[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C28 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{a (1 − b)}{a − b}_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ (B) \end{matrix} ; q, z] + \frac{(1 − a) b}{−a + b}_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C28[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(b − \frac{a}{q})}{b (1 − \frac{a}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, b, (A) \\ (B) \end{matrix} ; q, z] + \frac{a (1 − b)}{b (1 − \frac{a}{q}) q}_{r}ϕ_{s}[ \begin{matrix}a/q, bq, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C29[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C30 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ (−1)^{1−r+s} (−b + \frac{a}{q}) z \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}a, bq, (qA) \\ (qB) \end{matrix} ; q, q^{1−r+s}z] +_{r}ϕ_{s}[ \begin{matrix}a/q, bq, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C30[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C31 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s}q^{1−r+s}}{(\frac{a}{q} − \frac{b}{q} ) z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−2}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a, b/q, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s}q^{1−r+s}}{(\frac{a}{q} − \frac{b}{q} ) z} \frac{∏_{i=1}^{s}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−2}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, b, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C31[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

Alphabetic List: C35 27

C32 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}a/q, b/q, (A) \\ (B) \end{matrix} ; q, qz]

+ (−1)^{1−r+s} (1 − \frac{ab}{q}) z \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}a, b, ab, (qA) \\ ab/q, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C32[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C33 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, bq, (A) \\ (B) \end{matrix} ; q, \frac{z}{q} ]

− \frac{(−1)^{1−r+s} (1 − abq) z}{q} \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}aq, bq, abq^{2}, (qA) \\ abq, (qB) \end{matrix} ; q, q^{−r+s}z]

\underline{Usage}: Expr/.C33[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C34 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − \frac{b}{q} )}{a − \frac{b}{q}}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b/q, (B) \end{matrix} ; q, \frac{z}{q} ] + \frac{(1 − a)}{−a + \frac{b}{q}} _{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ b, (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C34[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C35 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(−a + b)}{(1 − \frac{a}{q}) q}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b, (B) \end{matrix} ; q, qz] + \frac{(1 − \frac{b}{q} )}{1 − \frac{a}{q}}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b/q, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C35[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(a − b)}{1 − b}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ bq, (B) \end{matrix} ; q, qz] + \frac{(1 − a)}{1 − b}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ bq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C36[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C37 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{a (1 − \frac{b}{q} )}{a − \frac{b}{q}}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b/q, (B) \end{matrix} ; q, z] + \frac{(1 − a) b}{(−a + \frac{b}{q} ) q}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C37[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C38 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(−a + b)}{b (1 − \frac{a}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b, (B) \end{matrix} ; q, z] + \frac{a (1 − \frac{b}{q} )}{b (1 − \frac{a}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b/q, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C38[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C39 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(a − b)}{a (1 − b)}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ bq, (B) \end{matrix} ; q, z] + \frac{(1 − a) b}{a (1 − b)}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ bq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C39[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s} (\frac{a}{q} − \frac{b}{q} ) z}{(1 − b) (1 − \frac{b}{q} )} \frac{∏_{i=1}^{s−1}(1 − B_{i})}{∏_{i=1}^{r−1}(1 − A_{i})}_{r}ϕ_{s}[ \begin{matrix}a, (qA) \\ bq, (qB) \end{matrix} ; q, q^{1−r+s}z] +_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b/q, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C40[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C41 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ bq, (B) \end{matrix} ; q, z] − \frac{(−1)^{1−r+s} (a − b) z}{(1 − b) (1 − bq)} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, (qA) \\ bq^{2}, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C41[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C42 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{1−r+s}}{(−\frac{b}{q^{2}} + \frac{a}{q}) z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a, (A/q) \\ b/q, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{1−r+s}}{(−\frac{b}{q^{2}} + \frac{a}{q}) z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, (A/q) \\ b/q^{2}, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C42[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C43 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ bq, (B) \end{matrix} ; q, qz] + \frac{(−1)^{1−r+s} (1 − ab) z}{(1 − b) (1 − bq)} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}a, abq, (qA) \\ bq^{2}, ab, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C43[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ b/q, (B) \end{matrix} ; q, \frac{z}{q} ] − \frac{(−1)^{1−r+s} (1 − ab) z}{(1 − b) (1 − \frac{b}{q} ) q} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}aq, abq, (qA) \\ bq, ab, (qB) \end{matrix} ; q, q^{−r+s}z]

\underline{Usage}: Expr/.C44[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C45 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − \frac{b}{q} ) q}{a − b}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, b/q, (B) \end{matrix} ; q, \frac{z}{q} ] + \frac{(1 − \frac{a}{q}) q}{−a + b}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, b, (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C45[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C46 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ \frac{(−a + \frac{b}{q} )}{1 − a}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b, (B) \end{matrix} ; q, qz] + \frac{(1 − \frac{b}{q} )}{1 − a}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b/q, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C46[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C47 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ \frac{a (1 − \frac{b}{q} )}{a − b}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, b/q, (B) \end{matrix} ; q, z] + \frac{b (1 − \frac{a}{q})}{−a + b}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C47[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ \frac{(−a + \frac{b}{q} ) q}{(1 − a) b}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b, (B) \end{matrix} ; q, z] + \frac{a (1 − \frac{b}{q} ) q}{(1 − a) b}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b/q, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C48[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C49 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b/q, (B) \end{matrix} ; q, z]

+ \frac{(−1)^{1−r+s} (a − \frac{b}{q} ) z}{(1 − a) (1 − b) (1 − \frac{b}{q} ) (1 − aq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ aq^{2}, bq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C49[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C50 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s} (1 − \frac{a}{q^{2}} ) (1 − \frac{b}{q^{2}} ) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{1−r+s}}{(\frac{a}{q^{2}} − \frac{b}{q^{2}} ) z} \frac{∏_{i=1}^{s−2}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ a/q^{2}, b/q, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{a}{q^{2}} ) (1 − \frac{b}{q^{2}} ) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{1−r+s}}{(\frac{a}{q^{2}} − \frac{b}{q^{2}} ) z} \frac{∏_{i=1}^{s−2}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r}(1 − \frac{A_{i}}{q})}_{r}ϕ_{s}[ \begin{matrix}(A/q) \\ a/q, b/q^{2}, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C50[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

32 Alphabetic List: C54

C51 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, bq, (B) \end{matrix} ; q, qz]

+ \frac{(−1)^{1−r+s} (1 − abq) z}{(1 − a) (1 − b) (1 − aq) (1 − bq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}abq^{2}, (qA) \\ aq^{2}, bq^{2}, abq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C51[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C52 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, b/q, (B) \end{matrix} ; q, \frac{z}{q} ]

− \frac{(−1)^{1−r+s} (1 − \frac{ab}{q}) z}{(1 − a) (1 − b) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}ab, (qA) \\ aq, bq, ab/q, (qB) \end{matrix} ; q, q^{−r+s}z]

\underline{Usage}: Expr/.C52[n1,n2].

m1, m2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C53 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, c, (A) \\ (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − b) & (a − \frac{c}{q} ) \\ (a − b) & (1 − \frac{c}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, bq, c/q, (A) \\ (B) \end{matrix} ; q, z] + \begin{matrix}(1 − a) & (b − \frac{c}{q} ) \\ (−a + b) & (1 − \frac{c}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, b, c/q, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C53[m1,m2,m3].

m1, m2, m3 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C54 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − b) & (a − c) \\ (a − b) & (1 − c) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ cq, (B) \end{matrix} ; q, z] + \begin{matrix}(1 − a) & (b − c) \\ (−a + b) & (1 − c) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ cq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C54[m1,m2,n1].

m1, m2 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(−b + \frac{a}{q}) & (1 − \frac{c}{q} ) \\ (1 − \frac{a}{q}) & (−b + \frac{c}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, b, (A) \\ c/q, (B) \end{matrix} ; q, z] + \begin{matrix}(1 − b) & (\frac{a}{q} − \frac{c}{q} ) \\ (1 − \frac{a}{q}) & (b − \frac{c}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, bq, (A) \\ c, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C55[m1,m2,n1].

m1, m2 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C56 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, c, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(a − c) & (1 − \frac{b}{q} ) \\ (1 − c) & (a − \frac{b}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b/q, cq, (B) \end{matrix} ; q, z] + \begin{matrix}(1 − a) & (−c + \frac{b}{q} ) \\ (1 − c) & (−a + \frac{b}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ b, cq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C56[m1,n1,n2].

m1 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C57 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ b, c, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(\frac{a}{q} − \frac{b}{q} ) & (1 − \frac{c}{q} ) \\ (1 − \frac{a}{q}) & (−\frac{b}{q} + \frac{c}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b, c/q, (B) \end{matrix} ; q, z] + \begin{matrix}(1 − \frac{b}{q} ) & (\frac{a}{q} − \frac{c}{q} ) \\ (1 − \frac{a}{q}) & (\frac{b}{q} − \frac{c}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ b/q, c, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C57[m1,n1,n2].

m1 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C58 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, b, c, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(−c + \frac{a}{q}) & (1 − \frac{b}{q} ) \\ (1 − c) & (\frac{a}{q} − \frac{b}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, b/q, cq, (B) \end{matrix} ; q, z] + \begin{matrix}(1 − \frac{a}{q}) & (−c + \frac{b}{q} ) \\ (1 − c) & (−\frac{a}{q} + \frac{b}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, b, cq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C58[n1,n2,n3].

n1, n2, n3 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, cq^{2}/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(b − \frac{a}{q}) & (1 − \frac{cq}{ab}) \\ (1 − \frac{a}{q}) & (1 − \frac{cq}{a} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, cq/a, b, c/b, (A) \\ (B) \end{matrix} ; q, qz]

+ \begin{matrix}(1 − b) & (1 − \frac{c}{b}) \\ (1 − \frac{a}{q}) & (1 − \frac{cq}{a} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, cq/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C59[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C60 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) \\ (b − a) & (1 − \frac{c}{ab}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b, c/b, (A) \\ (B) \end{matrix} ; q, \frac{z}{q} ]

− \begin{matrix}(1 − b) & (1 − \frac{c}{b}) \\ (b − a) & (1 − \frac{c}{ab}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, c/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C60[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C61 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − b) c}{ab (1 − \frac{c}{aq} ) (1 − \frac{c}{bq} ) q}_{r}ϕ_{s}[ \begin{matrix}aq, c/aq, bq, c/bq, (A) \\ (B) \end{matrix} ; q, z]

+ \begin{matrix}(1 − \frac{c}{q} ) & (1 − \frac{c}{abq} ) \\ (1 − \frac{c}{aq} ) & (1 − \frac{c}{bq} ) \end{matrix}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}\sqrt{q}, −\sqrt{c}\sqrt{q}, a, c/aq, b, c/bq, (A) \\ \sqrt{c}/\sqrt{q}, −\sqrt{c}/\sqrt{q}, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C61[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, c/a, b, cq^{2}/b, (A) \\ (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b/q, c ∗ q/b, (A) \\ (B) \end{matrix} ; q, z]

+ (−1)^{1−r+s} (\frac{b}{q} − a) (1 − \frac{c ∗ q}{ab} ) (1 − cq) z \frac{∏_{i=1}^{r−4}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}aq, cq/a, b, cq^{2}/b, cq^{2}, (qA) \\ cq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C62[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C63 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, cq^{2}/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − b) & (1 − \frac{c}{b}) & (1 − \frac{a}{dq} ) & (1 − \frac{cq}{ad}) \\ (1 − \frac{b}{d}) & (1 − \frac{c}{bd}) & (1 − \frac{a}{q}) & (1 − \frac{cq}{a} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, cq/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, z]

+ \frac{(1 − \frac{c}{d}) (1 − d) (−b + \frac{a}{q}) (1 − \frac{cq}{ab})}{(1 − \frac{b}{d}) (1 − \frac{c}{bd}) d (1 − \frac{a}{q}) (1 − \frac{cq}{a} )}_{2+r}ϕ_{2+s}[ \begin{matrix}a/q, cq/a, b, c/b, dq, cq/d, (A) \\ d, c/d, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C63[m1,m2,m3,m4,d].

m1, m2, m3, m4 are the positions of the special upper parameters, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C64 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, cq/a, b, (A) \\ c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{ab}) & (b − \frac{a}{q}) \\ (1 − \frac{c}{a}) & (1 − \frac{a}{q}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, c/a, b, (A) \\ c/b, (B) \end{matrix} ; q, qz]

+ \begin{matrix}(1 − b) & (1 − \frac{c}{bq} ) \\ (1 − \frac{c}{a}) & (1 − \frac{a}{q}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, c/a, bq, (A) \\ c/bq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C64[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively. \underline{Example(s)}:

In[1]:= ph[{A,B,q*C,D},{E,A*C/D,F},q,z]

             A, B, C q, D
Out[1]= ϕ  A C     ; q, z
        4 3   E, ---, F
                  D
In[2]:= %/.C64[3,1,4,2]
                  A
               C, -, D, B
                  q                  C        A
        D ϕ  ; q, q z   (-; q)  (---; q)
          4 3  A C                   D    1  D q    1
               ---, E, F
                D
Out[2]= --------------------------------------------- +
                                A
                       (C; q)  (-; q)
                             1  q    1
             A
          C, -, D q, B
             q                          A C
      ϕ  ; q, z   (D; q)  (---; q)
     4 3   A C                       1  D q    1
           ---, E, F
           D q
⟩  -------------------------------------------
                            A
                   (C; q)  (-; q)
                         1  q    1
In[3]:= %/.pqaufl

                                    A
                                 C, -, D, B
             C          A           q
        (1 - -) D (1 - ---) ϕ  ; q, q z
             D         D q  4 3  A C
                                 ---, E, F
                                  D
Out[3]= --------------------------------------------- +
                                    A
                       (1 - C) (1 - -)
                                    q
                               A
                            C, -, D q, B
                  A C          q
     (1 - D) (1 - ---) ϕ  ; q, z
                  D q  4 3   A C
                             ---, E, F
                             D q
⟩  -------------------------------------------
                                A
                   (1 - C) (1 - -)
                                q

The third, first, and fourth upper parameters in Out[1] are q C, A, and D, respectively, the second lower parameter in Out[1] is \frac{A C}{D}. Hence C64 can be applied with the replacements a → q C, b → D and c → A C.

\underline{See also}: ContigListe, Ers, PosListe.

C65

\underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) \\ (1 − \frac{b}{q} ) & (1 − \frac{cq}{b} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b/q, (A) \\ cq^{2}/b, (B) \end{matrix} ; q, z]

− \begin{matrix}(\frac{b}{q} − a) & (1 − \frac{cq}{ab}) \\ (1 − \frac{b}{q} ) & (1 − \frac{cq}{b} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, c/a, b/q, (A) \\ cq^{2}/b, (B) \end{matrix} ; q, qz]

\underline{Usage}: Expr/.C65[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C66

\underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) \\ (b − a) & (1 − \frac{c}{ab}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, \frac{z}{q} ]

− \begin{matrix}(1 − b) & (1 − \frac{c}{b}) \\ (b − a) & (1 − \frac{c}{ab}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, c/a, bq, (A) \\ c/b, (B) \end{matrix} ; q, \frac{z}{q} ]

38 Alphabetic List: C69

\underline{Usage}: Expr/.C66[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C67 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, cq/a, b, (A) \\ c/b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − b) c}{ab (1 − \frac{c}{a}) (1 − \frac{c}{b})}_{r}ϕ_{s}[ \begin{matrix}aq, c/a, bq, (A) \\ cq/b, (B) \end{matrix} ; q, z]

+ \begin{matrix}(1 − c) & (1 − \frac{c}{ab}) \\ (1 − \frac{c}{a}) & (1 − \frac{c}{b}) \end{matrix}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, c/a, b, (A) \\ \sqrt{c}, −\sqrt{c}, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C67[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C68 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] ⟶ \frac{ab (1 − \frac{c}{a}) (1 − \frac{c}{b})}{c (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q}_{r}ϕ_{s}[ \begin{matrix}a/q, cq/a, b/q, (A) \\ c/b, (B) \end{matrix} ; q, z]

− \frac{ab (1 − \frac{c}{q} ) (1 − \frac{cq}{ab})}{c (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}\sqrt{q}, −\sqrt{c}\sqrt{q}, a/q, c/a, b/q, (A) \\ \sqrt{c}/\sqrt{q}, −\sqrt{c}/\sqrt{q}, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C68[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C69 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b/q, (A) \\ cq^{2}/b, (B) \end{matrix} ; q, z]

+ \frac{(−1)^{1−r+s} (\frac{b}{q} − a) (1 − cq) (1 − \frac{cq}{ab}) z}{(1 − \frac{cq}{b} ) (1 − \frac{cq^{2}}{b} )} \frac{∏_{i=1}^{r−3}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}aq, cq/a, b, cq^{2}, (qA) \\ cq^{3}/b, cq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C69[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, cq/a, b, (A) \\ c/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}a/q, c/a, bq, (A) \\ c/bq, (B) \end{matrix} ; q, z]

− \frac{(−1)^{1−r+s} (1 − c) (1 − \frac{c}{ab}) (b − \frac{a}{q}) z}{(1 − \frac{c}{b}) (1 − \frac{c}{bq} )} \frac{∏_{i=1}^{r−3}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}a, cq/a, bq, cq, (qA) \\ cq/b, c, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C70[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C71 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, cq/a, b, (A) \\ c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − b) & (1 − \frac{c}{ad}) & (1 − \frac{c}{bq} ) & (1 − \frac{a}{dq} ) \\ (1 − \frac{c}{a}) & (1 − \frac{b}{d}) & (1 − \frac{a}{q}) & (1 − \frac{c}{bdq} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, c/a, bq, (A) \\ c/bq, (B) \end{matrix} ; q, z]

+ \frac{(1 − \frac{c}{ab}) (1 − d) (−b + \frac{a}{q}) (1 − \frac{c}{dq} )}{(1 − \frac{c}{a}) (1 − \frac{b}{d}) d (1 − \frac{a}{q}) (1 − \frac{c}{bdq} )}_{2+r}ϕ_{2+s}[ \begin{matrix}a/q, c/a, b, dq, c/d, (A) \\ c/b, d, c/dq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C71[m1,m2,m3,n1,d].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C72 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) & (1 − \frac{b}{dq} ) & (1 − \frac{cq}{bd}) \\ (1 − \frac{a}{d}) & (1 − \frac{c}{ad}) & (1 − \frac{b}{q} ) & (1 − \frac{cq}{b} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b/q, (A) \\ cq^{2}/b, (B) \end{matrix} ; q, z]

− \frac{(1 − \frac{c}{d}) (1 − d) (a − \frac{b}{q} ) (1 − \frac{cq}{ab})}{(1 − \frac{a}{d}) (1 − \frac{c}{ad}) d (1 − \frac{b}{q} ) (1 − \frac{cq}{b} )}_{2+r}ϕ_{2+s}[ \begin{matrix}a, c/a, b/q, dq, cq/d, (A) \\ cq^{2}/b, d, c/d, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C72[m1,m2,m3,n1,d].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, b, ab, (A) \\ ab/q, (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s}q^{1−r+s}}{(1 − \frac{ab}{q}) z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−3}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a, b, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s}q^{1−r+s}}{(1 − \frac{ab}{q}) z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−3}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a/q, b/q, (A/q) \\ (B/q) \end{matrix} ; q, q^{r−s}z]

\underline{Usage}: Expr/.C73[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C74 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − \frac{ab}{c}) c (−b + \frac{a}{q})}{ab (1 − \frac{c}{a}) (1 − \frac{a}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, b, (A) \\ cq/a, c/b, (B) \end{matrix} ; q, qz]

+ \begin{matrix}(1 − b) & (1 − \frac{c}{bq} ) \\ (1 − \frac{c}{a}) & (1 − \frac{a}{q}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, bq, (A) \\ cq/a, c/bq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C74[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C75 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) ab (1 − \frac{c}{aq} ) q}{(a − b) c (1 − \frac{abq}{c} )}_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ c/aq, c/b, (B) \end{matrix} ; q, \frac{z}{q} ]

− \frac{a (1 − b) b (1 − \frac{c}{bq} ) q}{(a − b) c (1 − \frac{abq}{c} )}_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ c/a, c/bq, (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C75[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − b) c}{ab (1 − \frac{c}{a}) (1 − \frac{c}{b})}_{r}ϕ_{s}[ \begin{matrix}aq, bq, (A) \\ cq/a, cq/b, (B) \end{matrix} ; q, z]

+ \begin{matrix}(1 − c) & (1 − \frac{c}{ab}) \\ (1 − \frac{c}{a}) & (1 − \frac{c}{b}) \end{matrix}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, b, (A) \\ \sqrt{c}, −\sqrt{c}, cq/a, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C76[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C77 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{ab (1 − \frac{c}{aq} ) (1 − \frac{c}{bq} )}{c (1 − \frac{a}{q}) (1 − \frac{b}{q} )}_{r}ϕ_{s}[ \begin{matrix}a/q, b/q, (A) \\ c/aq, c/bq, (B) \end{matrix} ; q, z]

− \frac{ab (1 − \frac{c}{ab}) (1 − \frac{c}{q^{2}} )}{c (1 − \frac{a}{q}) (1 − \frac{b}{q} )}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}, −\sqrt{c}, a/q, b/q, (A) \\ \sqrt{c}/q, −\sqrt{c}/q, c/a, c/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C77[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C78 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, b/q, (A) \\ c/aq, cq/b, (B) \end{matrix} ; q, z]

+ \frac{(−1)^{1−r+s} (1 − c) (1 − \frac{c}{ab}) (−a + \frac{b}{q} ) z}{(1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{c}{aq} ) (1 − \frac{cq}{b} )} \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}aq, b, cq, (qA) \\ cq/a, cq^{2}/b, c, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C78[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C79

\underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − b) & (1 − \frac{c}{ad}) & (1 − \frac{c}{bq} ) & (1 − \frac{a}{dq} ) \\ (1 − \frac{c}{a}) & (1 − \frac{b}{d}) & (1 − \frac{a}{q}) & (1 − \frac{c}{bdq} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, bq, (A) \\ cq/a, c/bq, (B) \end{matrix} ; q, z]

+ \frac{(1 − \frac{c}{ab}) (1 − d) (−b + \frac{a}{q}) (1 − \frac{c}{dq} )}{(1 − \frac{c}{a}) (1 − \frac{b}{d}) d (1 − \frac{a}{q}) (1 − \frac{c}{bdq} )}_{2+r}ϕ_{2+s}[ \begin{matrix}a/q, b, dq, c/d, (A) \\ cq/a, c/b, d, c/dq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C79[m1,m2,n1,n2,d].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{Example(s)}:

In[1]:= ph[{A,B,B*F/E,D},{E,F},q,z]
                   B F
             A, B, ---, D
Out[1]= ϕ  E    ; q, z
        4 2
                 E, F
In[2]:= %/.C79[2,3,2,1,x]
                                                 B  B F q
                                                 -, -----, A, D
             B F       E       F        B        q    E
        (1 - ---) (1 - -) (1 - -) (1 - ---) ϕ  ; q, z
              E        q       x       q x  4 2           E
                                                     F q, -
                                                          q
Out[2]= --------------------------------------------------------------- +
                                   B       B F        E
                      (1 - F) (1 - -) (1 - ---) (1 - ---)
                                   q       E x       q x

Alphabetic List: C81 43

⟩

                                                  B  B F       B F
                                                  -, ---, q x, ---, A, D
           E     B F    B       B F               q   E         x
      (1 - -) (-(---) + -) (1 - ---) (1 - x) ϕ  ; q, z
           B      E     q       q x          6 4                 B F
                                                      F q, E, x, ---
                                                                 q x
⟩  --------------------------------------------------------------------------
                                     B       B F        E
                        (1 - F) (1 - -) (1 - ---) (1 - ---) x
                                     q       E x       q x

The second and third upper parameters in Out[1] are B, and \frac{B F}{E}, respectively, the second and first lower parameters in Out[1] are F and E, respectively. Hence C79 can be applied with the replacements a → B, b → \frac{B F}{E} and c → B F. The x replaces d in the right-hand side expression.

\underline{See also}: ContigListe, Ers, PosListe.

C80 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{ab}) & (−\frac{a}{q} + \frac{b}{q} ) \\ (1 − \frac{a}{q}) & (1 − \frac{c}{aq} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, c/aq, (A) \\ b, c/b, (B) \end{matrix} ; q, qz]

+ \begin{matrix}(1 − \frac{b}{q} ) & (1 − \frac{c}{bq} ) \\ (1 − \frac{a}{q}) & (1 − \frac{c}{aq} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, c/aq, (A) \\ b/q, c/bq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C80[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C81 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) \\ (1 − b) & (1 − \frac{c}{b}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, z]

− \begin{matrix}(b − a) & (1 − \frac{c}{ab}) \\ (1 − b) & (1 − \frac{c}{b}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, c/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, qz]

\underline{Usage}: Expr/.C81[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, cq^{2}/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) \\ (\frac{b}{q} − a) & (1 − \frac{cq}{ab}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ b, cq^{2}/b, (B) \end{matrix} ; q, \frac{z}{q} ]

− \begin{matrix}(1 − \frac{b}{q} ) & (1 − \frac{cq}{b} ) \\ (\frac{b}{q} − a) & (1 − \frac{cq}{ab}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b/q, cq/b, (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C82[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C83 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) c (1 − \frac{b}{q} )}{ab (1 − \frac{c}{b}) (1 − \frac{c}{aq} )}_{r}ϕ_{s}[ \begin{matrix}aq, c/aq, (A) \\ b/q, cq/b, (B) \end{matrix} ; q, z]

+ \begin{matrix}(1 − \frac{c}{ab}) & (1 − \frac{c}{q} ) \\ (1 − \frac{c}{b}) & (1 − \frac{c}{aq} ) \end{matrix}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}\sqrt{q}, −\sqrt{c}\sqrt{q}, a, c/aq, (A) \\ \sqrt{c}/\sqrt{q}, −\sqrt{c}/\sqrt{q}, b, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C83[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C84 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, z]

+ \frac{(−1)^{1−r+s} (b − a) (1 − \frac{c}{ab}) (1 − cq) z}{(1 − b) (1 − \frac{c}{b}) (1 − bq) (1 − \frac{cq}{b} )} \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}aq, cq/a, cq^{2}, (qA) \\ bq^{2}, cq^{2}/b, cq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C84[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}a/q, c/aq, (A) \\ b/q, c/bq, (B) \end{matrix} ; q, z]

− \frac{(−1)^{1−r+s} (1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z}{(1 − b) (1 − \frac{c}{b}) (1 − \frac{b}{q} ) (1 − \frac{c}{bq} )} \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}a, c/a, c, (qA) \\ bq, cq/b, c/q, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C85[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C86 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{b}{q} ) & (1 − \frac{c}{bq} ) & (1 − \frac{a}{dq} ) & (1 − \frac{c}{adq} ) \\ (1 − \frac{a}{q}) & (1 − \frac{c}{aq} ) & (1 − \frac{b}{dq} ) & (1 − \frac{c}{bdq} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, c/aq, (A) \\ b/q, c/bq, (B) \end{matrix} ; q, z]

+ \frac{(1 − \frac{c}{ab}) (1 − d) (1 − \frac{c}{dq^{2}} ) (\frac{a}{q} − \frac{b}{q} )}{d (1 − \frac{a}{q}) (1 − \frac{c}{aq} ) (1 − \frac{b}{dq} ) (1 − \frac{c}{bdq} )}_{2+r}ϕ_{2+s}[ \begin{matrix}a/q, c/aq, dq, c/dq, (A) \\ b, c/b, d, c/dq^{2}, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C86[m1,m2,n1,n2,d].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C87 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) & (1 − \frac{b}{d}) & (1 − \frac{c}{bd}) \\ (1 − b) & (1 − \frac{c}{b}) & (1 − \frac{a}{d}) & (1 − \frac{c}{ad}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, z]

− \frac{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}{(1 − b) (1 − \frac{c}{b}) (1 − \frac{a}{d}) (1 − \frac{c}{ad}) d}_{2+r}ϕ_{2+s}[ \begin{matrix}a, c/a, dq, cq/d, (A) \\ bq, cq/b, d, c/d, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C87[m1,m2,n1,n2,d].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, ab/q, (A) \\ b, ab/q^{2}, (B) \end{matrix} ; q, z] ⟶ \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{1−r+s}}{(1 − \frac{ab}{q^{2}}) z} \frac{∏_{i=1}^{s−2}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−2}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a, (A/q) \\ b/q^{2}, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) q^{1−r+s}}{(1 − \frac{ab}{q^{2}}) z} \frac{∏_{i=1}^{s−2}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−2}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a/q, (A/q) \\ b/q, (B/q) \end{matrix} ; q, q^{r−s}z]

\underline{Usage}: Expr/.C88[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C89 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ c/a, b, cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(−\frac{a}{q} + \frac{b}{q} ) & (1 − \frac{cq}{ab}) \\ (1 − \frac{c}{a}) & (1 − \frac{a}{q}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ cq/a, b, cq/b, (B) \end{matrix} ; q, qz]

+ \begin{matrix}(1 − \frac{c}{b}) & (1 − \frac{b}{q} ) \\ (1 − \frac{c}{a}) & (1 − \frac{a}{q}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ cq/a, b/q, c/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C89[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C90 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) \\ (1 − b) & (1 − \frac{c}{b}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ c/a, bq, cq/b, (B) \end{matrix} ; q, z]

− \begin{matrix}(b − a) & (1 − \frac{c}{ab}) \\ (1 − b) & (1 − \frac{c}{b}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, bq, cq/b, (B) \end{matrix} ; q, qz]

\underline{Usage}: Expr/.C90[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ c/a, b, cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{aq} ) \\ (1 − \frac{c}{ab}) & (\frac{b}{q} − a) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ c/aq, b, cq/b, (B) \end{matrix} ; q, \frac{z}{q} ]

− \begin{matrix}(1 − \frac{c}{b}) & (1 − \frac{b}{q} ) \\ (1 − \frac{c}{ab}) & (\frac{b}{q} − a) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ c/a, b/q, c/b, (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C91[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C92 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ c/a, b, cq/b, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) c (1 − \frac{b}{q} ) q}{ab (1 − \frac{c}{a}) (1 − \frac{cq}{b} )}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ cq/a, b/q, cq^{2}/b, (B) \end{matrix} ; q, z]

+ \begin{matrix}(1 − c) & (1 − \frac{cq}{ab}) \\ (1 − \frac{c}{a}) & (1 − \frac{cq}{b} ) \end{matrix}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, (A) \\ \sqrt{c}, −\sqrt{c}, cq/a, b, cq^{2}/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C92[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C93 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{ab (1 − \frac{c}{a}) (1 − \frac{c}{bq} )}{(1 − b) c (1 − \frac{a}{q})}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ c/a, bq, c/bq, (B) \end{matrix} ; q, z]

− \frac{ab (1 − \frac{c}{ab}) (1 − \frac{c}{q} )}{(1 − b) c (1 − \frac{a}{q})}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}\sqrt{q}, −\sqrt{c}\sqrt{q}, a/q, (A) \\ \sqrt{c}/\sqrt{q}, −\sqrt{c}/\sqrt{q}, cq/a, bq, c/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C93[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ c/a, bq, cq/b, (B) \end{matrix} ; q, z]

+ \frac{(−1)^{1−r+s} (b − a) (1 − \frac{c}{ab}) (1 − cq) z}{(1 − b) (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − bq) (1 − \frac{cq}{a} ) (1 − \frac{cq}{b} )} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−3}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}aq, cq^{2}, (qA) \\ cq^{2}/a, bq^{2}, cq^{2}/b, cq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C94[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C95 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ c/a, b, cq/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ cq/a, b/q, c/b, (B) \end{matrix} ; q, z]

− \frac{(−1)^{1−r+s} (1 − c) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{cq}{ab}) z}{(1 − b) (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{b}{q} ) (1 − \frac{cq}{a} ) (1 − \frac{cq}{b} )} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−3}(1 − B_{i})}_{1+r}ϕ_{1+s}[ \begin{matrix}a, cq, (qA) \\ cq^{2}/a, bq, cq^{2}/b, c, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C95[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C96 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, (A) \\ c/a, b, cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{b}) & (1 − \frac{c}{ad}) & (1 − \frac{b}{q} ) & (1 − \frac{a}{dq} ) \\ (1 − \frac{c}{a}) & (1 − \frac{c}{bd}) & (1 − \frac{a}{q}) & (1 − \frac{b}{dq} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}a/q, (A) \\ cq/a, b/q, c/b, (B) \end{matrix} ; q, z]

+ \frac{(1 − d) (\frac{a}{q} − \frac{b}{q} ) (1 − \frac{c}{dq} ) (1 − \frac{cq}{ab})}{(1 − \frac{c}{a}) (1 − \frac{c}{bd}) d (1 − \frac{a}{q}) (1 − \frac{b}{dq} )}_{2+r}ϕ_{2+s}[ \begin{matrix}a/q, dq, c/d, (A) \\ cq/a, b, cq/b, d, c/dq, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C96[m1,n1,n2,n3,d].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − a) & (1 − \frac{c}{a}) & (1 − \frac{b}{d}) & (1 − \frac{c}{bd}) \\ (1 − b) & (1 − \frac{c}{b}) & (1 − \frac{a}{d}) & (1 − \frac{c}{ad}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ c/a, bq, cq/b, (B) \end{matrix} ; q, z]

− \frac{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}{(1 − b) (1 − \frac{c}{b}) (1 − \frac{a}{d}) (1 − \frac{c}{ad}) d}_{2+r}ϕ_{2+s}[ \begin{matrix}a, dq, cq/d, (A) \\ cq/a, bq, cq/b, d, c/d, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C97[m1,n1,n2,n3,d].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C98 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}ab/q^{2}, (A) \\ a, b, ab/q^{3}, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s} (1 − \frac{a}{q^{2}} ) (1 − \frac{b}{q^{2}} ) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{1−r+s}}{(1 − \frac{ab}{q^{3}}) z} \frac{∏_{i=1}^{s−3}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}(A/q) \\ a/q^{2}, b/q^{2}, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{a}{q^{2}} ) (1 − \frac{b}{q^{2}} ) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{1−r+s}}{(1 − \frac{ab}{q^{3}}) z} \frac{∏_{i=1}^{s−3}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}(A/q) \\ a/q, b/q, (B/q) \end{matrix} ; q, q^{r−s}z]

\underline{Usage}: Expr/.C98[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C99 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, c/a, b, cq^{2}/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(\frac{b}{q} − a) & (1 − \frac{cq}{ab}) \\ (1 − a) & (1 − \frac{c}{a}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, b, cq^{2}/b, (B) \end{matrix} ; q, qz]

+ \begin{matrix}(1 − \frac{b}{q} ) & (1 − \frac{cq}{b} ) \\ (1 − a) & (1 − \frac{c}{a}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, b/q, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C99[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}(A) \\ a, c/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{a}{q}) & (1 − \frac{c}{aq} ) \\ (1 − \frac{c}{ab}) & (−\frac{a}{q} + \frac{b}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, c/aq, b, c/b, (B) \end{matrix} ; q, \frac{z}{q} ]

− \begin{matrix}(1 − \frac{b}{q} ) & (1 − \frac{c}{bq} ) \\ (1 − \frac{c}{ab}) & (−\frac{a}{q} + \frac{b}{q} ) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, c/a, b/q, c/bq, (B) \end{matrix} ; q, \frac{z}{q} ]

\underline{Usage}: Expr/.C100[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C101 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, c/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ \frac{c (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q}{ab (1 − \frac{c}{a}) (1 − \frac{c}{b})}_{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, cq/a, b/q, cq/b, (B) \end{matrix} ; q, z]

+ \begin{matrix}(1 − \frac{c}{q} ) & (1 − \frac{cq}{ab}) \\ (1 − \frac{c}{a}) & (1 − \frac{c}{b}) \end{matrix}_{2+r}ϕ_{2+s}[ \begin{matrix}\sqrt{c}\sqrt{q}, −\sqrt{c}\sqrt{q}, (A) \\ \sqrt{c}/\sqrt{q}, −\sqrt{c}/\sqrt{q}, a, cq/a, b, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C101[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C102 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}(A) \\ a, cq^{2}/a, b, c/b, (B) \end{matrix} ; q, z] ⟶ _{r}ϕ_{s}[ \begin{matrix}(A) \\ a/q, cq/a, bq, cq/b, (B) \end{matrix} ; q, z]

+ \frac{(−1)^{1−r+s} (b − \frac{a}{q}) (1 − cq) (1 − \frac{cq}{ab}) z}{(1 − a) (1 − b) (1 − \frac{c}{b}) (1 − \frac{a}{q}) (1 − bq) (1 − \frac{cq}{a} ) (1 − \frac{cq}{b} ) (1 − \frac{cq^{2}}{a} )} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−4}(1 − B_{i})}

1+rϕ_{1+s}[ \begin{matrix}cq^{2}, (qA) \\ aq, cq^{3}/a, bq^{2}, cq^{2}/b, cq, (qB) \end{matrix} ; q, q^{1−r+s}z]

\underline{Usage}: Expr/.C102[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}(A) \\ a, c/a, b, cq^{2}/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{a}{d}) & (1 − \frac{c}{ad}) & (1 − \frac{b}{q} ) & (1 − \frac{cq}{b} ) \\ (1 − a) & (1 − \frac{c}{a}) & (1 − \frac{b}{dq} ) & (1 − \frac{cq}{bd}) \end{matrix}_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, b/q, cq/b, (B) \end{matrix} ; q, z]

+ \frac{(1 − \frac{c}{d}) (1 − d) (a − \frac{b}{q} ) (1 − \frac{cq}{ab})}{(1 − a) (1 − \frac{c}{a}) d (1 − \frac{b}{dq} ) (1 − \frac{cq}{bd})}_{2+r}ϕ_{2+s}[ \begin{matrix}dq, cq/d, (A) \\ aq, cq/a, b, cq^{2}/b, d, c/d, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C103[n1,n2,n3,n4,d].

n1, n2, n3, n4 are the positions of the special lower parameters, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C104 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, c/b, c, (A) \\ c/q, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s}q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−5}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a/q, c/aq, b, c/b, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s}q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−1}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−5}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a, c/a, b/q, c/bq, (A/q) \\ (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C104[m1,m2,m3,m4,m5,n1].

m1, m2, m3, m4, m5 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C105 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, c, (A) \\ cq/b, c/q, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s} (1 − \frac{c}{b}) (1 − \frac{c}{bq} ) q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−2}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−4}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a/q, c/aq, b, (A/q) \\ c/bq, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{c}{b}) (1 − \frac{c}{bq} ) q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−2}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−4}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a, c/a, b/q, (A/q) \\ c/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C105[m1,m2,m3,m4,n1,n2].

m1, m2, m3, m4 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, b, c, (A) \\ cq/a, cq/b, c/q, (B) \end{matrix} ; q, z] ⟶

\frac{(−1)^{1−r+s} (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{c}{aq} ) (1 − \frac{c}{bq} ) q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−3} 1 − \frac{B_{i}}{q}}{∏_{i=1}^{r−3} 1 − \frac{A_{i}}{q}} _{r−1}ϕ_{s−1}[ \begin{matrix}a/q, b, (A/q) \\ c/a, c/bq, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{c}{aq} ) (1 − \frac{c}{bq} ) q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−3} 1 − \frac{B_{i}}{q}}{∏_{i=1}^{r−3} 1 − \frac{A_{i}}{q}} _{r−1}ϕ_{s−1}[ \begin{matrix}a, b/q, (A/q) \\ c/aq, c/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C106[m1,m2,m3,n1,n2,n3]. m1, m2, m3 and n1, n2, n3 are the positions of the special upper and

lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C107 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, c, (A) \\ bq, cq/b, c/q, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s} (1 − b) (1 − \frac{c}{b}) (1 − \frac{b}{q} ) (1 − \frac{c}{bq} ) q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−3}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−3}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a/q, c/aq, (A/q) \\ b/q, c/bq, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − b) (1 − \frac{c}{b}) (1 − \frac{b}{q} ) (1 − \frac{c}{bq} ) q^{1−r+s}}{(1 − \frac{c}{ab}) (−\frac{a}{q} + \frac{b}{q} ) (1 − \frac{c}{q} ) z} \frac{∏_{i=1}^{s−3}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−3}(1 − \frac{A_{i}}{q})}_{r−1}ϕ_{s−1}[ \begin{matrix}a, c/a, (A/q) \\ b, c/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C107[m1,m2,m3,n1,n2,n3].

m1, m2, m3 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}a, c, (A) \\ cq/a, b, cq^{2}/b, c/q, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s} (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) (1 − \frac{c}{aq} ) q^{1−r+s} (1 − \frac{cq}{b} )}{(\frac{b}{q^{2}} − \frac{a}{q}) (1 − \frac{c}{q} ) (1 − \frac{cq}{ab}) z} \frac{∏_{i=1}^{s−4}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−2}(1 − \frac{A_{i}}{q})}

r−1ϕ_{s−1}[ \begin{matrix}a/q, (A/q) \\ c/a, b/q^{2}, c/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{b}{q^{2}} ) (1 − \frac{b}{q} ) (1 − \frac{c}{aq} ) q^{1−r+s} (1 − \frac{cq}{b} )}{(\frac{b}{q^{2}} − \frac{a}{q}) (1 − \frac{c}{q} ) (1 − \frac{cq}{ab}) z} \frac{∏_{i=1}^{s−4}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−2}(1 − \frac{A_{i}}{q})}

r−1ϕ_{s−1}[ \begin{matrix}a, (A/q) \\ c/aq, b/q, cq/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C108[m1,m2,n1,n2,n3,n4].

m1, m2 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C109 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}c, (A) \\ a, cq^{2}/a, b, cq^{2}/b, c/q, (B) \end{matrix} ; q, z]

⟶ \frac{(−1)^{1−r+s} (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{a}{q^{2}} ) (1 − \frac{b}{q^{2}} ) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{1−r+s} (1 − \frac{cq}{a} ) (1 − \frac{cq}{b} )}{(−\frac{a}{q^{2}} + \frac{b}{q^{2}} ) (1 − \frac{c}{q} ) (1 − \frac{cq^{2}}{ab} ) z} \frac{∏_{i=1}^{s−5}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}

r−1ϕ_{s−1}[ \begin{matrix}(A/q) \\ a/q, cq/a, b/q^{2}, c/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

− \frac{(−1)^{1−r+s} (1 − \frac{c}{a}) (1 − \frac{c}{b}) (1 − \frac{a}{q^{2}} ) (1 − \frac{b}{q^{2}} ) (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{1−r+s} (1 − \frac{cq}{a} ) (1 − \frac{cq}{b} )}{(−\frac{a}{q^{2}} + \frac{b}{q^{2}} ) (1 − \frac{c}{q} ) (1 − \frac{cq^{2}}{ab} ) z} \frac{∏_{i=1}^{s−5}(1 − \frac{B_{i}}{q})}{∏_{i=1}^{r−1}(1 − \frac{A_{i}}{q})}

r−1ϕ_{s−1}[ \begin{matrix}(A/q) \\ a/q^{2}, c/a, b/q, cq/b, (B/q) \end{matrix} ; q, q^{−1+r−s}z]

\underline{Usage}: Expr/.C109[m1,n1,n2,n3,n4,n5].

m1 and n1, n2, n3, n4, n5 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

54 Alphabetic List: C113

C110 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, c/a, b, c/b, (A) \\ \sqrt{c}, −\sqrt{c}, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{a}) & (1 − \frac{c}{b}) \\ (1 − c) & (1 − \frac{c}{ab}) \end{matrix}_{r−2}ϕ_{s−2}[ \begin{matrix}a, cq/a, b, cq/b, (A) \\ (B) \end{matrix} ; q, z]

− \frac{(1 − a) (1 − b) c}{ab (1 − c) (1 − \frac{c}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, c/a, bq, c/b, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C110[m1,m2,m3,m4,m5,m6,n1,n2].

m1, m2, m3, m4, m5, m6 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C111 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, c/b, dq, cq/d, (A) \\ d, c/d, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − \frac{c}{a}) (1 − \frac{b}{d}) (1 − \frac{c}{bd}) d}{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, cq/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z]

− \frac{(1 − b) (1 − \frac{c}{b}) (1 − \frac{a}{d}) (1 − \frac{c}{ad}) d}{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}_{r−2}ϕ_{s−2}[ \begin{matrix}a, c/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C111[m1,m2,m3,m4,m5,m6,n1,n2].

m1, m2, m3, m4, m5, m6 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C112 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, c/a, b, (A) \\ \sqrt{c}, −\sqrt{c}, cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{a}) & (1 − \frac{c}{b}) \\ (1 − c) & (1 − \frac{c}{ab}) \end{matrix}_{r−2}ϕ_{s−2}[ \begin{matrix}a, cq/a, b, (A) \\ c/b, (B) \end{matrix} ; q, z]

− \frac{(1 − a) (1 − b) c}{ab (1 − c) (1 − \frac{c}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, c/a, bq, (A) \\ cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C112[m1,m2,m3,m4,m5,n1,n2,n3].

m1, m2, m3, m4, m5 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C113 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, b, dq, cq/d, (A) \\ cq/b, d, c/d, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − \frac{c}{a}) (1 − \frac{b}{d}) (1 − \frac{c}{bd}) d}{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, cq/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z]

− \frac{(1 − b) (1 − \frac{c}{b}) (1 − \frac{a}{d}) (1 − \frac{c}{ad}) d}{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}_{r−2}ϕ_{s−2}[ \begin{matrix}a, c/a, bq, (A) \\ c/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C113[m1,m2,m3,m4,m5,n1,n2,n3].

m1, m2, m3, m4, m5 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

Alphabetic List: C117 55

C114 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, b, (A) \\ \sqrt{c}, −\sqrt{c}, cq/a, cq/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{a}) & (1 − \frac{c}{b}) \\ (1 − c) & (1 − \frac{c}{ab}) \end{matrix}_{r−2}ϕ_{s−2}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z]

− \frac{(1 − a) (1 − b) c}{ab (1 − c) (1 − \frac{c}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, bq, (A) \\ cq/a, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C114[m1,m2,m3,m4,n1,n2,n3,n4].

m1, m2, m3, m4 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C115 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, c/a, (A) \\ \sqrt{c}, −\sqrt{c}, b, cq^{2}/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{a}) & (1 − \frac{cq}{b} ) \\ (1 − c) & (1 − \frac{cq}{ab}) \end{matrix}_{r−2}ϕ_{s−2}[ \begin{matrix}a, cq/a, (A) \\ b, cq/b, (B) \end{matrix} ; q, z]

− \frac{(1 − a) c (1 − \frac{b}{q} ) q}{ab (1 − c) (1 − \frac{cq}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, c/a, (A) \\ b/q, cq^{2}/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C115[m1,m2,m3,m4,n1,n2,n3,n4].

m1, m2, m3, m4 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C116 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, b, dq, cq/d, (A) \\ cq/a, cq/b, d, c/d, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − \frac{c}{a}) (1 − \frac{b}{d}) (1 − \frac{c}{bd}) d}{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, b, (A) \\ c/a, cq/b, (B) \end{matrix} ; q, z]

− \frac{(1 − b) (1 − \frac{c}{b}) (1 − \frac{a}{d}) (1 − \frac{c}{ad}) d}{(a − b) (1 − \frac{c}{ab}) (1 − \frac{c}{d}) (1 − d)}_{r−2}ϕ_{s−2}[ \begin{matrix}a, bq, (A) \\ cq/a, c/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C116[m1,m2,m3,m4,n1,n2,n3,n4].

m1, m2, m3, m4 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C117 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, c/a, dq, cq/d, (A) \\ b, cq^{2}/b, d, c/d, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − \frac{c}{a}) d (1 − \frac{b}{dq} ) (1 − \frac{cq}{bd})}{(1 − \frac{c}{d}) (1 − d) (a − \frac{b}{q} ) (1 − \frac{cq}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, cq/a, (A) \\ b, cq^{2}/b, (B) \end{matrix} ; q, z]

− \frac{(1 − \frac{a}{d}) (1 − \frac{c}{ad}) d (1 − \frac{b}{q} ) (1 − \frac{cq}{b} )}{(1 − \frac{c}{d}) (1 − d) (a − \frac{b}{q} ) (1 − \frac{cq}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}a, c/a, (A) \\ b/q, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C117[m1,m2,m3,m4,n1,n2,n3,n4].

m1, m2, m3, m4 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, a, (A) \\ \sqrt{c}, −\sqrt{c}, cq/a, b, cq^{2}/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{c}{a}) & (1 − \frac{cq}{b} ) \\ (1 − c) & (1 − \frac{cq}{ab}) \end{matrix}_{r−2}ϕ_{s−2}[ \begin{matrix}a, (A) \\ c/a, b, cq/b, (B) \end{matrix} ; q, z]

− \frac{(1 − a) c (1 − \frac{b}{q} ) q}{ab (1 − c) (1 − \frac{cq}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, (A) \\ cq/a, b/q, cq^{2}/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C118[m1,m2,m3,n1,n2,n3,n4,n5].

m1, m2, m3 and n1, n2, n3, n4, n5 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C119 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}a, dq, cq/d, (A) \\ cq/a, b, cq^{2}/b, d, c/d, (B) \end{matrix} ; q, z] ⟶ \frac{(1 − a) (1 − \frac{c}{a}) d (1 − \frac{b}{dq} ) (1 − \frac{cq}{bd})}{(1 − \frac{c}{d}) (1 − d) (a − \frac{b}{q} ) (1 − \frac{cq}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}aq, (A) \\ c/a, b, cq^{2}/b, (B) \end{matrix} ; q, z]

− \frac{(1 − \frac{a}{d}) (1 − \frac{c}{ad}) d (1 − \frac{b}{q} ) (1 − \frac{cq}{b} )}{(1 − \frac{c}{d}) (1 − d) (a − \frac{b}{q} ) (1 − \frac{cq}{ab})}_{r−2}ϕ_{s−2}[ \begin{matrix}a, (A) \\ cq/a, b/q, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C119[m1,m2,m3,n1,n2,n3,n4,n5].

m1, m2, m3 and n1, n2, n3, n4, n5 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C120 \underline{Description}: Contiguous relation in form of a rule.

rϕ_{s}[ \begin{matrix}\sqrt{c}q, −\sqrt{c}q, (A) \\ \sqrt{c}, −\sqrt{c}, a, cq^{2}/a, b, cq^{2}/b, (B) \end{matrix} ; q, z] ⟶ \begin{matrix}(1 − \frac{cq}{a} ) & (1 − \frac{cq}{b} ) \\ (1 − c) & (1 − \frac{cq^{2}}{ab} ) \end{matrix}_{r−2}ϕ_{s−2}[ \begin{matrix}(A) \\ a, cq/a, b, cq/b, (B) \end{matrix} ; q, z]

− \frac{c (1 − \frac{a}{q}) (1 − \frac{b}{q} ) q^{2}}{ab (1 − c) (1 − \frac{cq^{2}}{ab} )}_{r−2}ϕ_{s−2}[ \begin{matrix}(A) \\ a/q, cq^{2}/a, b/q, cq^{2}/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C120[m1,m2,n1,n2,n3,n4,n5,n6].

m1, m2 and n1, n2, n3, n4, n5, n6 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rϕ_{s}[ \begin{matrix}dq, cq/d, (A) \\ a, cq^{2}/a, b, cq^{2}/b, d, c/d, (B) \end{matrix} ; q, z]

⟶ \frac{d (1 − \frac{a}{q}) (1 − \frac{b}{dq} ) (1 − \frac{cq}{a} ) (1 − \frac{cq}{bd})}{(1 − \frac{c}{d}) (1 − d) (\frac{a}{q} − \frac{b}{q} ) (1 − \frac{cq^{2}}{ab} )}_{r−2}ϕ_{s−2}[ \begin{matrix}(A) \\ a/q, cq/a, b, cq^{2}/b, (B) \end{matrix} ; q, z]

− \frac{d (1 − \frac{b}{q} ) (1 − \frac{a}{dq} ) (1 − \frac{cq}{b} ) (1 − \frac{cq}{ad})}{(1 − \frac{c}{d}) (1 − d) (\frac{a}{q} − \frac{b}{q} ) (1 − \frac{cq^{2}}{ab} )}_{r−2}ϕ_{s−2}[ \begin{matrix}(A) \\ a, cq^{2}/a, b/q, cq/b, (B) \end{matrix} ; q, z]

\underline{Usage}: Expr/.C121[m1,m2,n1,n2,n3,n4,n5,n6].

m1, m2 and n1, n2, n3, n4, n5, n6 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

ContigListe \underline{Description}: List of all contiguous relations.

\underline{Usage}: ContigListe.

Div \underline{Description}: Function that divides Gleichung by Expr.

\underline{Usage}: Div[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= Div[a^{∧}n]
                 -n
         ϕ a, q  ; q, q       c
        2 1                    (-; q)
               c                a    n
Out[2]= ------------------- == -------
                 n             (c; q)
                a                    n

In[3]:= Gleichung

                 -n
         ϕ a, q  ; q, q       c
        2 1                    (-; q)
               c                a    n
Out[3]= ------------------- == -------
                 n             (c; q)
                a                    n

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Sub, Hoch, GlTausche, Ers.

Drucke

\underline{Description}: Function that directly sends an expression Expr in the Form PrintedForm to the printer. This function only works for DOS-machines with a printer directly connected. PrintedForm is an optional parameter which can be any of the format types (InputForm, OutputForm, TeXForm, …). The default is OutputForm.

\underline{Usage}: Drucke[Expr,PrintedForm].
\underline{See also}: TeXMat, TeX, TeXphW.

Ers

\underline{Description}: Function for controlled application of rules and functions.

\underline{Usage}: Ers[Expr,Rules,PosList].

Rules can be a rule, a list of rules, or a function. PosList must be a list of positions in Expr to which Rules should be applied. For instance, if PosList={{1, 2}, {4}}, then Rules is applied to Expr[[1,2]] and Expr[[4]] in Expr. If PosList={2, 3}, then Rules is applied to Expr[[2]] and Expr[[3]] in Expr. The positions of subexpressions can be determined by the function PosListe. If Ers is applied to an equation then the new left-hand and right-hand sides are automatically stored in the variables LS and RS.

There is an exceptional usage of Ers, namely

       Ers[Rules].

In this case the Rules are applied to both sides of the equation that is currently stored in Gleichung. Again, the new left-hand and right-hand sides are automatically stored in the variables LS and RS.

\underline{Example(s)}:

In[1]:= (-1)^{∧}n*pq[a*q,n]*pq[c,k]/pq[1/b,m]/pq[q/d,l]
            n
        (-1)  (c; q)  (a q; q)
                    k         n
Out[1]= -----------------------
             1       q
            (-; q)  (-; q)
             b    m  d    l
In[2]:= PosListe[%]

              n              1
Out[2]= {{(-1) , {{1}}}, {-------, {{2}}}, {(c; q) , {{3}}},
                           1                      k
                          (-; q)
                           b    m
                             1
⟩  {(a q; q) , {{4}}}, {-------, {{5}}}}
             n            q
                         (-; q)
                          d    l
In[3]:= Ers[%%,neg1,{5}]
                                  1 + l
            n                    q
        (-1)  (c; q)  (a q; q)  (------; q)
                    k         n    d       -l
Out[3]= -------------------------------------
                        1
                       (-; q)
                        b    m
In[4]:= Ers[%%%,neg1,{2,4}]
                        m
            n          q
        (-1)  (c; q)  (--; q)
                    k  b     -m
Out[4]= -----------------------
         q          1 + n
        (-; q)  (a q     ; q)
         d    l              -n
In[5]:= SUM[%,{k,0,Infinity}]
                                       1 + l
          ∞  n                    q
             (-1)  (c; q)  (a q; q)  (------; q)
         \  k         n    d       -l
Out[5]= ⟩ -------------------------------------
         /                   1
                            (-; q)
         k=0                 b    m
In[6]:= PosListe[%,2]

                             n
Out[6]= {{0, {{2, 2}}}, {(-1) , {{1, 1}}}, {k, {{2, 1}}}, {∞, {{2, 3}}},
         1
⟩  {-------, {{1, 2}}}, {(c; q) , {{1, 3}}}, {(a q; q) , {{1, 4}}},
       1                         k                      n
      (-; q)
       b    m
        1 + l
       q
⟩  {(------; q)  , {{1, 5}}}}
         d       -l
In[7]:= Ers[%%,neg1,{{1,3}}]
                               1 + l
          ∞  n            q
             (-1)  (a q; q)  (------; q)
         \  n    d       -l
Out[7]= ⟩ -----------------------------
         /         1          k
                  (-; q)  (c q ; q)
         k=0       b    m          -k
In[8]:= Ers[%%%,neg1,{{1,2},{1,5}}]
                                       m
          ∞  n                    q
             (-1)  (c; q)  (a q; q)  (--; q)
         \  k         n  b     -m
Out[8]= ⟩ ---------------------------------
         /                 q
                          (-; q)
         k=0               d    l
In[9]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[9]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[10]:= Ers[a-⟩q/a]

                                 n  a c
              q   -n            q  (---; q)
              -, q                   q     n
Out[10]= ϕ a     ; q, q   == ------------
         2 1                      n
                c                a  (c; q)
                                          n
In[11]:= Gleichung
                                 n  a c
              q   -n            q  (---; q)
              -, q                   q     n
Out[11]= ϕ a     ; q, q   == ------------
         2 1                      n
                c                a  (c; q)
                                          n
In[12]:= PQ
In[13]:= pq[a,4]+1/pq[b,3]
                      1                                           2          3
Out[13]= ---------------------------- + (1 - a) (1 - a q) (1 - a q ) (1 - a q )
                                   2
         (1 - b) (1 - b q) (1 - b q )
In[14]:= Ers[%,Expand,{2}]
                        2        2    2  2      3      2  3    3  3    2  4
Out[14]= 1 - a - a q + a  q - a q  + a  q  - a q  + 2 a  q  - a  q  + a  q  -
      3  4    2  5    3  5    3  6    4  6                1
⟩  a  q  + a  q  - a  q  - a  q  + a  q  + ----------------------------
                                                                       2
                                             (1 - b) (1 - b q) (1 - b q )
\underline{See also}: PosListe, ManipulationsListe, Subst.

erw1

\underline{Description}: \begin{matrix}(a; q)_{n} → (a; q)_{m+n}/(aq^{n}; q)_{m}, \\ (a; q)_{n} → (a; q)_{∞}/(aq^{n}; q)_{∞}. \end{matrix}

The parameter m has to be entered on request. To apply the second rule, m has to be Infinity.

\underline{Usage}: Expr/.erw1.

\underline{Example(s)}:

In[1]:= pq[a,n,q^{∧}2]

             2
Out[1]= (a; q )
               n
In[2]:= %/.erw1
top-extend by:  m
             2
        (a; q )
               m + n
Out[2]= -------------
            2 n   2
        (a q   ; q )
                    m
In[3]:= pq[a,m,1/q]
            1
Out[3]= (a; -)
            q m
In[4]:= %/.erw1
top-extend by:  Infinity
           1
        (a;-)
           q ∞
Out[4]= -------
         a  1
        (--;-)
          m q ∞
         q
\underline{See also}: erw2, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

erw2

\underline{Description}: \begin{matrix}(a; q)_{n} → (a/q^{m}; q)_{m+n}/(a/q^{m}; q)_{m}, \\ (a; q)_{∞} → (a/q^{m}; q)_{∞}/(a/q^{m}; q)_{m}. \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.erw2.

\underline{Example(s)}:

In[1]:= pq[a,n,q^{∧}2]
             2
Out[1]= (a; q )
               n

In[2]:= %/.erw2
bottom-extend by:  m
          a     2
        (----; q )
          2 m     m + n
         q
Out[2]= ---------------
            a     2
          (----; q )
            2 m     m
           q
In[3]:= pqinf[b,1/q]
           1
Out[3]= (b;-)
           q ∞
In[4]:= %/.erw2
bottom-extend by:  n
            n 1
        (b q ;-)
              q ∞
Out[4]= ----------
            n  1
        (b q ; -)
               q n
\underline{See also}: erw1, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

Expandq

\underline{Description}: Rule that expands all the exponents in powers.

\underline{Usage}: Expr/.Expandq.

\underline{Example(s)}:

In[1]:= pq[a*q^{∧}3,n]/.trans
                              2      -2 - n
            n  n  3 n - (n - n )/2  q
Out[1]= (-1)  a  q                 (-------; q)
                                       a       n
In[2]:= %/.Expandq

                             2     -2 - n
            n  n  (5 n)/2 + n /2  q
Out[2]= (-1)  a  q               (-------; q)
                                     a       n
\underline{See also}: SimplifyPQ, MinusOne, SUMExpand, Ers, PosListe.

Factorialpq

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

\underline{Usage}: Factorialpq[n,q]
   or: Factorialpq[n].

\underline{Example(s)}:

In[1]:= Factorialpq[n]
        (q; q)
              n
Out[1]= --------
               n
        (1 - q)
In[2]:= Factorialpq[5]
        (q; q)
              5
Out[2]= --------
               5
        (1 - q)
\underline{See also}: Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialq.

Factorialq

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

\underline{Usage}: Factorialq[n,q]
   or: Factorialq[n].

\underline{Example(s)}:

In[1]:= Factorialq[n]

Out[1]=   n   !
             q
In[2]:= Factorialq[5]

                     2       3       4       5       6       7      8
Out[2]= 1 + 4 q + 9 q  + 15 q  + 20 q  + 22 q  + 20 q  + 15 q  + 9 q  +
        9    10
⟩  4 q  + q
\underline{See also}: Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialpq.

Gleichung

\underline{Description}: Is a variable which stores equations. The equation Gleichung can be manipulated using the functions Add, Sub, Mal, Div, Hoch, GlTausche, Ers, and SUM[k,m,n], where m and n are integers or variables. The last command causes the equation to be summed over k from m to n. The parameter k is optional. It will be set kk, ii, jj, ll, mm, or nn, if it is omitted.

\underline{Usage}: Gleichung.

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= Gleichung
                                n  c
                               a  (-; q)
                 -n                a    n
Out[2]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[3]:= Add[1]
                                        n  c
                                       a  (-; q)
                     -n                    a    n
Out[3]= 1 + ϕ a, q  ; q, q   == 1 + ----------
            2 1                         (c; q)
                   c                          n

                                        n  c
                                       a  (-; q)
                     -n                    a    n
Out[4]= 1 + ϕ a, q  ; q, q   == 1 + ----------
            2 1                         (c; q)
                   c                          n
In[5]:= LS=5
Out[5]= 5
In[6]:= Gleichung
                  n  c
                 a  (-; q)
                     a    n
Out[6]= 5 == 1 + ----------
                  (c; q)
                        n
In[7]:= SUM[n,0,m]
          m         m        m   n  c
                                a  (-; q)

\  \  \  a    n

Out[7]= ⟩ 5 == ⟩ 1 + ⟩ ----------
         /         /        /    (c; q)
                                       n
         n=0       n=0      n=0
In[8]:= SUM[0,M]
          M    m         M    m        M    m   n  c
                                               a  (-; q)

\  \  \  \  \  \  a    n

Out[8]= ⟩  ⟩ 5 == ⟩  ⟩ 1 + ⟩  ⟩ ----------
         /    /         /    /        /    /    (c; q)
                                                      n
        kk=0  n=0      kk=0  n=0     kk=0  n=0

          M    m         M    m        M    m   n  c
                                               a  (-; q)

\  \  \  \  \  \  a    n

Out[9]= ⟩  ⟩ 5 == ⟩  ⟩ 1 + ⟩  ⟩ ----------
         /    /         /    /        /    /    (c; q)
                                                      n
        kk=0  n=0      kk=0  n=0     kk=0  n=0
\underline{See also}:  SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers, Subst,
  PQSort.

GlTausche

\underline{Description}: GlTausche interchanges right-hand and left-hand sides in Gleichung.

\underline{Usage}: GlTausche.

\underline{Example(s)}:

In[1]:= Sgl3201
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                      -n                c       c
               a, b, q                 (-; q)  (-; q)
                                        a    n  b    n
Out[1]= ϕ  1 - n; q, q   == -----------------
        3 2     a b q                           c
             c, ----------            (c; q)  (---; q)
                    c                       n  a b    n
In[2]:= Gleichung
                      -n                c       c
               a, b, q                 (-; q)  (-; q)
                                        a    n  b    n
Out[2]= ϕ  1 - n; q, q   == -----------------
        3 2     a b q                           c
             c, ----------            (c; q)  (---; q)
                    c                       n  a b    n
In[3]:= GlTausche
          c       c                        -n
         (-; q)  (-; q)             a, b, q
          a    n  b    n
Out[3]= ----------------- == ϕ  1 - n; q, q
                  c          3 2     a b q
        (c; q)  (---; q)          c, ----------
              n  a b    n                c

In[4]:= Gleichung
          c       c                        -n
         (-; q)  (-; q)             a, b, q
          a    n  b    n
Out[4]= ----------------- == ϕ  1 - n; q, q
                  c          3 2     a b q
        (c; q)  (---; q)          c, ----------
              n  a b    n                c

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Sub, Hoch, Ers, Subst.

Hoch

\underline{Description}: Function that takes Gleichung to the Expr-th power.

\underline{Usage}: Hoch[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= Hoch[3]
                                 3 n  c     3
                                a    (-; q)
                 -n        3          a    n
Out[2]= ϕ a, q  ; q, q    == -------------
        2 1                              3
               c                  (c; q)
                                        n
In[3]:= Gleichung

                                 3 n  c     3
                                a    (-; q)
                 -n        3          a    n
Out[3]= ϕ a, q  ; q, q    == -------------
        2 1                              3
               c                  (c; q)
                                        n

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Sub, GlTausche, Ers.

hypqAttributes

\underline{Description}: Shows the current setup of the session. The setup can be changed by the switches PQ, phCancel, TeX, and TeXphW. The default-setup is shown in the following Example.

\underline{Usage}: hypqAttributes.

\underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.

\underline{See also}: PQ, phCancel, TeX, TeXphW.

inv1

\underline{Description}: (a; q)_{n} → (1/a; 1/q)_{n}(−a)^{n}q^{\binom{n}{2}}.

\underline{Usage}: Expr/.inv1.

\underline{Example(s)}:

In[1]:= pq[a^{∧}2,n]
          2
Out[1]= (a ; q)
               n
In[2]:= %/.inv1

            n  2 n   -2  1
        (-1)  a    (a  ; -)
                         q n
Out[2]= --------------------
                   2
             (n - n )/2
            q
\underline{See also}: inv2, Ers, PosListe, ManipulationsListe.

inv2

\underline{Description}: (a; q)_{n} → (aq^{n−1}; 1/q)_{n}.

\underline{Usage}: Expr/.inv2.

\underline{Example(s)}:

In[1]:= pq[a^{∧}2,n]
          2
Out[1]= (a ; q)
               n
In[2]:= %/.inv2
          2  -1 + n  1
Out[2]= (a  q      ; -)
                     q n
\underline{See also}: inv1, Ers, PosListe, ManipulationsListe.

LaTeX

\underline{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 AMS-TEX.

\underline{Usage}: LaTeX.

\underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AMSTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[2]:= TeXForm[ph[{a,b},{c},q,z]]
Out[2]//TeXForm=
{} {2} \phi {1} \! \left [ \matrix \let \over / a, b\\ \let \over /
 c\endmatrix ;q, {\displaystyle z} \right ]

In[3]:= LaTeX
In[4]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[5]:= TeXForm[ph[{a,b},{c},q,z]]
Out[5]//TeXForm=
{} {2} \phi {1} \! \left [ \matrix { \let \over / a, b\cr \let \over / c}
 ;q, {\displaystyle z} \right ]

\underline{See also}: AmSTeX, AmSLaTeX, TeX, TeXMat, TeXphW.

Limes

\underline{Description}: Function for doing formal limits of basic hypergeometric expressions. If required for taking the limit, you will be asked whether or not the absolute value of some variable or expression is smaller than 1. You will be offered three options, [y|n|u]. If the absolute value of the variable is smaller than 1 then enter y, if it is greater than 1 then enter n, if you do not want to make an explicit declaration then enter 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 AbsGreater, AbsSmaller, or AbsUndetermined, respectively. By default the absolute value of q is defined to be smaller than 1. Also this can be changed by AbsGreater, AbsSmaller, or AbsUndetermined, respectively.

If you want to let a base q tend to 1, then you should also have the basic file hyp.m of the MATHEMATICA package HYP in your MAHEMATICA input directory, since in these situations the file hyp.m will be loaded automatically. The $ContextPath (cf. [4]) is set that objects of the package HYP do not override the respective objects of the package HYPQ. The situation is just as if you would have loaded hyp.m first and then hyp.q. Concerning a simultaneous use of the packages HYPQ and HYP confer the section Simultaneous use of HYP and HYPQ at the beginning of this handbook.

Hint: When using Limes you sometimes have to do a little bit of “preparation”. For instance, when applying Limes[ …,Regel] to expressions of the form (a; q)_{n}, where a and n tend to ∞ under Regel, then the package will be unable to do it, unless at least 1/aq^{n} tends to a definite limit. Hence, you will sometimes have to use zerl before being able to apply Limes (see the last example).

\underline{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.

\underline{Usage}: Limes[Expr, x->x0].

\underline{Example(s)}:

The derivation of the identity [1, (2.7.6)] that leads to the Rogers–Ramanujan identities, starting from Watson’s transformation Tgl8702.

In[1]:= Tgl8702
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n

                                       -n        2  2 + n
a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e, q         a  q
Out[1]= ϕ  ; q, ---------
8 7                     a q  a q  a q  a q     1 + n      b c d e
Sqrt[a], -Sqrt[a], ---, ---, ---, ---, a q
                b    c    d    e
a q         -n               a q
---, d, e, q           (a q, ---; q)
b c                          d e    n
⟩  == ϕ  ; q, q   --------------
4 3  a q  a q  d e           a q  a q
---, ---, ----         (---, ---; q)
 b    c      n           d    e     n
          a q
In[2]:= Limes[%,b-⟩Infinity]
                                    -n         2  2 + n
a, Sqrt[a] q, -(Sqrt[a] q), c, d, e, q          a  q
Out[2]= ϕ  ; q, ---------   ==
7 7                        a q  a q  a q     1 + n       c d e
Sqrt[a], -Sqrt[a], 0, ---, ---, ---, a q
                   c    d    e
-n
d, e, q
                      a q
ϕ a q  d e ; q, q   (a q; q)  (---; q)
3 2  ---, ----                 n  d e    n
c      n
a q
⟩  ------------------------------------------
a q       a q
(---; q)  (---; q)
 d     n   e     n
In[3]:= Limes[%,c-⟩Infinity]
                                  -n         2  2 + n
a, Sqrt[a] q, -(Sqrt[a] q), d, e, q          a  q
Out[3]= ϕ  ; q, ---------   ==
6 7                           a q  a q     1 + n        d e
Sqrt[a], -Sqrt[a], 0, 0, ---, ---, a q
                      d    e

                 -n
          d, e, q
                                       a q
      ϕ  d e  ; q, q   (a q; q)  (---; q)
     3 2   0, ----                  n  d e    n
                 n
              a q
⟩  ------------------------------------------
                 a q       a q
                (---; q)  (---; q)
                  d     n   e     n
In[4]:= Limes[%,d-⟩Infinity]
                                                -n          2  2 + n
                a, Sqrt[a] q, -(Sqrt[a] q), e, q           a  q
Out[4]= ϕ  ; q, ---------   ==
        5 7                              a q     1 + n         e
             Sqrt[a], -Sqrt[a], 0, 0, 0, ---, a q
                                          e
                        1 + n
              -n     a q
      ϕ e, q  ; q, --------   (a q; q)
     2 1                e               n
            0
⟩  ------------------------------------
                   a q
                  (---; q)
                    e     n
In[5]:= Limes[%,e-⟩Infinity]
                                              -n
                 a, Sqrt[a] q, -(Sqrt[a] q), q            2  2 + n
Out[5]= ϕ  ; q, a  q        ==
        4 7                                    1 + n
             Sqrt[a], -Sqrt[a], 0, 0, 0, 0, a q

           -n        1 + n
⟩  ϕ q  ; q, a q        (a q; q)
     1 1                             n
           0
In[6]:= Limes[%,n-⟩Infinity]

                a, Sqrt[a] q, -(Sqrt[a] q)         2  2
Out[6]= ϕ  ; q, a  q    ==
        3 7  Sqrt[a], -Sqrt[a], 0, 0, 0, 0, 0

                   -
⟩  (a q;q) ϕ  ; q, a q
           ∞ 0 1  0

In[7]:= %/.phSUM/.SUMRegeln

∞

                                         2
         \  kk  2 kk  -kk/2 + (5 kk )/2
Out[7]= ⟩ ((-1)   a     q                  (a; q)   (-(Sqrt[a] q); q)
         /                                          kk                  kk
        kk=0
⟩  (Sqrt[a] q; q)  ) / ((-Sqrt[a]; q)   (Sqrt[a]; q)   (q; q)  ) ==
                        kk                  kk             kk       kk
                ∞  2
                     kk  kk
                \ a   q
⟩  (a q;q)  ( ⟩ --------)
           ∞ /   (q; q)
                          kk
               kk=0
In[8]:= PosListe[%,3]
                                kk                  2 kk
Out[8]= {{0, {{1, 2, 2}}}, {(-1)  , {{1, 1, 1}}}, {a    , {{1, 1, 2}}},
⟩  {kk, {{1, 2, 1}}}, {q, {{2, 1, 2}}}, {a q, {{2, 1, 1}}},
                    2
       -kk/2 + (5 kk )/2
⟩  {q                 , {{1, 1, 3}}}, {∞, {{1, 2, 3}}},
                                        1
⟩  {{kk, 0, ∞}, {{2, 2, 2}}}, {---------------, {{1, 1, 4}}},
                                 (-Sqrt[a]; q)
                                              kk

Alphabetic List: Limes 75

            1
⟩  {--------------, {{1, 1, 5}}}, {(a; q)  , {{1, 1, 6}}},
      (Sqrt[a]; q)                         kk
                  kk
                                      2
                                kk  kk
         1                     a   q
⟩  {--------, {{1, 1, 7}}}, {--------, {{2, 2, 1}}},
      (q; q)                   (q; q)
            kk                       kk
⟩  {(-(Sqrt[a] q); q)  , {{1, 1, 8}}}, {(Sqrt[a] q; q)  , {{1, 1, 9}}}}
                       kk                               kk
In[9]:= Ers[%%,lina1,{{1,1,4},{1,1,5}}]

∞

                                         2
         \  kk  2 kk  -kk/2 + (5 kk )/2
Out[9]= ⟩ ((-1)   a     q                  (a; q)   (-(Sqrt[a] q); q)
         /                                          kk                  kk
        kk=0
⟩  (Sqrt[a] q; q)  ) /
                        kk
⟩  ((1 - Sqrt[a]) (1 + Sqrt[a]) (q; q)   (-(Sqrt[a] q); q)
                                           kk                  -1 + kk
                                               ∞  2
                                                    kk  kk
                                               \ a   q
⟩  (Sqrt[a] q; q)       ) == (a q;q)  ( ⟩ --------)
                        -1 + kk  ∞ /   (q; q)
                                                         kk
                                              kk=0
In[10]:= PosListe[%,3]
                                 kk
Out[10]= {{0, {{1, 2, 2}}}, {(-1)  , {{1, 1, 1}}},
           1                           1
⟩  {-----------, {{1, 1, 2}}}, {-----------, {{1, 1, 3}}},
      1 - Sqrt[a]                 1 + Sqrt[a]

       2 kk
⟩  {a    , {{1, 1, 4}}}, {kk, {{1, 2, 1}}}, {q, {{2, 1, 2}}},
                                        2
                           -kk/2 + (5 kk )/2
⟩  {a q, {{2, 1, 1}}}, {q                 , {{1, 1, 5}}}, {∞, {{1, 2, 3}}},
⟩  {{kk, 0, ∞}, {{2, 2, 2}}}, {(a; q)  , {{1, 1, 6}}},
                                       kk
                                      2
                                kk  kk
         1                     a   q
⟩  {--------, {{1, 1, 7}}}, {--------, {{2, 2, 1}}},
      (q; q)                   (q; q)
            kk                       kk
                 1
⟩  {------------------------, {{1, 1, 8}}},
      (-(Sqrt[a] q); q)
                       -1 + kk
⟩  {(-(Sqrt[a] q); q)  , {{1, 1, 9}}},
                       kk
                1
⟩  {---------------------, {{1, 1, 10}}}, {(Sqrt[a] q; q)  , {{1, 1, 11}}}}
      (Sqrt[a] q; q)                                       kk
                    -1 + kk
In[11]:= Ers[%%,lina2,{{1,1,9},{1,1,11}}]

∞

                                          2
          \  kk  2 kk  -kk/2 + (5 kk )/2               kk
Out[11]= ⟩ ((-1)   a     q                  (1 - Sqrt[a] q  )
          /
         kk=0
                        kk
⟩  (1 + Sqrt[a] q  ) (a; q)  ) / ((1 - Sqrt[a]) (1 + Sqrt[a]) (q; q)  )
                                  kk                                       kk

                      ∞  2

                          kk  kk
                     \ a   q
⟩  == (a q;q)  ( ⟩ --------)
                 ∞ /   (q; q)
                               kk
                    kk=0
In[12]:= Ers[%,ExpandAll,{{1,1}}]
                                 2
           ∞  kk  2 kk  (5 kk )/2
              (-1)   a     q          (a; q)
          \  kk
Out[12]= ⟩ --------------------------------- -
          /    kk/2               kk/2
              q     (q; q)   - a q     (q; q)
         kk=0             kk                 kk
                                      2
            kk  1 + 2 kk  2 kk + (5 kk )/2
        (-1)   a         q                 (a; q)
                                                 kk
⟩  ------------------------------------------- ==
              kk/2               kk/2
             q     (q; q)   - a q     (q; q)
                         kk                 kk

                 ∞  2

                     kk  kk
                \ a   q
⟩  (a q;q)  ( ⟩ --------)
            ∞ /   (q; q)
                          kk
               kk=0
In[13]:= Ers[%,Factor,{{1,1}}]
                                         2
           ∞  kk  2 kk  -kk/2 + (5 kk )/2          2 kk
              (-1)   a     q                  (-1 + a q    ) (a; q)
          \  kk
Out[13]= ⟩ ------------------------------------------------------- ==
          /                      (-1 + a) (q; q)
                                                kk
         kk=0

                 ∞  2

                     kk  kk
                \ a   q
⟩  (a q;q)  ( ⟩ --------)
            ∞ /   (q; q)
                          kk
               kk=0

Examples for letting the base q tend to 1.

In[1]:= Sgl3201
Do you want to set values for the equation?  [y|n]:  y
a=q^{∧}a
b=q^{∧}b
c=q^{∧}c
n=n
Do you want to set a value for q in the equation?  [y|n]:  n
                                                 -a + c        -b + c
                   a   b   -n                  (q      ; q)  (q      ; q)
                  q , q , q                                n             n
Out[1]= ϕ  ; q, q   == ---------------------------
        3 2   c   1 + a + b - c - n              c        -a - b + c
             q , q                             (q ; q)  (q          ; q)
                                                      n                 n
In[2]:= Limes[%,q-⟩1]
                                          (-a + c)  (-b + c)
                   a, b, -n                       n         n
Out[2]=  F                       ; 1   == -------------------
        3 2  c, 1 + a + b - c - n         (c)  (-a - b + c)
                                             n             n
In[3]:= Sgl2103
Do you want to set values for the equation?  [y|n]:  y
a=q^{∧}a
b=q^{∧}b
c=q^{∧}c
Do you want to set a value for q in the equation?  [y|n]:  n
                                           -a + c   -b + c
              a   b                      (q      , q      ; q)
             q , q       -a - b + c  ∞
Out[3]= ϕ  ; q, q             == ----------------------
        2 1     c                          c   -a - b + c
               q                         (q , q          ; q)

∞

In[4]:= Limes[%,q-⟩1]

             a, b  Γ(c) Γ(-a - b + c)
Out[4]=  F       ; 1   == -------------------
        2 1   c  Γ(-a + c) Γ(-b + c)

An example that should illustrate the hint in the Description of Limes.

In[1]:= pq[a/q^{∧}n,2*n]/a^{∧}n*(-1)^{∧}n*q^{∧}Binomial[n+1,2]
            n  (n (1 + n))/2  a
        (-1)  q              (--; q)
                               n    2 n
                              q
Out[1]= -------------------------------
                       n
                      a
In[2]:= Limes[%,n-⟩Infinity]
The expression
Indeterminate
was obtained.
Therefore the limit n -⟩ ∞ could not be determined.
Here is your expression:
            n  (n (1 + n))/2  a
        (-1)  q              (--; q)
                               n    2 n
                              q
Out[2]= -------------------------------
                       n
                      a
In[3]:= %/.zerl
bottom-split by:  n

            n  (n (1 + n))/2          a
        (-1)  q              (a; q)  (--; q)
                                   n   n    n
                                      q
Out[3]= -------------------------------------
                          n
                         a
In[4]:= Limes[%,n-⟩Infinity]
Is 2 n even, odd, or neither of both?
[e|o|n]:  e
                q
Out[4]= (a;q)  (-;q)

∞ a ∞

\underline{See also}: AbsGreater, AbsSmaller, AbsUndetermined, MinusOne.

lina1

\underline{Description}: \begin{matrix}(a; q)_{n} → (1 − a)(aq; q)_{n−1}, \\ (a; q)_{∞} → (1 − a)(aq; q)_{∞}. \end{matrix}

\underline{Usage}: Expr/.lina1.

\underline{Example(s)}:

In[1]:= pq[a,m,q^{∧}2]
             2
Out[1]= (a; q )
               m
In[2]:= %/.lina1
                    2   2
Out[2]= (1 - a) (a q ; q )
                          -1 + m
In[3]:= 1/pqinf[a,q^{∧}2]
           1
Out[3]= -------
            2
        (a;q )

∞

In[4]:= %/.lina1

                1
Out[4]= ------------------
                    2  2
        (1 - a) (a q ;q )

∞

\underline{See also}: lina2, linz, Ers, PosListe, ManipulationsListe.

lina2

\underline{Description}: (a; q)_{n} → (1 − aq^{n−1})(a; q)_{n−1}.

\underline{Usage}: Expr/.lina2.

\underline{Example(s)}:

In[1]:= pq[a,m,q^{∧}2]
             2
Out[1]= (a; q )
               m
In[2]:= %/.lina2
                -2 + 2 m       2
Out[2]= (1 - a q        ) (a; q )
                                 -1 + m
\underline{See also}: lina1, linz, Ers, PosListe, ManipulationsListe.

linz

\underline{Description}: Rule that absorbs linear terms.

\underline{Usage}: Expr/.linz.

\underline{Example(s)}:

In[1]:= (1-a)*pq[a*q^{∧}2,m,q^{∧}2]/(1-a*q^{∧}(2*m-2))/pq[a,m-1,q^{∧}2]
                          2   2
              (1 - a) (a q ; q )
                                m
Out[1]= -------------------------------
                -2 + 2 m       2
        (1 - a q        ) (a; q )
                                 -1 + m
In[2]:= %/.linz

                      2
                 (a; q )
                        1 + m
Out[2]= -------------------------------
                -2 + 2 m       2
        (1 - a q        ) (a; q )
                                 -1 + m
In[3]:= %/.linz
             2
        (a; q )
               1 + m
Out[3]= ------------
               2
          (a; q )
                 m
In[4]:= 1/(1-b/q)/pqinf[b]
              1
Out[4]= --------------
             b
        (1 - -) (b;q)
             q  ∞
In[5]:= %/.linz
          1
Out[5]= ------
         b
        (-;q)
         q ∞

\underline{See also}: lina1, lina2, Ers, PosListe, ManipulationsListe.

LS

\underline{Description}: LS is the left-hand side in Gleichung.

\underline{Usage}: LS.

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n

                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= LS

                 -n
Out[2]= ϕ a, q  ; q, q
        2 1
               c
In[3]:= Add[1]
                                        n  c
                                       a  (-; q)
                     -n                    a    n
Out[3]= 1 + ϕ a, q  ; q, q   == 1 + ----------
            2 1                         (c; q)
                   c                          n
In[4]:= LS

                     -n
Out[4]= 1 + ϕ a, q  ; q, q
            2 1
                   c
In[5]:= LS=pq[a,m]
Out[5]= (a; q)
              m
In[6]:= Gleichung
                        n  c
                       a  (-; q)
                           a    n
Out[6]= (a; q)  == 1 + ----------
              m         (c; q)
                              n
\underline{See also}:  Gleichung, SumListe$gl, TransListe$gl, RS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers,
  Subst.

Mal

\underline{Description}: Function that multiplies Gleichung by Expr.

\underline{Usage}: Mal[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= Mal[pq[c,n,q]]

                 -n                     n  c
Out[2]= ϕ a, q  ; q, q   (c; q)  == a  (-; q)
        2 1                       n        a    n
               c
In[3]:= Gleichung

                 -n                     n  c
Out[3]= ϕ a, q  ; q, q   (c; q)  == a  (-; q)
        2 1                       n        a    n
               c
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Add, Div, Sub, Hoch, GlTausche, Ers.

ManipulationsListe

\underline{Description}: Gives a list of all available rules for manipulating finite and infinite q-factorial symbols.

\underline{Usage}: ManipulationsListe.

MinusOne

\underline{Description}: : Rule for getting rid of expressions of the form (−1)^{N} where N is an even or odd integer.

\underline{Usage}: Expr/.MinusOne.
In[1]:= pq[a,2*n]
Out[1]= (a; q)
              2 n

In[2]:= %/.trans
                       1 - 2 n
            2 n  2 n  q
        (-1)    a    (--------; q)
                         a        2 n
Out[2]= -----------------------------
                          2
                (2 n - 4 n )/2
               q
In[3]:= %/.MinusOne
Is 2 n even, odd, or neither of both?
[e|o|n]:  e
               1 - 2 n
         2 n  q
        a    (--------; q)
                 a        2 n
Out[3]= ---------------------
                      2
            (2 n - 4 n )/2
           q
\underline{See also}: SimplifyPQ, Expandq, SUMExpand.

Multinomialpq

\underline{Description}: : Multinomialq[n1,n2, …,q] is the q-multinomial coefficient [ \begin{matrix}∑_{i} n_{i} \\ n_{1},n_{2},... \end{matrix} ]_{q}, written in terms of qfactorial symbols pq.

\underline{Usage}: Multinomialpq[n1,n2, . . . ,q].
In[1]:= Multinomialpq[a,b,c,s]
            (s; s)
                  a + b + c
Out[1]= -----------------------
        (s; s)  (s; s)  (s; s)
              a       b       c
In[2]:= Multinomialpq[2,3,1,q]
                (q; q)
                      6
Out[2]= -----------------------
        (q; q)  (q; q)  (q; q)
              1       2       3
\underline{See also}: Multinomialq, Binomialq, Binomialpq, Factorialq, Factorialpq.

Multinomialq

\underline{Description}: : Multinomialq[n1,n2, …,q] is the q-multinomial coefficient [ \begin{matrix}∑_{i} n_{i} \\ n_{1},n_{2},... \end{matrix} ]_{q}, expanded into a q-series, if possible.

\underline{Usage}: Multinomialq[n1,n2, . . . ,q].
In[1]:= Multinomialq[2,3,1,s]
                     2      3      4      5      6      7      8      9
Out[1]= 1 + 2 s + 4 s  + 6 s  + 8 s  + 9 s  + 9 s  + 8 s  + 6 s  + 4 s  +
        10    11
⟩  2 s   + s
In[2]:= Multinomialq[a,b,c,q]

          a + b + c
Out[2]=
           a, b, c
                     q
\underline{See also}: Multinomialpq, Binomialq, Binomialpq, Factorialq, Factorialpq.

neg1

\underline{Description}: (a; q)_{n} → 1/(aq^{n}; q)_{−n}.

\underline{Usage}: Expr/.neg1.

\underline{Example(s)}:

In[1]:= pq[a^{∧}2,-n]
          2
Out[1]= (a ; q)
               -n
In[2]:= %/.neg1
            1
Out[2]= ------------
          2  -n
        (a  q  ; q)
                   n
\underline{See also}: neg2, Ers, PosListe, ManipulationsListe.

Alphabetic List: Ph 87

neg2 \underline{Description}: (a; q)_{n} → q^{\binom{n+1}{2}}/((−q/a)^{n}(q/a; q)_{−n}).

\underline{Usage}: Expr/.neg2.

\underline{Example(s)}:

In[1]:= pq[a^{∧}2,-n]
          2
Out[1]= (a ; q)
               -n
In[2]:= %/.neg2
                        2
                 n/2 + n /2
            n   q
Out[2]= (-1)  --------------
               2 n  q
              a    (--; q)
                     2    n
                    a
\underline{See also}: neg1, Ers, PosListe, ManipulationsListe.

Ph

\underline{Description}: Ph[List1A,List1B,q1,List2A,List2B,q2, . . . ,ListkA,ListkB,qk,z] is the multibasic basic hyper-

geometric series with upper parameters List1A and lower parameters List1B for base q1, …, with upper parameters ListkA and lower parameters ListkB for base qk, and argument z.

\underline{Usage}: Ph[List1A,List1B,q1,List2A,List2B,q2, . . . ,ListkA,ListkB,qk,z].

\underline{Example(s)}:

In[1]:= Ph[{a,b},{c},q,{e,f,g},{E,F,G},p,z]
           a, b:  e, f, g
Out[1]= ϕ  ; q, p; z
            c:  E, F, G
\underline{See also}:  SListe, TListe, SUMRegeln, SUMph, PhSUM, ph, ps, pq, pqinf, phCancel, phOrdne, phPerm,
  phTausche, PQ, phFormat.

88 Alphabetic List: phCancel

ph \underline{Description}: ph[List1,List2,q,z] is the basic hypergeometric series with upper parameters List1, lower parameters

List2, base q, and argument z.

\underline{Usage}: ph[List1,List2,q,z].

\underline{Example(s)}:

In[1]:= ph[{a,b},{c},q,z]
             a, b
Out[1]= ϕ  ; q, z
        2 1   c
In[2]:= ph[{a,b,c},{d,e,0},q,z]
             a, b, c
Out[2]= ϕ  ; q, z
        3 3  d, e, 0
\underline{See also}: SListe, TListe, SUMRegeln, SUMph, phSUM, Ph, ph, pq, pqinf, phCancel, phOrdne, phPerm, phTau-
  sche, PQ, phFormat.

phCancel \underline{Description}: Switch that activates automatic cancelling of the upper and lower parameters in ph[], Ph[] and ps[],

or makes it inactive, respectively. By default automatic cancelling is active.

\underline{Usage}: phCancel.

\underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[2]:= ph[{a,b},{a,c},q,z]
             b
Out[2]= ϕ  ; q, z
        1 1  c
In[3]:= pq[{a,b},{a,c},n,q^{∧}2]

             2
        (b; q )
               n
Out[3]= --------
             2
        (c; q )
               n
In[4]:= pqinf[{a,b},{a,c}]
        (b; q)

∞

Out[4]= -------
        (c; q)

∞

In[5]:= phCancel
In[6]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph inactive.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[7]:= ph[{a,b},{a,c},q,z]

             a, b
Out[7]= ϕ  ; q, z
        2 2  a, c

In[8]:= pq[{a,b},{a,c},n,q^{∧}2]
             2
        (b; q )
               n
Out[8]= --------
             2
        (c; q )
               n
In[9]:= pqinf[{a,b},{a,c}]

        (b; q)

∞

Out[9]= -------
        (c; q)

∞

In[10]:= phCancel
In[11]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\underline{See also}: ph, W, hypqAttributes.

PhEinf

\underline{Description}: Rule that inactivates automatic cancelling in Ph[] and then adds a parameter which has to be entered on request, together with the information to which base it belongs, to the upper and lower parameters of Ph[].

\underline{Usage}: Expr/.PhEinf.

\underline{Example(s)}:

In[1]:= Ph[{a,b},{c},q,{e,f,g},{E,F,G},p,z]

           a, b:  e, f, g
Out[1]= ϕ  ; q, p; z
            c:  E, F, G

In[2]:= %/.PhEinf
Add the parameter:  A
to the parameters belonging to the i-th base, where i=1

           A, a, b:  e, f, g
Out[2]= ϕ  ; q, p; z
            A, c:  E, F, G

In[3]:= %/.PhEinf
Add the parameter:  B
to the parameters belonging to the i-th base, where i=2

           A, a, b:  B, e, f, g
Out[3]= ϕ  ; q, p; z
            A, c:  B, E, F, G
\underline{See also}: phCancel, phOrdne, PhPerm, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe.

phEinf \underline{Description}: Rule that inactivates automatic cancelling in ph[] and then adds a parameter which has to be entered

on request to the upper and lower parameters of ph[].

\underline{Usage}: Expr/.phEinf.

\underline{Example(s)}:

In[1]:= ph[{b,c,q},{q*a/c,q*a/b},q,z]
             b, c, q
Out[1]= ϕ a q  a q; q, z
        3 2  ---, ---
              c    b
In[2]:= %/.phEinf
Add the parameter:  a
             a, b, c, q
Out[2]= ϕ  a q  a q; q, z
        4 3  a, ---, ---
                 c    b
\underline{See also}: phOrdne, phPerm, phTausche, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe.

phFormat \underline{Description}: Switch that activates basic hypergeometric output, or makes it inactive, respectively. By default basic

hypergeometric output is active.

\underline{Usage}: phFormat.

\underline{Example(s)}:

In[1]:= pq[a,n]/pqinf[b,q^{∧}2]*ph[{c,d},{c*d},q,z]

             c, d
         ϕ  ; q, z   (a; q)
        2 1  c d                n
Out[1]= -------------------------
                     2
                 (b;q )

∞

In[2]:= Tgl5402
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n

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

In[3]:= phFormat
In[4]:= %%
                                1 - n   1 - n   1 - n
             -n                q       q       q        2  2  2    -2 + 2 n
Out[4]= ph[{q  , b, c, d, e}, {------, ------, ------, b  c  d  e q  },
                                 b       c       d
                    1 - 2 n          3/2 - n
                   q                q
⟩  q, q] == ph[{--------, -----------------------,
                    b c d    Sqrt[b] Sqrt[c] Sqrt[d]
                  3/2 - n            1 - n   1 - n   1 - n
                 q                  q       q       q        -n/2    -n/2
⟩  -(-----------------------), ------, ------, ------, q    , -q    ,
          Sqrt[b] Sqrt[c] Sqrt[d]    c d     b d     b c
                                      3 - 3 n
         1/2 - n/2    1/2 - n/2      q
⟩  q         , -q         , e, ----------},
                                     2  2  2
                                    b  c  d  e
                1/2 - n                    1/2 - n            1 - n   1 - n
               q                          q                  q       q
⟩  {-----------------------, -(-----------------------), ------, ------,
        Sqrt[b] Sqrt[c] Sqrt[d]    Sqrt[b] Sqrt[c] Sqrt[d]     b       c

         1 - n   2 - (3 n)/2     2 - (3 n)/2    3/2 - (3 n)/2
        q       q               q              q
⟩  ------, ------------, -(------------), --------------,
          d        b c d           b c d           b c d
           3/2 - (3 n)/2    2 - 2 n
          q                q                  -1 + n
⟩  -(--------------), --------, b c d e q  }, q, q]
              b c d        b c d e
           2 - 2 n   3 - 3 n     2 - 2 n    3 - 3 n
          q         q           q          q
⟩  pq[{--------, --------}, {--------, ----------}, n, q]
          b c d e    2  2  2     b c d     2  2  2
                    b  c  d               b  c  d  e
In[5]:= %1
        ph[{c, d}, {c d}, q, z] pq[a, n, q]
Out[5]= -----------------------------------
                              2
                   pqinf[b, q ]
In[6]:= phFormat
In[7]:= %1

             c, d
         ϕ  ; q, z   (a; q)
        2 1  c d                n
Out[7]= -------------------------
                     2
                 (b;q )

∞

\underline{See also}: ph.

Phinv

\underline{Description}:  Rule  that  transforms  a  multibasic  hypergeometric  series

  Ph[List1A,List1B,q1,List2A,List2B,q2, . . . ,ListkA,ListkB,qk,z] with bases q1, . . . , into a multibasic hy-

pergeometric series Ph[ …,1/q1,, …,1/q2, …, …,1/qk,z] with bases 1/q1, …

\underline{Usage}: Expr/.Phinv.

\underline{Example(s)}:

In[1]:= Ph[{a,b},{c},q,{e,f,g},{E,F,G},p,z]

           a, b:  e, f, g
Out[1]= ϕ  ; q, p; z
            c:  E, F, G

In[2]:= %/.Phinv
           1  1   1  1  1
           -, -:  -, -, -
           a  b   e  f  g  1  1  a b e f g z
Out[2]= ϕ  ; -, -; -----------
            1   1  1  1    q  p   c E F G q
            -:  -, -, -
            c   E  F  G
In[3]:= Ph[{a,b},{c,0},q,{e,f,g},{E,F},p,z]

           a, b:  e, f, g
Out[3]= ϕ  ; q, p; z
            c, 0:  E, F

In[4]:= %/.Phinv
           1  1      1  1  1
           -, -, 0:  -, -, -
           a  b      e  f  g  1  1  a b e f g z
Out[4]= ϕ  ; -, -; -----------
              1   1  1        q  p    c E F q
              -:  -, -, 0
              c   E  F

\underline{See also}: SUMRegeln, Ers, PosListe, phinv, psinv.

phinv

\underline{Description}: Rule that transforms a basic hypergeometric series ph[List1,List2,q,z] with base q into a basic hypergeometric series ph[ …,1/q,..] with base 1/q.

\underline{Usage}: Expr/.phinv.

\underline{Example(s)}:

In[1]:= ph[{a,b},{c},q,z]

             a, b
Out[1]= ϕ  ; q, z
        2 1   c

In[2]:= %/.phinv
             1  1
             -, -
             a  b  1  a b z
Out[2]= ϕ  ; -, -----
        2 1   1    q   c q
              -
              c
In[3]:= ph[{a,b,c},{d,0},q,z]

             a, b, c
Out[3]= ϕ  ; q, z
        3 2   d, 0

In[4]:= %/.phinv
             1  1  1
             -, -, -
             a  b  c  1  a b c z
Out[4]= ϕ  ; -, -------
        3 1     1     q    d q
                -
                d
\underline{See also}: SUMRegeln, Ers, PosListe.

phOrdne

\underline{Description}: Rule that tries to order the parameters of a basic hypergeometric series in “well-poised” order. If there is an upper parameter of the form q^{−n}, where n might be a nonnegative integer, then it is put at the very last place in the upper list. If the parameters could be paired such that the product of each pair equals Aq, however A is missing in the upper parameters, then you have to add A to the upper and lower parameters by phEinf before applying

  phOrdne.
\underline{Usage}: Expr/.phOrdne.

\underline{Example(s)}:

In[1]:= ph[{q^{∧}-n,b,q*Sqrt[a],a,-q*Sqrt[a]},{a*q/b,-Sqrt[a],Sqrt[a],
        a*q^{∧}(1+n)},q,z]

              -n
             q  , b, Sqrt[a] q, a, -(Sqrt[a] q)
Out[1]= ϕ  ; q, z
        5 4   a q                        1 + n
              ---, -Sqrt[a], Sqrt[a], a q
               b
In[2]:= %/.phOrdne
                                             -n
             a, Sqrt[a] q, -(Sqrt[a] q), b, q
Out[2]= ϕ  ; q, z
        5 4                      a q     1 + n
              Sqrt[a], -Sqrt[a], ---, a q
                                  b
In[3]:= ph[{q^{∧}-n,b,c},{d,e},q^{∧}2,z]

              -n         2
Out[3]= ϕ q  , b, c; q , z
        3 2
               d, e
In[4]:= %/.phOrdne

                    -n   2
Out[4]= ϕ b, c, q  ; q , z
        3 2
               e, d
In[5]:= ph[{b,c,q},{a*q/c,a*q/b},q,z]
             b, c, q
Out[5]= ϕ a q  a q; q, z
        3 2  ---, ---
              c    b
In[6]:= %/.phOrdne
             b, c, q
Out[6]= ϕ a q  a q; q, z
        3 2  ---, ---
              c    b

In[7]:= %/.phEinf
Add the parameter:  a
             a, b, c, q
Out[7]= ϕ  a q  a q; q, z
        4 3  a, ---, ---
                 c    b
In[8]:= %/.phOrdne
             a, b, c, q
Out[8]= ϕ a q  a q   ; q, z
        4 3  ---, ---, a
              b    c
In[9]:= phCancel
In[10]:= %%/.phOrdne
              b, c, q
Out[10]= ϕ a q  a q; q, z
         3 2  ---, ---
               b    c

\underline{See also}: phEinf, phPerm, phTausche, ph, W, PQSort, Ers, PosListe.

phPerm

\underline{Description}: Rule for permuting parameters in basic hypergeometric series.

\underline{Usage}: Expr/.phPerm[<Permutation>,x].

x can be u, l, b. u causes a permutation of upper parameters, l causes a permutation of lower parameters, b causes a simultaneous permutation of respective upper and lower parameters. Permutation must be a sequence of positive numbers forming a permutation. Under the options u and l the effect is that the new parameter at position i is the old parameter from position Permutation[i]. However, the behaviour of FPerm under the option b is special. The option b is especially designed for the permutation of parameters of well-poised series. Hence, the first upper parameter is not moved, whereas the new upper parameter at position i+1 is the old upper parameter from position Permutation[i]+1, and the new lower parameter at position i is the old lower parameter from position

  Permutation[i].

\underline{Example(s)}:

In[1]:= ph[{a,b,c,d},{e,f,g},q,z]

98 Alphabetic List: Phph

             a, b, c, d
Out[1]= ϕ  ; q, z
        4 3   e, f, g
In[2]:= %1/.phPerm[3,2,1,u]
             c, b, a, d
Out[2]= ϕ  ; q, z
        4 3   e, f, g
In[3]:= %1/.phPerm[3,2,1,l]
             a, b, c, d
Out[3]= ϕ  ; q, z
        4 3   g, f, e
In[4]:= %1/.phPerm[2,1,b]
             a, c, b, d
Out[4]= ϕ  ; q, z
        4 3   f, e, g
\underline{See also}: phTausche, phOrdne, ph, W, PQSort, Ers, PosListe.

Phph \underline{Description}: Rule that transforms a Ph[] into “ordinary” hypergeometric notation, if possible.

\underline{Usage}: Expr/.Phph.

\underline{Example(s)}:

In[1]:= Ph[{a,b},{c},q,{e,f,g},{E,F,G},q^{∧}2,z]
           a, b:  e, f, g      2
Out[1]= ϕ  ; q, q ; z
            c:  E, F, G
In[2]:= %/.Phph

Out[2]= ϕ
        8 7

      a, b, Sqrt[e], -Sqrt[e], Sqrt[f], -Sqrt[f], Sqrt[g], -Sqrt[g]
⟩  ; q, z
       c, Sqrt[E], -Sqrt[E], Sqrt[F], -Sqrt[F], Sqrt[G], -Sqrt[G]
\underline{See also}: Ph, ph, phPh, Ers, PosListe.

phPh \underline{Description}: Rule that transforms a ph[] into multibasic notation.

\underline{Usage}: Expr/.phPh.

\underline{Example(s)}:

In[1]:= ph[{a,d,e^{∧}(1/2),-e^{∧}(1/2),f^{∧}(1/2),-f^{∧}(1/2),g^{∧}(1/2),-g^{∧}(1/2)},
        {c,E^{∧}(1/2),-E^{∧}(1/2),F^{∧}(1/2),-F^{∧}(1/2),G^{∧}(1/2),-G^{∧}(1/2)},q,z]

Out[1]= ϕ
        8 7

      a, d, Sqrt[e], -Sqrt[e], Sqrt[f], -Sqrt[f], Sqrt[g], -Sqrt[g]
⟩  ; q, z
       c, Sqrt[E], -Sqrt[E], Sqrt[F], -Sqrt[F], Sqrt[G], -Sqrt[G]
In[2]:= %/.phPh
           a, d:  e, f, g      2
Out[2]= ϕ  ; q, q ; z
            c:  E, F, G
\underline{See also}: ph, Ph, Phph, Ers, PosListe.

phps

\underline{Description}: Rule that transforms a ph[] into a difference of a ps[] and a ph[].

\underline{Usage}: Expr/.phps.

\underline{Example(s)}:

In[1]:= ph[{a,b},{c},q,z]

             a, b
Out[1]= ϕ  ; q, z
        2 1   c

In[2]:= %/.phps
                2
               q                     c
               --, q, q             (-, 1; q)
               c             c q     q       1
           ϕ  ; q, -----   ----------
          3 2    2   2      a b z    a  b
                q   q               (-, -; q)
                --, --               q  q    1
                a   b                                   a, b
Out[2]= -(------------------------------------) +  ps       ; q, z
                           z                      2  2  c, q
In[3]:= %/.pqaufl

              a, b
Out[3]=  ps       ; q, z
        2  2  c, q

In[4]:= ph[{a,b,q},{c,d},q,z]

             a, b, q
Out[4]= ϕ  ; q, z
        3 2   c, d

In[5]:= %/.phps

                2   2
               q   q                  c  d
               --, --, q             (-, -; q)
               c   d          c d     q  q    1
           ϕ  ; q, -----   ----------
          3 2    2   2       a b z    a  b
                q   q                (-, -; q)
                --, --                q  q    1
                a   b                                    a, b
Out[5]= -(-------------------------------------) +  ps       ; q, z
                            z                      2  2  c, d

\underline{See also}: ph, ps, psph, Ers, PosListe.

PhSUM

\underline{Description}: Rule that transforms a Ph[] into a SUM[].

\underline{Usage}: Expr/.PhSUM.

\underline{Example(s)}:

In[1]:= Ph[{a,b},{c},q,{e,f,g},{E,F,G},p,z]

           a, b:  e, f, g
Out[1]= ϕ  ; q, p; z
            c:  E, F, G

In[2]:= %/.PhSUM
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k

          ∞ k

             z  (a; q)  (b; q)  (e; p)  (f; p)  (g; p)
         \  k       k       k       k       k
Out[2]= ⟩ ------------------------------------------
         /    (c; q)  (E; p)  (F; p)  (G; p)  (q; q)
                    k       k       k       k       k
         k=0
In[3]:= Ph[{a,b},{c},q,{e,f,p^{∧}-n},{E,F,G},p,z]

                         -n
Out[3]= ϕ a, b:  e, f, p  ; q, p; z
             c:  E, F, G

In[4]:= %/.PhSUM
Is n a nonnegative integer?
[y|n]:    y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k
          n   k                                   -n
             z  (a; q)  (b; q)  (e; p)  (f; p)  (p  ; p)
         \  k       k       k       k         k
Out[4]= ⟩ --------------------------------------------
         /     (c; q)  (E; p)  (F; p)  (G; p)  (q; q)
                     k       k       k       k       k
         k=0

\underline{See also}: Ph, SUM, SUMPh, Ers, PosListe.

phSUM

\underline{Description}: Rule that transforms a ph[] into a SUM[].

\underline{Usage}: Expr/.phSUM.

\underline{Example(s)}:

In[1]:= ph[{a,b},{c},q^{∧}2,z]

             a, b   2
Out[1]= ϕ  ; q , z
        2 1   c

In[2]:= %/.phSUM
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k

          ∞ k      2        2

             z  (a; q )  (b; q )
         \  k        k
Out[2]= ⟩ --------------------
         /         2     2   2
              (c; q )  (q ; q )
         k=0         k         k
In[3]:= ph[{a,q^{∧}(-2*n)},{c},q^{∧}2,z]

                 -2 n   2
Out[3]= ϕ a, q    ; q , z
        2 1
                c
In[4]:= %/.phSUM
Is n a nonnegative integer?
[y|n]:    y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  j
          n   j      2     -2 n   2
             z  (a; q )  (q    ; q )
         \  j            j
Out[4]= ⟩ ------------------------
         /           2     2   2
                (c; q )  (q ; q )
         j=0           j         j
In[5]:= ph[{a,b},{c,d},q,z]

             a, b
Out[5]= ϕ  ; q, z
        2 2  c, d

In[6]:= %/.phSUM
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k
                            2
          ∞  k  -k/2 + k /2  k
             (-1)  q            z  (a; q)  (b; q)
         \  k       k
Out[6]= ⟩ -------------------------------------
         /          (c; q)  (d; q)  (q; q)
                          k       k       k
         k=0
\underline{See also}: ph, SUM, SUMph, Ers, PosListe.

phTausche

\underline{Description}: Rule for reordering parameters in basic hypergeometric series.

\underline{Usage}: Expr/.phTausche[n1,n2,x].

x can be u, l, b. u causes a reordering of upper parameters, l causes a reordering of lower parameters, b causes a simultaneous reordering of respective upper and lower parameters. n1 is the position of the parameter to be reordered, n2 is the new position.

\underline{Example(s)}:

In[1]:= ph[{q^{∧}-n,b,q*Sqrt[a],a,-q*Sqrt[a]},{a*q/b,-Sqrt[a],Sqrt[a],
        a*q^{∧}(1+n)},q,z]
              -n
             q  , b, Sqrt[a] q, a, -(Sqrt[a] q)
Out[1]= ϕ  ; q, z
        5 4   a q                        1 + n
              ---, -Sqrt[a], Sqrt[a], a q
               b
In[2]:= %/.phTausche[1,3,u]
                            -n
             b, Sqrt[a] q, q  , a, -(Sqrt[a] q)
Out[2]= ϕ  ; q, z
        5 4   a q                        1 + n
              ---, -Sqrt[a], Sqrt[a], a q
               b
In[3]:= %/.phTausche[4,2,l]
                            -n
             b, Sqrt[a] q, q  , a, -(Sqrt[a] q)
Out[3]= ϕ  ; q, z
        5 4   a q     1 + n
              ---, a q     , -Sqrt[a], Sqrt[a]
               b
In[4]:= %/.phTausche[1,4,b]
                 -n
             b, q  , a, -(Sqrt[a] q), Sqrt[a] q
Out[4]= ϕ  ; q, z
        5 4      1 + n                     a q
              a q     , -Sqrt[a], Sqrt[a], ---
                                            b
\underline{See also}: phPerm, phOrdne, ph, W, PQSort, Ers, PosListe.

PosListe

\underline{Description}: Function that provides a list of subexpressions of Expr together with the respective positions in Expr. This helps to use controlled application of rules or functions by means of Ers.

\underline{Usage}: PosListe[Expr].

\underline{Example(s)}:

In[1]:= pq[a,n]/pq[q,n]*SUM[pq[b,k]/pq[c,k+1]*q^{∧}k,{k,0,Infinity}]

           ∞ k

              q  (b; q)
          \  k
        ( ⟩ -----------) (a; q)
          /   (c; q)             n
                    1 + k
          k=0
Out[1]= --------------------------
                 (q; q)
                       n
In[2]:= PosListe[%]

            ∞ k

               q  (b; q)
           \  k                                 1
Out[2]= {{ ⟩ -----------, {{1}}}, {(a; q) , {{2}}}, {-------, {{3}}}}
           /   (c; q)                      n           (q; q)
                     1 + k                                   n
           k=0
In[3]:= PosListe[%%,2]
Out[3]= {{-1, {{3, 2}}}, {a, {{2, 1}}}, {n, {{2, 2}}}, {q, {{2, 3}}},
                              k
                             q  (b; q)
                                      k
⟩  {{k, 0, ∞}, {{1, 2}}}, {-----------, {{1, 1}}}, {(q; q) , {{3, 1}}}}
                             (c; q)                         n
                                   1 + k
In[4]:= PosListe[%%%,3]
Out[4]= {{0, {{1, 2, 2}}}, {k, {{1, 2, 1}}}, {n, {{3, 1, 2}}},
                                    k
⟩  {q, {{3, 1, 1}, {3, 1, 3}}}, {q , {{1, 1, 1}}}, {∞, {{1, 2, 3}}},
                                   1
⟩  {(b; q) , {{1, 1, 2}}}, {-----------, {{1, 1, 3}}}}
            k                 (c; q)
                                    1 + k
\underline{See also}: Ers, Subst.

pq

\underline{Description}: pq[x,n,q] is the q-factorial symbol (x; q)_{n}. pq[List1,List2,n,q] is also provided as the usual abbreviation for the quotient of q-factorial symbols (see [1, (1.2.41)]). In both cases the parameter q is optional. It will be set equal q if it is omitted.

\underline{Usage}: pq[x,n,q]
   or: pq[x,n]
   or: pq[List1,List2,n,q]
   or: pq[List1,List2,n].

\underline{Example(s)}:

In[1]:= pq[a,n]
Out[1]= (a; q)
              n
In[2]:= pq[a,n,q^{∧}2]
             2
Out[2]= (a; q )
               n
In[3]:= pq[{a,b},{c,d},2*m]
        (a, b; q)
                 2 m
Out[3]= ------------
        (c, d; q)
                 2 m
In[4]:= pq[{a,b},{c,d},2*m,1/q]
               1
        (a, b; -)
               q 2 m
Out[4]= ------------
               1
        (c, d; -)
               q 2 m
\underline{See also}: pqinf, Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialq, Factorialpq, PQ, pqaufl,
  pqzerl, pqzus,
  phFormat.

PQ

\underline{Description}: Is a switch that activates automatic evaluating of q-factorial symbols pq and basic hypergeometric series ph, Ph, ps, or makes it inactive, respectively. By default automatic evaluating is inactive.

\underline{Usage}: PQ.

\underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[2]:= pq[a,5]
Out[2]= (a; q)
              5
In[3]:= ph[{a,b},{c},q,z]

             a, b
Out[3]= ϕ  ; q, z
        2 1   c

In[4]:= ph[{q^{∧}-n,b},{c},q,z]

              -n
Out[4]= ϕ q  , b; q, z
        2 1
               c
In[5]:= ph[{q^{∧}-3,b},{c},q,z]

              -3
Out[5]= ϕ q  , b; q, z
        2 1
               c
In[6]:= PQ
In[7]:= hypqAttributes
Automatic evaluation of pq and ph is active.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[8]:= pq[a,5]

                                  2          3          4
Out[8]= (1 - a) (1 - a q) (1 - a q ) (1 - a q ) (1 - a q )
In[9]:= ph[{a,b},{c},q,z]
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k

          ∞ k

             z  (a; q)  (b; q)
         \  k       k
Out[9]= ⟩ ------------------
         /    (c; q)  (q; q)
                    k       k
         k=0
In[10]:= ph[{q^{∧}-n,b},{c},q,z]
Is n a nonnegative integer?
[y|n]:    y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  j
           n   j           -n
              z  (b; q)  (q  ; q)
          \  j         j
Out[10]= ⟩ --------------------
          /     (c; q)  (q; q)
                      j       j
          j=0
In[11]:= ph[{q^{∧}-3,b},{c},q,z]
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  s
                           -3                    -3        -2             2
             (1 - b) (1 - q  ) z   (1 - b) (1 - q  ) (1 - q  ) (1 - b q) z
Out[11]= 1 + ------------------- + ---------------------------------------- +
               (1 - c) (1 - q)                                        2
                                      (1 - c) (1 - q) (1 - c q) (1 - q )
                   -3        -2       1                    2   3
     (1 - b) (1 - q  ) (1 - q  ) (1 - -) (1 - b q) (1 - b q ) z
                                      q
⟩  -----------------------------------------------------------
                                       2          2        3
       (1 - c) (1 - q) (1 - c q) (1 - q ) (1 - c q ) (1 - q )
In[12]:= PQ

In[13]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.

\underline{See also}: ph, Ph, ps, pq, hypqAttributes.

pqaufl

\underline{Description}: Rule that writes (x; q)_{n} as the defining product ∏_{i=0}^{n−1}(1 − xq^{i}), if n is an integer.

\underline{Usage}: Expr/.pqaufl.

\underline{Example(s)}:

In[1]:= pq[a,-3]/pq[b,2]*pq[c,1]
        (a; q)   (c; q)
              -3       1
Out[1]= ----------------
            (b; q)
                  2
In[2]:= %/.pqaufl
                           1 - c
Out[2]= -------------------------------------------
                     a        a        a
        (1 - b) (1 - --) (1 - --) (1 - -) (1 - b q)
                      3        2       q
                     q        q
In[3]:= ph[{a,b,c},{d,e,f},q,z]

             a, b, c
Out[3]= ϕ  ; q, z
        3 3  d, e, f

In[4]:= %/.C01

                   q, a q, b q, c q
            z ϕ  ; q, q z   (a; q)  (b; q)  (c; q)
              4 4   2                                1       1       1
                   q , d q, e q, f q
Out[4]= 1 - ----------------------------------------------------------
                         (d; q)  (e; q)  (f; q)  (q; q)
                               1       1       1       1
In[5]:= %/.pqaufl

                                           q, a q, b q, c q
            (1 - a) (1 - b) (1 - c) z ϕ  ; q, q z
                                      4 4   2
                                           q , d q, e q, f q
Out[5]= 1 - ----------------------------------------------------------
                         (1 - d) (1 - e) (1 - f) (1 - q)

\underline{See also}: pqzerl, pqzus, pq, pqinfzerl, pqinfzus, Ers, PosListe.

pqinf

\underline{Description}: pqinf[x,q] is the infinite q-factorial symbol (x; q)_{∞}. pqinf[List1,List2,q] is also provided as the usual abbreviation for the quotient of infinite q-factorial symbols (see [1, (1.2.42)]). In both cases the parameter q is optional. If it is omitted it is set equal q.

\underline{Usage}: pqinf[x,q]
   or: pqinf[x]
   or: pqinf[List1,List2,q]
   or: pqinf[List1,List2].

\underline{Example(s)}:

In[1]:= pqinf[a^{∧}2*q]
          2
Out[1]= (a  q;q)

∞

In[2]:= pqinf[a^{∧}2*q,q^{∧}3]
          2    3
Out[2]= (a  q;q )

∞

In[3]:= pqinf[{a,b},{c,d}]

        (a, b; q)

∞

Out[3]= ----------
        (c, d; q)

∞

In[4]:= pqinf[{a,b},{c,d},1/q]
               1
        (a, b; -)
               q ∞
Out[4]= ----------
               1
        (c, d; -)
               q ∞
\underline{See also}: pq, PQ, pqinfzerl, pqinfzus, phFormat.

pqinfzerl

\underline{Description}: Rule that splits pqinf[List1,List2,q] into a quotient of products of infinite q-factorial symbols.

\underline{Usage}: Expr/.pqinfzerl.

\underline{Example(s)}:

In[1]:= pqinf[{a,b},{c,d}]
        (a, b; q)

∞

Out[1]= ----------
        (c, d; q)

∞

In[2]:= %/.pqinfzerl
        (a;q)  (b;q)

∞  ∞

Out[2]= -------------
        (c;q)  (d;q)

∞  ∞

\underline{See also}: pqaufl, pqzerl, pqzus, pqinf, pqinfzus, Ers, PosListe.

pqinfzus

\underline{Description}: Rule that collects several infinite q-factorial symbols pqinf[xi,q] to an expression pqinf[List1,List2,q]. The parameter q is optional. It is set equal q if it is omitted.

\underline{Usage}: Expr/.pqinfzus[q]
   or: Expr/.pqinfzus[].

\underline{Example(s)}:

In[1]:= pqinf[a]*pqinf[b]/pqinf[c]/pqinf[d]
        (a;q)  (b;q)

∞  ∞

Out[1]= -------------
        (c;q)  (d;q)

∞  ∞

In[2]:= %/.pqinfzus[]
        (a, b; q)

∞

Out[2]= ----------
        (c, d; q)

∞

In[3]:= pqinf[a,q]*pqinf[b,q^{∧}2]/pqinf[c,q^{∧}2]/pqinf[d,q]
                   2
        (a;q)  (b;q )

∞  ∞

Out[3]= --------------
            2
        (c;q )  (d;q)

∞  ∞

In[4]:= %/.pqinfzus[]
                (a; q)
            2  ∞
        (b;q )  -------
              ∞ (d; q)

∞

Out[4]= ---------------
                2
            (c;q )

∞

In[5]:= %/.pqinfzus[q^{∧}2]

Alphabetic List: pqzerl 113

                     2
        (a; q)  (b; q )

∞  ∞

Out[5]= ------- --------
        (d; q)       2
              ∞ (c; q )

∞

\underline{See also}: pqaufl, pqzerl, pqzus, pqinf, pqinfzerl, Ers, PosListe.

PQSort \underline{Description}: Rule that orders the parameters of basic hypergeometric series ph[List1,List2,q,z], Ph[ …],

ps[ …], W[ …], of “multiple” upper q-factorials pq[List1,List2,n,q], and of “multiple” infinite q-factorials pqinf[List1,List2,q] in a standard order. For instance, this function can be used for a quick test if two expressions agree. It is recommended to apply pqinfzus[q] and pqzus[n,q] first.

\underline{Usage}: Expr/.PQSort.

\underline{Example(s)}:

In[1]:= pq[{c,c/a,b},{a/b/c,b*q},n,q^{∧}2]*pqinf[{b/a,c},{a*b*c,1/b},q]*
        ph[{b/c,b*a,a},{b,a},1/q,z]
           b                                   c      2
          (-, c; q)         b              (c, -, b; q )
           a  ∞  -, a b  1          a        n
Out[1]= -------------- ϕ c     ; -, z   ---------------
                1      2 1          q        a         2
        (a b c, -; q)         b            (---, b q; q )
                b ∞  b c          n
In[2]:= %/.PQSort
           b                                      c   2
          (-, c; q)              b         (b, c, -; q )
           a  ∞  a b, -  1             a     n
Out[2]= -------------- ϕ  c; -, z   ---------------
         1             2 1          q        a         2
        (-, a b c; q)         b            (---, b q; q )
         b  ∞  b c          n
\underline{See also}: SimplifyPQ, SUMExpand, phEinf, phOrdne, phPerm, phTausche, ph, Ph, ps, W, pq, pqinf.

pqzerl \underline{Description}: Rule that splits pq[List1,List2,n,q] into a quotient of products of q-factorial symbols.

\underline{Usage}: Expr/.pqzerl.

\underline{Example(s)}:

In[1]:= pq[{a,b},{c,d},m,q^{∧}2]

                2
        (a, b; q )
                  m
Out[1]= -----------
                2
        (c, d; q )
                  m
In[2]:= %/.pqzerl
             2        2
        (a; q )  (b; q )
               m        m
Out[2]= -----------------
             2        2
        (c; q )  (d; q )
               m        m
\underline{See also}: pqaufl, pqzus, pq, pqinfzerl, pqinfzus, Ers, PosListe.

pqzus

\underline{Description}: Rule that collects several q-factorial symbols pq[xi,n,q] to an expression pq[List1,List2,n,q]. The parameter q is optional. It is set equal q if it is omitted.

\underline{Usage}: Expr/.pqzus[n,q]
   or: Expr/.pqzus[n].

\underline{Example(s)}:

In[1]:= pq[a,n]*pq[b,m,q^{∧}2]/pq[c,n]/pq[d,m,q^{∧}2]
                     2
        (a; q)  (b; q )
              n        m
Out[1]= ----------------
                     2
        (c; q)  (d; q )
              n        m
In[2]:= %/.pqzus[n,q]
                 (a; q)
             2         n
        (b; q )  -------
               m (c; q)
                       n
Out[2]= ----------------
                 2
            (d; q )

In[3]:= %/.pqzus[m,q^{∧}2]
                     2
        (a; q)  (b; q )
              n        m
Out[3]= ------- --------
        (c; q)       2
              n (d; q )
                       m
\underline{See also}: pqaufl, pqzerl, pq, pqinfzus, pqinfzerl, Ers, PosListe.

ps

\underline{Description}: ps[List1,List2,q,z] is the bilateral basic hypergeometric series with upper parameters List1, lower parameters List2, base q, and argument z.

\underline{Usage}: ps[List1,List2,q,z].

\underline{Example(s)}:

In[1]:= ps[{a,b,c},{d,e,f},q,z]

              a, b, c
Out[1]=  ps          ; q, z
        3  3  d, e, f

\underline{See also}:  SListe, TListe, SUMRegeln, SUMps, psSUM, ph, Ph, pq, pqinf, phCancel, phOrdne, phPerm,
  phTausche, PQ, phFormat.

psEinf

\underline{Description}: Rule that inactivates automatic cancelling in ps[] and then adds a parameter which has to be entered on request to the upper and lower parameters of ps[].

\underline{Usage}: Expr/.psEinf.

\underline{Example(s)}:

In[1]:= ps[{a,b,c},{d,e,f},q,z]

              a, b, c
Out[1]=  ps          ; q, z
        3  3  d, e, f

In[2]:= %/.psEinf
Add the parameter:  A

              A, a, b, c
Out[2]=  ps             ; q, z
        4  4  A, d, e, f

\underline{See also}: phCancel, psOrdne, psPerm, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe.

psinv

\underline{Description}: Rule that transforms a bilateral basic hypergeometric series ps[List1,List2,q,z] with base q into a bilateral basic hypergeometric series ps[ …,1/q,..] with base 1/q.

\underline{Usage}: Expr/.psinv.

\underline{Example(s)}:

In[1]:= ps[{a,b},{c,d},q,z]

              a, b
Out[1]=  ps       ; q, z
        2  2  c, d

In[2]:= %/.psinv
              1  1
              -, -
              a  b  1  a b z
Out[2]=  ps       ; -, -----
        2  2  1  1  q   c d
              -, -
              c  d
In[3]:= ps[{a,b},{c,d,0},q,z]

               a, b
Out[3]=  ps          ; q, z
        2  3  c, d, 0

In[4]:= %/.psinv

              1  1
              -, -, 0
              a  b     1  a b z
Out[4]=  ps          ; -, -----
        3  2   1  1    q   c d
               -, -
               c  d
\underline{See also}: SUMRegeln, Ers, PosListe, phinv, Phinv.

psOrdne

\underline{Description}: Rule that tries to order the parameters of a bilateral basic hypergeometric series in “well-poised” order. If there is an upper parameter of the form q^{−n}, where n is a nonnegative integer, then it is put at the very last place in the upper list.

\underline{Usage}: Expr/.psOrdne.

\underline{Example(s)}:

In[1]:= ps[{q^{∧}-n,b,q*Sqrt[a],-q*Sqrt[a]},{a*q/b,-Sqrt[a],a*q^{∧}(1+n),
        Sqrt[a]},q,z]
               -n
              q  , b, Sqrt[a] q, -(Sqrt[a] q)
Out[1]=  ps                                   ; q, z
        4  4  a q               1 + n
              ---, -Sqrt[a], a q     , Sqrt[a]
               b
In[2]:= %/.psOrdne
                                           -n
              Sqrt[a] q, -(Sqrt[a] q), b, q
Out[2]=  ps                                   ; q, z
        4  4                     a q     1 + n
              Sqrt[a], -Sqrt[a], ---, a q
                                  b
\underline{See also}: psEinf, psPerm, ps, PQSort, Ers, PosListe.

psPerm

\underline{Description}: Rule for permuting parameters in bilateral basic hypergeometric series.

\underline{Usage}: Expr/.psPerm[⟨Permutation⟩,x].

x can be u, l, b. u causes a permutation of upper parameters, l causes a permutation of lower parameters, b causes a simultaneous permutation of respective upper and lower parameters. Permutation must be a sequence of positive numbers forming a permutation. The effect is that the new parameter at position i is the old parameter from

  position Permutation[i].

118 Alphabetic List: psph

\underline{Example(s)}:

In[1]:= ps[{a,b,c},{e,f,g},q,z]
              a, b, c
Out[1]=  ps          ; q, z
        3  3  e, f, g
In[2]:= %1/.psPerm[3,2,1,u]
              c, b, a
Out[2]=  ps          ; q, z
        3  3  e, f, g
In[3]:= %1/.psPerm[3,2,1,l]
              a, b, c
Out[3]=  ps          ; q, z
        3  3  g, f, e
In[4]:= %1/.psPerm[3,2,1,b]
              c, b, a
Out[4]=  ps          ; q, z
        3  3  g, f, e
\underline{See also}: psOrdne, ps, PQSort, Ers, PosListe, phPerm, PhPerm.

psph \underline{Description}: Rule that transforms a ps[] into a sum of two ph[]’s.

rψ_{s}[ \begin{matrix}a_{1}, …, a_{r} \\ b_{1}, …, b_{s} \end{matrix}; q, z] → ∑_{n=−∞}^{m−1} \frac{(a_{1}; q)_{n} ⋯ (a_{r}; q)_{n}}{(b_{1}; q)_{n} ⋯ (b_{s}; q)_{n}} ((−1)^{n}q^{\binom{n}{2}})^{s−r+1} z^{n}

+ ∑_{n=m}^{∞} \frac{(a_{1}; q)_{n} ⋯ (a_{r}; q)_{n}}{(b_{1}; q)_{n} ⋯ (b_{s}; q)_{n}} ((−1)^{n}q^{\binom{n}{2}})^{s−r+1} z^{n}.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.psph.

\underline{Example(s)}:

In[1]:= ps[{a,b,c},{e,f,g},q,z]

              a, b, c
Out[1]=  ps          ; q, z
        3  3  e, f, g

In[2]:= %/.psph
Split at:  0

             a, b, c, q
Out[2]= ϕ  ; q, z   +
        4 3   e, f, g

           2   2   2
          q   q   q
          --, --, --, q               (a, b, c; q)
          e   f   g          e f g                -1
      ϕ  ; q, -------   --------------
     4 3    2   2   2       a b c z   (e, f, g; q)
           q   q   q                              -1
           --, --, --
           a   b   c
⟩  -----------------------------------------------
                            z
In[3]:= %1/.psph
Split at:  4

                 1     1     1
                ----, ----, ----, q
                   2     2     2                  (a, b, c; q)
         3      e q   f q   g q          e f g                3
Out[3]= z ϕ  ; q, -------   ------------- +
           4 3    1     1     1         a b c z   (e, f, g; q)
                 ----, ----, ----                             3
                    2     2     2
                 a q   b q   c q
                4     4     4            (a, b, c; q)
      4      a q , b q , c q , q                     4
⟩  z ϕ  ; q, z   -------------
        4 3      4     4     4           (e, f, g; q)
              e q , f q , g q                        4
\underline{See also}: ps, ph, phps, Ers, PosListe.

psShift

\underline{Description}: Rule that shifts the summation index in a bilateral basic hypergeometric series.

rψ_{s}[ \begin{matrix}a_{1}, …, a_{r} \\ b_{1}, …, b_{s} \end{matrix}; q, z] → z^{m} ((−1)^{m}q^{\binom{m}{2}})^{s−r} \frac{∏_{i=1}^{r}(a_{i}; q)_{m}}{∏_{i=1}^{s}(b_{i}; q)_{m}}_{r}ψ_{s}[ \begin{matrix}a_{1}q^{m}, …, a_{r}q^{m} \\ b_{1}q^{m}, …, b_{s}q^{m} \end{matrix}; q, zq^{m(s−r)}].

The parameter m has to be entered on request.

\underline{Usage}: Expr/.psShift.

\underline{Example(s)}:

In[1]:= ps[{a,b,c},{e,f,g},q,z]

              a, b, c
Out[1]=  ps          ; q, z
        3  3  e, f, g

In[2]:= %/.psShift
shift by:  5
           (a, b, c; q)           5     5     5
         5             5       a q , b q , c q
Out[2]= z  -------------  ps                   ; q, z
           (e, f, g; q)  3  3     5     5     5
                       5       e q , f q , g q
\underline{See also}: psEinf, psPerm, ps, PSort, Ers, PosListe.

psSUM

\underline{Description}: Rule that transforms a ps[] into a SUM[].

\underline{Usage}: Expr/.psSUM.

\underline{Example(s)}:

In[1]:= ps[{a,b,c,q^{∧}-n},{e,f,q^{∧}m},q,z]
                        -n
              a, b, c, q
Out[1]=  ps               ; q, z
        4  3           m
                e, f, q
In[2]:= %/.psSUM
Is n a nonnegative integer?
[y|n]:    n
Is m a nonnegative integer?
[y|n]:    n
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k
                     2
          ∞ k/2 - k /2  k                           -n
             q           z  (a; q)  (b; q)  (c; q)  (q  ; q)
         \  k       k       k         k
Out[2]= ⟩ ------------------------------------------------
         /                k                   m
                      (-1)  (e; q)  (f; q)  (q ; q)
        k=-∞  k       k        k
In[3]:= %1/.psSUM
Is n a nonnegative integer?
[y|n]:    y
Is m a nonnegative integer?
[y|n]:    y
A basic hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k

                       2
           n    k/2 - k /2  k                           -n
               q           z  (a; q)  (b; q)  (c; q)  (q  ; q)
          \  k       k       k         k
Out[3]=  ⟩ ------------------------------------------------
          /                 k                   m
                        (-1)  (e; q)  (f; q)  (q ; q)
        k=1 - m                     k       k        k
\underline{See also}: ps, SUM, SUMps, Ers, PosListe.

RS

\underline{Description}: RS is the right-hand side in Gleichung.

\underline{Usage}: RS.

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= RS
         n  c
        a  (-; q)
            a    n
Out[2]= ----------
         (c; q)
               n
In[3]:= Add[1]
                                        n  c
                                       a  (-; q)
                     -n                    a    n
Out[3]= 1 + ϕ a, q  ; q, q   == 1 + ----------
            2 1                         (c; q)
                   c                          n
In[4]:= RS

             n  c
            a  (-; q)
                a    n
Out[4]= 1 + ----------
             (c; q)
                   n
In[5]:= RS=1/pq[q,m]
           1
Out[5]= -------
        (q; q)
              m
In[6]:= Gleichung

                     -n               1
Out[6]= 1 + ϕ a, q  ; q, q   == -------
            2 1                    (q; q)
                   c                     m
\underline{See also}:  Gleichung, SumListe$gl, TransListe$gl, LS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers,
  Subst.

S1001

\underline{Description}: Summation formula ([1], (1.3.2); Appendix (II.3)) in form of a rule.

1ϕ_{0}[ \begin{matrix}a \\ − \end{matrix}; q, z] ⟶ \frac{(az; q)_{∞}}{(z; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S0110

\underline{Description}: Summation formula ([1], (1.6.1); Appendix (II.28)) in form of a rule.

0ψ_{1}[ \begin{matrix}− \\ 0 \end{matrix} ; q, z] ⟶ (q, q/z, z; q)_{∞}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S1101

\underline{Description}: Summation formula ([1], Ex. 1.6(ii); Appendix (II.5)) in form of a rule.

1ϕ_{1}[ \begin{matrix}a \\ c \end{matrix}; q, \frac{c}{a}] ⟶ \frac{(c/a; q)_{∞}}{(c; q)_{∞}}

S1102 \underline{Description}: Summation formula ([1], (1.8.1); Appendix (II.9); b → ∞) in form of a rule.

1ϕ_{1}[ \begin{matrix}a \\ 0 \end{matrix} ; q, −q] ⟶ (−q; q)_{∞}(aq; q^{2})_{∞}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S1110 \underline{Description}: Summation formula ([1], (5.2.1); Appendix (II.29)) in form of a rule.

1ψ_{1}[ \begin{matrix}a \\ b \end{matrix}; q, z] ⟶ \frac{(q, b/a, az, q/az; q)_{∞}}{(b, q/a, z, b/az; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2101 \underline{Description}: Summation formula ([1], (1.5.3); Appendix (II.6)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, q^{−n} \\ c \end{matrix}; q, q] ⟶ \begin{matrix}a^{n} & (c/a; q) & n \\ (c; q) & n \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2102 \underline{Description}: Summation formula ([1], (1.5.2); Appendix (II.7)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, q^{−n} \\ c \end{matrix}; q, \frac{cq^{n}}{a} ] ⟶ \begin{matrix}(c/a; q) & n \\ (c; q) & n \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2103 \underline{Description}: Summation formula ([1], (1.5.1); Appendix (II.8)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, \frac{c}{ab}] ⟶ \frac{(c/a, c/b; q)_{∞}}{(c, c/ab; q)_{∞}}

\underline{Example(s)}:

In[1]:= ph[{a,b},{q^{∧}2*a*b},q,q^{∧}2]
              a, b       2
Out[1]= ϕ  ; q, q
        2 1       2
             a b q

Alphabetic List: S2107 125

In[2]:= %/.S2103
            2     2
        (b q , a q ; q)

∞

Out[2]= ----------------
              2   2
        (a b q , q ; q)

∞

\underline{See also}: S3201, SListe, SumListe, Ers, PosListe.

S2104 \underline{Description}: Summation formula ([1], (1.8.1); Appendix (II.9)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ aq/b \end{matrix}; q, −\frac{q}{b} ] ⟶ \frac{(−q; q)_{∞}}{(−q/b, aq/b; q)_{∞}} (aq, aq^{2}/b^{2}; q^{2})_{∞}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2105 \underline{Description}: Summation formula ([1], Ex 1.6(i)) in form of a rule.

2ϕ_{1}[ \begin{matrix}q^{−n}, q^{1−n} \\ b^{2}q \end{matrix}; q^{2}, q^{2}] ⟶ q^{−\binom{n}{2}} \frac{(b^{2}; q^{2})_{n}}{(b^{2}; q)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2106 \underline{Description}: Summation formula ([1], Ex 1.7) in form of a rule.

2ϕ_{1}[ \begin{matrix}a^{2}, aq \\ a \end{matrix}; q, z] ⟶ \frac{(1 + az) (a^{2}qz; q)_{∞}}{(z; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2107 \underline{Description}: Summation formula ([1], Ex. 1.8) in form of a rule.

2ϕ_{1}[ \begin{matrix}a^{2}, a^{2}/b \\ b \end{matrix}; q^{2}, \frac{bq}{a^{2}} ] ⟶ \frac{1}{2}(\frac{(−b/a; q)_{∞}}{(−a; q)_{∞}} + \frac{(b/a; q)_{∞}}{(a; q)_{∞}} ) \frac{(a^{2}, q; q^{2})_{∞}}{(b, bq/a^{2}; q^{2})_{∞}}

126 Alphabetic List: S3201

S2161 \underline{Description}: Summation formula ([1], (2.10.13); Appendix (II.23)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, q] ⟶ \frac{(q/c, abq/c; q)_{∞}}{(aq/c, bq/c; q)_{∞}} − \frac{(q/c, a, b; q)_{∞}}{(c/q, aq/c, bq/c; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}aq/c, bq/c \\ q^{2}/c \end{matrix}; q, q]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2201 \underline{Description}: Summation formula ([1], Ex. 1.19(i); Appendix (II.10)) in form of a rule.

2ϕ_{2}[ \begin{matrix}a, q/a \\ −q, b \end{matrix}; q, −b] ⟶ \begin{matrix}(ab, bq/a; q^{2}) & ∞ \\ (b; q) & ∞ \end{matrix}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2202 \underline{Description}: Summation formula ([1], Ex. 1.19(ii); Appendix (II.11)) in form of a rule.

2ϕ_{2}[ \begin{matrix}a^{2}, b^{2} \\ ab\sqrt{q}, −ab\sqrt{q} \end{matrix}; q, −q] ⟶ \frac{(a^{2}q, b^{2}q; q^{2})_{∞}}{(q, a^{2}b^{2}q; q^{2})_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2210 \underline{Description}: Summation formula ([1], (5.3.4); Appendix (II.30)) in form of a rule.

2ψ_{2}[ \begin{matrix}b, c \\ aq/b, aq/c \end{matrix}; q, −\frac{aq}{bc} ] ⟶ \frac{(aq/bc; q)_{∞}}{(aq/b, aq/c, q/b, q/c, −aq/bc; q)_{∞}} (aq^{2}/b^{2}, aq^{2}/c^{2}, q^{2}, aq, q/a; q^{2})_{∞}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3201 \underline{Description}: Summation formula ([1], (1.7.2); Appendix (II.12)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, q^{−n} \\ c, abq^{1−n}/c \end{matrix}; q, q] ⟶ \begin{matrix}(c/a; q) & n & (c/b; q) & n \\ (c; q) & n & (c/ab; q) & n \end{matrix}

where n is a nonnegative integer. \underline{Example(s)}:

In[1]:= ph[{a,b,q^{∧}-n},{a*b/q^{∧}n,q},q,q]
                    -n
             a, b, q
Out[1]= ϕ  a b     ; q, q
        3 2   ---, q
               n
              q

Alphabetic List: S3201 127

In[2]:= %/.S3201
Is n a nonnegative integer?
[y|n]:    y
          a        b
         (--; q)  (--; q)
           n    n   n    n
          q        q
Out[2]= -------------------
          -n       a b
        (q  ; q)  (---; q)
                n   n     n
                   q
In[3]:= %%/.S3201
Is n a nonnegative integer?
[y|n]:    n
                    -n
             a, b, q
Out[3]= ϕ  a b     ; q, q
        3 2   ---, q
               n
              q
In[4]:= ph[{a,q^{∧}-m,q^{∧}-n},{a/q^{∧}(n+m),q},q,q]
                 -m   -n
             a, q  , q
Out[4]= ϕ  ; q, q
        3 2     -m - n
             a q      , q
In[5]:= %/.S3201
Is n a nonnegative integer?
[y|n]:    n
Is m a nonnegative integer?
[y|n]:    y

128 Alphabetic List: S3261

          a         -m - n
         (--; q)  (q      ; q)
           m    m             m
          q
Out[5]= -------------------------
          -m          -m - n
        (q  ; q)  (a q      ; q)
                m               m
\underline{See also}: S2103, SListe, SumListe, Ers, PosListe.

S3202 \underline{Description}: Summation formula ([1], Ex. 2.7; Appendix (II.15)) in form of a rule.

3ϕ_{2}[ \begin{matrix}q^{−2n}, b, c \\ q^{1−2n}/b, q^{1−2n}/c \end{matrix}; q, \frac{q^{2−n}}{bc} ] ⟶ \begin{matrix}(b; q) & n & (c; q) & n & (bc; q) & 2n & (q; q) & 2n \\ (b; q) & 2n & (c; q) & 2n & (bc; q) & n & (q; q) & n \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3203 \underline{Description}: Summation formula ([1], Ex. 2.1) in form of a rule.

3W_{2}(a; −; q, t) ⟶ (1 − aqt^{2}) \frac{(aq^{2}t; q)_{∞}}{(t; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3204 \underline{Description}: Summation formula ([1], Ex. 3.9) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, λq, b \\ λ, λ^{2}q/b \end{matrix}; q, \frac{λ^{2}}{ab^{2}} ] ⟶ \frac{(1 − λ + \frac{λ}{b} (1 − \frac{λ}{a}))}{(1 − λ) (1 + \frac{λ}{b})} \frac{(λ^{2}/b^{2}, λ^{2}q/ab; q)_{∞}}{(λ^{2}q/b, λ^{2}/ab^{2}; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3261 \underline{Description}: Summation formula ([1], (2.10.12); Appendix (II.24)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ e, abcq/e \end{matrix}; q, q]

⟶ \frac{(q/e, bcq/e, acq/e, abq/e; q)_{∞}}{(aq/e, bq/e, cq/e, abcq/e; q)_{∞}} − \frac{(q/e, a, b, c, abcq^{2}/e^{2}; q)_{∞}}{(e/q, aq/e, bq/e, cq/e, abcq/e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}aq/e, bq/e, cq/e \\ q^{2}/e, abcq^{2}/e^{2} \end{matrix} ; q, q]

S3310 \underline{Description}: Summation formula ([1], Ex. 5.18(i); Appendix (II.31)) in form of a rule.

3ψ_{3}[ \begin{matrix}b, c, d \\ q/b, q/c, q/d \end{matrix}; q, \frac{q}{bcd}] ⟶ \frac{(q, q/bc, q/bd, q/cd; q)_{∞}}{(q/b, q/c, q/d, q/bcd; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4301 \underline{Description}: Summation formula ([1], (2.7.2); Appendix (II.13)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, −\sqrt{a}q, b, c \\ −\sqrt{a}, aq/b, aq/c \end{matrix}; q, \frac{\sqrt{a}q}{bc} ] ⟶ \frac{(aq, \sqrt{a}q/b, \sqrt{a}q/c, aq/bc; q)_{∞}}{(aq/b, aq/c, \sqrt{a}q, \sqrt{a}q/bc; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4302 \underline{Description}: Summation formula ([1], (2.7.2), terminating form; Appendix (II.14)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, −\sqrt{a}q, b, q^{−n} \\ −\sqrt{a}, aq/b, aq^{1+n} \end{matrix}; q, \frac{\sqrt{a}q^{1+n}}{b} ] ⟶ \begin{matrix}(aq; q) & n & (\sqrt{a}q/b; q) & n \\ (\sqrt{a}q; q) & n & (aq/b; q) & n \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4303 \underline{Description}: Summation formula ([1], Ex. 2.8; Appendix (II.17)) in form of a rule.

4ϕ_{3}[ \begin{matrix}aq^{n}, c, −c, q^{−n} \\ \sqrt{a}\sqrt{q}, −\sqrt{a}\sqrt{q}, c^{2} \end{matrix} ; q, q] ⟶ \{ \begin{matrix}0 & if n is odd \\ \frac{c^{n}(q; q^{2})_{\frac{n}{2}} (aq/c^{2}; q^{2})_{\frac{n}{2}}}{(aq; q^{2})_{\frac{n}{2}} (c^{2}q; q^{2})_{\frac{n}{2}}} & if n is even \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4304 \underline{Description}: Summation formula (Andrews [1976a] in [1], Appendix (II.19), corrected) in form of a rule.

4ϕ_{3}[ \begin{matrix}q^{1+n}, c, −c, q^{−n} \\ e, c^{2}q/e, −q \end{matrix}; q, q] ⟶ q^{n(n+1)/2} \begin{matrix}(e/q^{n}, eq^{1+n}, c^{2}q^{1−n}/e, c^{2}q^{2+n}/e; q^{2}) & ∞ \\ (e, c^{2}q/e; q) & ∞ \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4305 \underline{Description}: Summation formula ([1], Ex. 2.6) in form of a rule.

4ϕ_{3}[ \begin{matrix}q^{−n}, b, c, −q^{1−n}/bc \\ q^{1−n}/b, q^{1−n}/c, −bc \end{matrix}; q, q] ⟶ \{ \begin{matrix}0 & if n is odd \\ \frac{(bc; q)_{n}}{(b, c; q)_{n}} \frac{(q, b^{2}, c^{2}; q^{2})\frac{n}{2}}{(b^{2}c^{2}; q^{2})\frac{n}{2}} & if n is even \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4306 \underline{Description}: Summation formula ([1], Ex. 2.14(i)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, \sqrt{a}q, b, q^{−n} \\ \sqrt{a}, aq/b, b^{2}q^{1−n} \end{matrix}; q, q] ⟶ \frac{(a/b^{2}, 1/b, −\sqrt{a}q/b; q)_{n}}{(b^{−2}, aq/b, −\sqrt{a}/b; q)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4307 \underline{Description}: Summation formula ([1], (2.7.2)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, \sqrt{a}q, b, c \\ \sqrt{a}, aq/b, aq/c \end{matrix}; q, −\frac{\sqrt{a}q}{bc} ] ⟶ \frac{(aq, aq/bc, −\sqrt{a}q/b, −\sqrt{a}q/c; q)_{∞}}{(aq/b, aq/c, −\sqrt{a}q, −\sqrt{a}q/bc; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4308 \underline{Description}: Summation formula ([1], (2.8.3), c → \sqrt{aq}, d → −q\sqrt{a}, sum the _{8}ϕ_{7} by [1], (2.6.2)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, −\sqrt{a}q, b, q^{−n} \\ −\sqrt{a}, aq/b, b^{2}q^{1−n} \end{matrix}; q, q] ⟶ \frac{(a/b^{2}, 1/b, \sqrt{a}q/b; q)_{n}}{(b^{−2}, aq/b, \sqrt{a}/b; q)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4361 \underline{Description}: Summation formula ([1], Ex. 2.9) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, −b, aq/c^{2} \\ aq/c, −aq/c, b^{2} \end{matrix} ; q, q] ⟶ \frac{(q/b^{2}, −q; q)_{∞}}{(aq/b^{2}, −aq; q)_{∞}} \frac{(a^{2}q^{2}, aq^{2}/b^{2}, aq^{2}/c^{2}, a^{2}q^{2}/b^{2}c^{2}; q^{2})_{∞}}{(q^{2}/b^{2}, a^{2}q^{2}/c^{2}, aq^{2}, aq^{2}/b^{2}c^{2}; q^{2})_{∞}}

− \frac{(q/b^{2}, a, b, −b, aq/c^{2}, aq^{2}/b^{2}c, −aq^{2}/b^{2}c; q)_{∞}}{(b^{2}/q, q/b, −q/b, aq/b^{2}, aq/c, −aq/c, aq^{2}/b^{2}c^{2}; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}q/b, −q/b, aq/b^{2}, aq^{2}/b^{2}c^{2} \\ aq^{2}/b^{2}c, −aq^{2}/b^{2}c, q^{2}/b^{2} \end{matrix} ; q, q]

S4410 \underline{Description}: Summation formula ([1], (5.3.3); Appendix (II.32)) in form of a rule.

4ψ_{4}[ \begin{matrix}−\sqrt{a}q, b, c, d \\ −\sqrt{a}, aq/b, aq/c, aq/d \end{matrix}; q, \frac{a^{\frac{3}{2}} q}{bcd} ] ⟶ \frac{(aq, aq/bc, aq/bd, aq/cd, \sqrt{a}q/b, \sqrt{a}q/c, \sqrt{a}q/d, q, q/a; q)_{∞}}{(aq/b, aq/c, aq/d, q/b, q/c, q/d, \sqrt{a}q, q/\sqrt{a}, a^{3/2}q/bcd; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S5401 \underline{Description}: Summation formula ([1], (2.7.1), d → \sqrt{aq}) in form of a rule.

5W_{4}(a; b, c; q, \frac{\sqrt{aq}}{bc}) ⟶ \frac{(aq, aq/bc, \sqrt{aq}/b, \sqrt{aq}/c; q)_{∞}}{(aq/b, aq/c, \sqrt{aq}, \sqrt{aq}/bc; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S5402 \underline{Description}: Summation formula ([1], (2.8.3), c → q\sqrt{a}, d → −q\sqrt{a}, sum the _{8}ϕ_{7} by [1], (2.6.2)) in form of a rule.

5ϕ_{4}[ \begin{matrix}a, \sqrt{a}q, −\sqrt{a}q, b, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, b^{2}q^{2−n} \end{matrix}; q, q] ⟶ \frac{(a/b^{2}q, 1/bq; q)_{n}}{(1/b^{2}q, aq/b; q)_{n}} \frac{(aq/b^{2}; q^{2})_{n}}{(a/b^{2}q; q^{2})_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S5501 \underline{Description}: Summation formula ([1], (2.7.1), d → ∞) in form of a rule.

5ϕ_{5}[ \begin{matrix}a, \sqrt{a}q, −\sqrt{a}q, b, c \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, 0 \end{matrix} ; q, \frac{aq}{bc} ] ⟶ \frac{(aq, aq/bc; q)_{∞}}{(aq/b, aq/c; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S6501 \underline{Description}: Summation formula ([1], (2.7.1); Appendix (II.20)) in form of a rule.

6W_{5}(a; b, c, d; q, \frac{aq}{bcd}) ⟶ \frac{(aq, aq/bc, aq/bd, aq/cd; q)_{∞}}{(aq/b, aq/c, aq/d, aq/bcd; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S6502 \underline{Description}: Summation formula ([1], (2.4.2); Appendix (II.21)) in form of a rule.

6W_{5}(a; b, c, q^{−n}; q, \frac{aq^{1+n}}{bc}) ⟶ \begin{matrix}(aq; q) & n & (aq/bc; q) & n \\ (aq/b; q) & n & (aq/c; q) & n \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

132 Alphabetic List: S8761

S6610 \underline{Description}: Summation formula ([1], (5.3.1); Appendix (II.33)) in form of a rule.

6ψ_{6}[ \begin{matrix}\sqrt{a}q, −\sqrt{a}q, b, c, d, e \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e \end{matrix}; q, \frac{a^{2}q}{bcde}] ⟶ \frac{(aq, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de, q, q/a; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, a^{2}q/bcde; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S8701 \underline{Description}: Summation formula ([1], (2.6.2); Appendix (II.22)) in form of a rule.

8ϕ_{7}[ \begin{matrix}a, \sqrt{a}q, −\sqrt{a}q, b, c, d, a^{2}q^{1+n}/bcd, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, bcd/aq^{n}, aq^{1+n} \end{matrix}; q, q] ⟶ \begin{matrix}(aq; q) & n & (aq/bc; q) & n & (aq/bd; q) & n & (aq/cd; q) & n \\ (aq/b; q) & n & (aq/c; q) & n & (aq/d; q) & n & (aq/bcd; q) & n \end{matrix}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S8702 \underline{Description}: Summation formula ([1], Ex. 2.17(i); Appendix (II.16)) in form of a rule.

8W_{7}(−\frac{c\sqrt{ab}}{\sqrt{q}}; a, b, c, −c, −\frac{\sqrt{abq}}{c}; q, \frac{c\sqrt{q}}{\sqrt{ab}}) ⟶ \frac{(−c\sqrt{abq}, −c\sqrt{q}/\sqrt{ab}; q)_{∞}}{(−c\sqrt{bq}/\sqrt{a}, −c\sqrt{aq}/\sqrt{b}; q)_{∞}} \frac{(aq, bq, c^{2}q/a, c^{2}q/b; q^{2})_{∞}}{(q, abq, c^{2}q, c^{2}q/ab; q^{2})_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S8703 \underline{Description}: Summation formula ([1], Ex. 2.17(ii); Appendix (II.18)) in form of a rule.

8W_{7}(−c; a, \frac{q}{a}, c, −d, −\frac{q}{d}; q, c) ⟶ \frac{(−c, −cq; q)_{∞}}{(cd, cq/d, −ac, −cq/a; q)_{∞}} (acd, acq/d, cdq/a, cq^{2}/ad; q^{2})_{∞}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S8704 \underline{Description}: Summation formula ([1], Ex. 3.10) in form of a rule.

8W_{7}(−λ; \sqrt{λ}q, −\sqrt{λ}q, a, b, −b; q, \frac{λ}{ab^{2}} ) ⟶ \frac{(λq, λ/a, −λq, λ/b^{2}, λq/ab, −λq/ab; q)_{∞}}{(λ, λq/a, −λq/a, λq/b, −λq/b, λ/ab^{2}; q)_{∞}}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S8761 \underline{Description}: Summation formula ([1], (2.11.7); Appendix (II.25)) in form of a rule.

8W_{7}(a; b, c, d, e, \frac{a^{2}q}{bcde}; q, q) ⟶ \frac{(aq, b/a, aq/cd, aq/ce, bde/a, aq/de, bce/a, bcd/a; q)_{∞}}{(aq/c, aq/d, aq/e, bcde/a, bc/a, bd/a, be/a, aq/cde; q)_{∞}}

− \frac{(aq, b/a, c, d, e, a^{2}q/bcde, bq/c, bq/d, bq/e, b^{2}cde/a^{2}; q)_{∞}}{(a/b, aq/c, aq/d, aq/e, bcde/a, bc/a, bd/a, be/a, aq/cde, b^{2}q/a; q)_{∞}}

8W_{7}(\frac{b^{2}}{a}; b, \frac{bc}{a}, \frac{bd}{a}, \frac{be}{a}, \frac{aq}{cde}; q, q)

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S10901

\underline{Description}: Summation formula ([1], Ex. 2.12) in form of a rule.

10W_{9}(a; \sqrt{b}, −\sqrt{b}, \sqrt{bq}, −\sqrt{bq}, \frac{a}{b}, \frac{a^{2}q^{1+n}}{b} , q^{−n}; q, q) ⟶ \frac{(aq, a^{2}q/b^{2}; q)_{n}}{(aq/b, a^{2}q/b; q)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

SchreibeZahl

\underline{Description}: Variable that counts the number of expressions already written by using TeXMat. Can be reset by defining a new value.

\underline{Usage}: SchreibeZahl=n Integer.

\underline{Example(s)}:

In[1]:= SchreibeZahl
Out[1]= 0
In[2]:= TeXMat[pq[a,n],file]
In[3]:= !type file.m
A[1]:=
pq[a, n, q]
In[3]:= !type file.tex
A[1]:=
({\let \over / a}; q) {n}
In[3]:= SchreibeZahl=4
Out[3]= 4
In[4]:= TeXMat[pq[a,n],file]
In[5]:= !type file.m
A[1]:=
pq[a, n, q]
A[5]:=
pq[a, n, q]
In[5]:= !type file.tex
A[1]:=
({\let \over / a}; q) {n}
A[5]:=
({\let \over / a}; q) {n}

\underline{See also}: TeXMat.

134 Alphabetic List: Sgl2101

Sgl0110 \underline{Description}: Summation formula ([1], (1.6.1); Appendix (II.28)) in form of an equation. It is the same summation as

that in S0110.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl1001 \underline{Description}: Summation formula ([1], (1.3.2); Appendix (II.3)) in form of an equation. It is the same summation as

that in S1001.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl1101 \underline{Description}: Summation formula ([1], Ex. 1.6(ii); Appendix (II.5)) in form of an equation. It is the same summation

as that in S1101.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl1102 \underline{Description}: Summation formula ([1], (1.8.1); Appendix (II.9); b → ∞) in form of an equation. It is the same

summation as that in S1102.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl1110 \underline{Description}: Summation formula ([1], (5.2.1); Appendix (II.29)) in form of an equation. It is the same summation as

that in S1110.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2101 \underline{Description}: Summation formula ([1], (1.5.3); Appendix (II.6)) in form of an equation. It is the same summation as

that in S2101. \underline{Example(s)}:

In[1]:= {a,c,q}
Out[1]= {a, c, q}
In[2]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[2]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n

In[3]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  y
a=a^{∧}2
c=1/b
n=n
Do you want to set a value for q in the equation?  [y|n]:  y
q=q^{∧}2
                                     2 n   1     2
                                    a    (----; q )
              2   -2 n                     2       n
             a , q       2   2            a  b
Out[3]= ϕ  ; q , q    == ----------------
        2 1      1                       1   2
                 -                      (-; q )
                 b                       b     n
In[4]:= {a,c,q}
Out[4]= {a, c, q}
In[5]:= a=b^{∧}2
         2
Out[5]= b
In[6]:= q=p
Out[6]= p
In[7]:= Sgl2101
Some variables have a value.  Should the variables
{a, c, n} be cleared?  Do you want to set
values for the equation (v)?  [y|n|yv|nv]:  nv
a=a
c=c
n=n
q has a value.  Should q be cleared?
Do you want to set a value for q in the equation (v)?  [y|n|yv|nv]:  yv
q=p^{∧}2

                                     2 n  c    2
                                    b    (--; p )
                                           2     n
              2   -2 n   2   2            b
Out[7]= ϕ b , p    ; p , p    == --------------
        2 1                                 2
                 c                     (c; p )
                                              n
In[8]:= {a,c,q}
          2
Out[8]= {b , c, q}
In[9]:= Sgl2101
Some variables have a value.  Should the variables
{a, c, n} be cleared?  Do you want to set
values for the equation (v)?  [y|n|yv|nv]:  y
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[9]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[10]:= {a,c,q}
Out[10]= {a, c, q}
\underline{See also}: SumListe$gl, Gleichung.

Sgl2102

\underline{Description}: Summation formula ([1], (1.5.2); Appendix (II.7)) in form of an equation. It is the same summation as that in S2102.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2103

\underline{Description}: Summation formula ([1], (1.5.1); Appendix (II.8)) in form of an equation. It is the same summation as that in S2103.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2104

\underline{Description}: Summation formula ([1], (1.8.1); Appendix (II.9)) in form of an equation. It is the same summation as that in S2104.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Alphabetic List: Sgl3202 137

Sgl2105 \underline{Description}: Summation formula ([1], Ex 1.6(i)) in form of an equation. It is the same summation as that in S2105.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2106 \underline{Description}: Summation formula ([1], Ex 1.7) in form of an equation. It is the same summation as that in S2106.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2107 \underline{Description}: Summation formula ([1], Ex. 1.8) in form of an equation. It is the same summation as that in S2107.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2161 \underline{Description}: Summation formula ([1], (2.10.13); Appendix (II.23)) in form of an equation. It is the same summation

as that in S2161.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2201 \underline{Description}: Summation formula ([1], Ex. 1.19(i); Appendix (II.10)) in form of an equation. It is the same summation

as that in S2201.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2202 \underline{Description}: Summation formula ([1], Ex. 1.19(ii); Appendix (II.11)) in form of an equation. It is the same summation

as that in S2202.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2210 \underline{Description}: Summation formula ([1], (5.3.4); Appendix (II.30)) in form of an equation. It is the same summation as

that in S2210.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3201 \underline{Description}: Summation formula ([1], (1.7.2); Appendix (II.12)) in form of an equation. It is the same summation as

that in S3201.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3202 \underline{Description}: Summation formula ([1], Ex. 2.7; Appendix (II.15)) in form of an equation. It is the same summation as

that in S3202.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

138 Alphabetic List: Sgl4305

Sgl3203 \underline{Description}: Summation formula ([1], Ex. 2.1) in form of an equation. It is the same summation as that in S3203.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3204 \underline{Description}: Summation formula ([1], Ex. 3.9) in form of an equation. It is the same summation as that in S3204.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3261 \underline{Description}: Summation formula ([1], (2.10.12); Appendix (II.24)) in form of an equation. It is the same summation

as that in S3261.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3310 \underline{Description}: Summation formula ([1], Ex. 5.18(i); Appendix (II.31)) in form of an equation. It is the same summation

as that in S3310.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4301 \underline{Description}: Summation formula ([1], (2.7.2); Appendix (II.13)) in form of an equation. It is the same summation as

that in S4301.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4302 \underline{Description}: Summation formula ([1], (2.7.2), terminating form; Appendix (II.14)) in form of an equation. It is the

same summation as that in S4302.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4303 \underline{Description}: Summation formula ([1], Ex. 2.8; Appendix (II.17)) in form of an equation. It is the same summation as

that in S4303.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4304 \underline{Description}: Summation formula (Andrews [1976a] in [1], Appendix (II.19)) in form of an equation. It is the same

summation as that in S4304.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4305 \underline{Description}: Summation formula ([1], Ex. 2.6) in form of an equation. It is the same summation as that in S4305.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4306 \underline{Description}: Summation formula ([1], Ex. 2.14(i)) in form of an equation. It is the same summation as that in S4306.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4307 \underline{Description}: Summation formula ([1], (2.7.2)) in form of an equation. It is the same summation as that in S4307.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4308 \underline{Description}: Summation formula ([1], (2.8.3), c → \sqrt{aq}, d → −q\sqrt{a}, sum the _{8}ϕ_{7} by [1], (2.6.2)) in form of an equation.

It is the same summation as that in S4308.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4361 \underline{Description}: Summation formula ([1], Ex. 2.9) in form of an equation. It is the same summation as that in S4361.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4410 \underline{Description}: Summation formula ([1], (5.3.3); Appendix (II.32)) in form of an equation. It is the same summation as

that in S4410.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl5401 \underline{Description}: Summation formula ([1], (2.7.1), d → \sqrt{aq}) in form of an equation. It is the same summation as that in

  S5401.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl5402 \underline{Description}: Summation formula ([1], (2.8.3), c → q\sqrt{a}, d → −q\sqrt{a}, sum the _{8}ϕ_{7} by [1], (2.6.2)) in form of an equation.

It is the same summation as that in S5402.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl5501 \underline{Description}: Summation formula ([1], (2.7.1), d → ∞) in form of an equation. It is the same summation as that in

  S5501.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl6501 \underline{Description}: Summation formula ([1], (2.7.1); Appendix (II.20)) in form of an equation. It is the same summation as

that in S6501.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

140 Alphabetic List: Sgl10901

Sgl6502 \underline{Description}: Summation formula ([1], (2.4.2); Appendix (II.21)) in form of an equation. It is the same summation as

that in S6502.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl6610 \underline{Description}: Summation formula ([1], (5.3.1); Appendix (II.33)) in form of an equation. It is the same summation as

that in S6610.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl8701 \underline{Description}: Summation formula ([1], (2.6.2); Appendix (II.22)) in form of an equation. It is the same summation as

that in S8701.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl8702 \underline{Description}: Summation formula ([1], Ex. 2.17(i); Appendix (II.16)) in form of an equation. It is the same summation

as that in S8702.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl8703 \underline{Description}: Summation formula ([1], Ex. 2.17(ii); Appendix (II.18)) in form of an equation. It is the same summation

as that in S8703.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl8704 \underline{Description}: Summation formula ([1], Ex. 3.10) in form of an equation. It is the same summation as that in S8704.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl8761 \underline{Description}: Summation formula ([1], (2.11.7); Appendix (II.25)) in form of an equation. It is the same summation

as that in S8761.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl10901 \underline{Description}: Summation formula ([1], Ex. 2.12) in form of an equation. It is the same summation as that in S10901.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

SimplifyPQ

\underline{Description}: Rule that simplifies arguments in pq, pqinf, ph, Ph, ps, W, SUM, and expands exponents in powers.

\underline{Usage}: Expr/.SimplifyPQ.

\underline{Example(s)}:

In[1]:= pq[a^{∧}2+a,(k-l)*2]/pqinf[1-f^{∧}2,p]*ph[{a^{∧}(1/2-(1-s)/2),-n},{a^{∧}(s/2)},q,
        (t-1)/(1-t)]
              1/2 + (-1 + s)/2
             a                , -n     -1 + t         2
         ϕ  ; q, ------   (a + a ; q)
        2 1           s/2              1 - t               2 (k - l)
                     a
Out[1]= ------------------------------------------------------------
                                      2
                                (1 - f ;p)

∞

In[2]:= %/.SimplifyPQ

             -n
         ϕ  ; q, -1   (a (1 + a); q)
        1 0  -                         2 k - 2 l
Out[2]= ----------------------------------------
                  ((1 - f) (1 + f);p)

∞

In[3]:= Simplify[%1]

             -n                2
         ϕ  ; q, -1   (a + a ; q)
        1 0  -                      2 (k - l)
Out[3]= -------------------------------------
                           2
                     (1 - f ;p)

∞

\underline{See also}: Expandq, MinusOne, SUMExpand, PQSort.

SListe

\underline{Description}: Rule that gives for a basic hypergeometric series a list of applicable summation formulas. Each entry

  of this list has the format {ArgumentPermutations,S⟨number⟩}, where ArgumentPermutations is a sequence of

reorderings of the parameters of the basic hypergeometric series (given in terms of phPerm and phTausche) and S⟨number⟩ is the name of the summation in form of a rule which can be applied subsequently. You should be aware that SListe automatically applies phOrdne before checking which summation could be applied.

\underline{Important Note}: If the value returned by SListe is the empty set this does 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 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.

\underline{Usage}: Expr/.SListe.

\underline{Example(s)}:

In[1]:= ph[{a,b},{c},q^{∧}2,c/a/b]

             a, b   2   c
Out[1]= ϕ  ; q , ---
        2 1   c        a b

In[2]:= %/.SListe
Be sure to apply "phOrdne" before using the following information!
Out[2]= {{S2103}}
In[3]:= ph[{q^{∧}-n,-c,c,q^{∧}(1+n)},{-q,e,c^{∧}2*q/e},q,q]
              -n          1 + n
             q  , -c, c, q
Out[3]= ϕ  2      ; q, q
        4 3            c  q
                -q, e, ----
                        e
In[4]:= %/.SListe
Is n a nonnegative integer?
[y|n]:    y
Is -1 - n a nonnegative integer?
[y|n]:    n
Be sure to apply "phOrdne" before using the following information!
Out[4]= {{phPerm[3,1,2,4,u], phTausche[1,3,l], S4304}}
In[5]:= %3/.phOrdne/.phPerm[3,1,2,4,u]/.phTausche[1,3,l]/.S4304
Is n a nonnegative integer?
[y|n]:    y

                        2  1 - n   2  2 + n
              1        c  q       c  q       e      1 + n   2
Out[5]= ------------- (---------, ---------, --, e q     ; q )
          2                e          e       n  ∞
         c  q                                q
        (----, e; q)
          e  ∞

Now we consider two examples illustrating the note above. Though none of the implemented summations can be applied, both series are special cases of a q-analogue of Dixon’s sum. This fact is also observed by using this package.

In[6]:= ph[{a,-q*Sqrt[a],q^{∧}-n},{-Sqrt[a],q*a*q^{∧}n},q,-Sqrt[q]*q^{∧}n]
                               -n
             a, -(Sqrt[a] q), q         1/2 + n
Out[6]= ϕ  ; q, -q
        3 2                1 + n
              -Sqrt[a], a q
In[7]:= %/.SListe
Out[7]= {}
In[8]:= Sgl4301
Do you want to set values for the equation?  [y|n]:  y
a=a
b=q^{∧}-n
c=-Sqrt[a*q]
Do you want to set a value for q in the equation?  [y|n]:  n
                               -n
             a, -(Sqrt[a] q), q         1/2 + n
Out[8]= ϕ  ; q, -q          ==
        3 2                1 + n
              -Sqrt[a], a q
                    1 + n                       1/2 + n
     (a q, Sqrt[a] q     , -Sqrt[q], -(Sqrt[a] q       ); q)

∞

⟩  --------------------------------------------------------
         1 + n                                   1/2 + n
     (a q     , -(Sqrt[a] Sqrt[q]), Sqrt[a] q, -q       ; q)

∞

In[9]:= ph[{-Sqrt[a]*q,q,c},{-Sqrt[a],a*q/c},q,Sqrt[a]/c]

             -(Sqrt[a] q), q, c
                                    Sqrt[a]
Out[9]= ϕ  a q   ; q, -------
        3 2    -Sqrt[a], ---           c
                          c
In[10]:= %/.SListe
Out[10]= {}
In[11]:= %%/.phEinf
Add the parameter:  a
              a, -(Sqrt[a] q), q, c
                                        Sqrt[a]
Out[11]= ϕ  a q   ; q, -------
         4 3    a, -Sqrt[a], ---           c
                              c
In[12]:= %/.SListe
Be sure to apply "phOrdne" before using the following information!
Out[12]= {{S4301}}

\underline{See also}: TListe, phPerm, phTausche, SumListe.

Sub

\underline{Description}: Function that subtracts Expr from Gleichung.

\underline{Usage}: Sub[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                n  c
                               a  (-; q)
                 -n                a    n
Out[1]= ϕ a, q  ; q, q   == ----------
        2 1                     (c; q)
               c                      n
In[2]:= Sub[pq[a,n]/pq[c/a,n]]

                                                       n  c
                              (a; q)       (a; q)     a  (-; q)
                 -n                 n            n        a    n
Out[2]= ϕ a, q  ; q, q   - ------- == -(-------) + ----------
        2 1                    c            c          (c; q)
               c              (-; q)       (-; q)            n
                               a    n       a    n
In[3]:= Gleichung
                                                       n  c
                              (a; q)       (a; q)     a  (-; q)
                 -n                 n            n        a    n
Out[3]= ϕ a, q  ; q, q   - ------- == -(-------) + ----------
        2 1                    c            c          (c; q)
               c              (-; q)       (-; q)            n
                               a    n       a    n

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Hoch, GlTausche, Ers.

Subst

\underline{Description}: Function that substitutes RS instead of LS at position Position in Expr. The parameters LS and RS are optional. If they are omitted, the right-hand side “RS” of Gleichung is substituted instead of the left-hand side

  “LS” of Gleichung.
\underline{Usage}: Subst[Expr,Position,LS,RS].

\underline{Example(s)}:

In[1]:= SUM[pq[a,k]/pq[b,k]/pq[q,k],{k,0,Infinity}]

∞

                 (a; q)
         \  k
Out[1]= ⟩ ---------------
         /   (b; q)  (q; q)
                   k       k
         k=0
In[2]:= Subst[%,{1},pq[a,k]/pq[b,k],ph[{b/a,q^{∧}-k},{b},q,a*q^{∧}k]]

146 Alphabetic List: Subst

                  b   -k
                  -, q          k
          ∞ ϕ a     ; q, a q
             2 1
         \  b
Out[2]= ⟩ ----------------------
         /          (q; q)
                          k
         k=0
In[3]:= Sgl2102
Do you want to set values for the equation?  [y|n]:  y
a=b/a
c=b
n=k
Do you want to set a value for q in the equation?  [y|n]:  n
             b   -k               (a; q)
             -, q          k            k
Out[3]= ϕ a     ; q, a q    == -------
        2 1                       (b; q)
               b                        k
In[4]:= GlTausche
        (a; q)          b   -k
              k         -, q          k
Out[4]= ------- == ϕ a     ; q, a q
        (b; q)     2 1
              k           b
In[5]:= Gleichung
        (a; q)          b   -k
              k         -, q          k
Out[5]= ------- == ϕ a     ; q, a q
        (b; q)     2 1
              k           b
In[6]:= Subst[%1,{1}]

Alphabetic List: SUMErw1 147

                  b   -k
                  -, q          k
          ∞ ϕ a     ; q, a q
             2 1
         \  b
Out[6]= ⟩ ----------------------
         /          (q; q)
                          k
         k=0
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, GlTausche, Ers, PosListe.

SUM \underline{Description}: This is HYPQ’s internal object for entering sums. It should be used instead of Mathematica’s Sum.

\underline{Usage}: SUM[Summand,summation-index,lower-bound,upper-bound].

\underline{Example(s)}: See the examples for SUMph and SUMInfinity.

\underline{See also}: SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMSammle, SUMTausche.

SUMErw1 \underline{Description}: Rule that extends a SUM[] at the top.

∑_{k=l}^{n} Expr → ∑_{k=l}^{n+m} Expr − ∑_{k=n+1}^{n+m} Expr.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SUMErw1.

\underline{Example(s)}:

In[1]:= SUM[a[k],{k,0,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=0
In[2]:= %/.SUMErw1
top-extend by:  3

148 Alphabetic List: SUMErw2

                                          3 + N

\

Out[2]= -a[1 + N] - a[2 + N] - a[3 + N] + ⟩ a[k]
                                           /
                                           k=0
In[3]:= %%/.SUMErw1
top-extend by:  M
        M + N        M + N

\  \

Out[3]= ⟩ a[k] -  ⟩ a[k]
         /            /
         k=0        k=1 + N
\underline{See also}: SUM, SUMErw2, SUMZerl, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe.

SUMErw2 \underline{Description}: Rule that extends a SUM[] at the bottom.

∑_{k=l}^{n} Expr → ∑_{k=l−m}^{n} Expr − ∑_{k=l−m}^{l−1} Expr.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SUMErw2.

\underline{Example(s)}:

In[1]:= SUM[a[k],{k,0,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=0
In[2]:= %/.SUMErw2
bottom-extend by:  3

Alphabetic List: SUMExpand 149

                                   N

\

Out[2]= -a[-3] - a[-2] - a[-1] + ⟩ a[k]
                                  /
                                 k=-3
In[3]:= %%/.SUMErw2
bottom-extend by:  M
           -1            N

\  \

Out[3]= -( ⟩ a[k]) + ⟩ a[k]
           /            /
          k=-M         k=-M
\underline{See also}: SUM, SUMErw1, SUMZerl, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe.

SUMExpand \underline{Description}: Rule that expands SUMs.

\underline{Usage}: Expr/.SUMExpand.

\underline{Example(s)}:

In[1]:= SUM[(x[k]-y[k])^{∧}2,{k,0,(m+n)/2}]
        m + n
        -----
          2
                          2
Out[1]= \ (x[k] - y[k])

⟩

         /
         k=0
In[2]:= %/.SUMExpand

        m   n        m   n               m   n
        - + -        - + -               - + -
        2   2        2   2               2   2
                 2                                2
Out[2]= \ x[k]  + \ -2 x[k] y[k] + \ y[k]

⟩  ⟩  ⟩

         /            /                   /
         k=0          k=0                 k=0
\underline{See also}: SUM, SimplifyP, Expandq, MinusOne, PSort.

SUMInfinity

\underline{Description}: Rule that changes the upper bound of a SUM[] to Infinity.

\underline{Usage}: Expr/.SUMInfinity.

\underline{Example(s)}:

In[1]:= SUM[pq[q^{∧}-n,k]/pq[q,k]*q^{∧}k,{k,0,n}]
          n   k   -n
             q  (q  ; q)
         \  k
Out[1]= ⟩ ------------
         /     (q; q)
                     k
         k=0
In[2]:= %/.SUMInfinity

          ∞ k   -n

             q  (q  ; q)
         \  k
Out[2]= ⟩ ------------
         /     (q; q)
                     k
         k=0
\underline{See also}: SUM, SUMph, SUMPh, SUMps, Ers, PosListe.

SumListe

\underline{Description}: List of all summation formulas.

\underline{Usage}: SumListe.
\underline{See also}: SumListe$gl, SListe.

Alphabetic List: SUMph 151

SumListe$gl \underline{Description}: List of all summation formulas.

\underline{Usage}: SumListe$gl.
\underline{See also}: SumListe.

SUMPh \underline{Description}: Rule that transforms a SUM[] into multibasic hypergeometric notation, if possible. If the upper bound is

not Infinity you have to apply SUMInfinity first (if allowed).

\underline{Usage}: Expr/.SUMPh.

\underline{Example(s)}:

In[1]:= SUM[pq[a,n]*pq[b,n,q^{∧}2]/pq[c,n,q^{∧}2]/pq[q,n]*z^{∧}n,{n,0,Infinity}]

          ∞ n              2

             z  (a; q)  (b; q )
         \  n        n
Out[1]= ⟩ -------------------
         /         2
              (c; q )  (q; q)
         n=0         n       n
In[2]:= %/.SUMPh
           a:  b      2
Out[2]= ϕ  ; q, q ; z
           -:  c
\underline{See also}: SUM, Ph, SUMRegeln, SUMph, SUMInfinity, PhSUM, Phph, phPh, Ers, PosListe.

SUMph \underline{Description}: Rule that transforms a SUM[] into basic hypergeometric notation, if possible. If the upper bound is not

Infinity you have to apply SUMInfinity first (if allowed).

\underline{Usage}: Expr/.SUMph.

\underline{Example(s)}:

In[1]:= SUM[pq[q^{∧}-n,k]/pq[q,k]*q^{∧}k,{k,0,Infinity}]

          ∞ k   -n

             q  (q  ; q)
         \  k
Out[1]= ⟩ ------------
         /     (q; q)
                     k
         k=0

In[2]:= %/.SUMph

              -n
Out[2]= ϕ q  ; q, q
        1 0
              -
In[3]:= SUM[(1-q^{∧}(k+2))*pq[{q^{∧}-n,a},{b,c,q},k+1]*q^{∧}(k*(k+1)/2),
        {k,0,Infinity}]
          ∞  -n
                                         (q  , a; q)
         \  (k (1 + k))/2       2 + k             1 + k
Out[3]= ⟩ q              (1 - q     ) -----------------
         /                               (b, c, q; q)
                                                     1 + k
         k=0
In[4]:= %/.SUMph
                                   1 - n      3
                             a q, q     , q, q                     -n
        (1 - q) (1 + q) ϕ  ; q, -q   (a; q)  (q  ; q)
                        4 4              2   2                 1         1
                              b q, c q, q , q
Out[4]= ------------------------------------------------------------------
                             (b; q)  (c; q)  (q; q)
                                   1       1       1

\underline{See also}: SUM, ph, SUMRegeln, SUMps, SUMInfinity, phSUM, Ers, PosListe.

SUMps

\underline{Description}: Rule that transforms a bilateral SUM[] into basic hypergeometric notation, if possible. If the upper bound is not Infinity you have to apply SUMInfinity first (if allowed). If the lower bound is not Infinity then SUMph is applied.

\underline{Usage}: Expr/.SUMps.

\underline{Example(s)}:

In[1]:= SUM[pq[a,n]/pq[b,n]*z^{∧}n,{n,-Infinity,Infinity}]

          ∞ n

             z  (a; q)
         \  n
Out[1]= ⟩ ----------
         /    (b; q)
                    n
        n=-∞
In[2]:= %/.SUMps

              a
Out[2]=  ps    ; q, z
        1  1  b

\underline{See also}: SUM, ps, SUMRegeln, SUMph, SUMInfinity, psSUM, Ers, PosListe.

SUMRegeln

\underline{Description}: Rule that transforms the expressions in a SUM[] into a form that could also be expressed in basic hypergeometric notation. This is useful, if you want to convert a SUM[] into basic hypergeometric notation but without using the ph[]-notation. In particular, expressions of the form (−1)^{dk}, where d is an integer and k is the summation index, will simplify.

\underline{Usage}: Expr/.SUMRegeln.

\underline{Example(s)}:

In[1]:= SUM[Binomialq[n,i]*Binomialq[m,k-i]*q^{∧}((n-i)*(k-i)),{i,0,Infinity}]

∞

         \  (-i + k) (-i + n)     m         n

Out[1]= ⟩ q
         /                        -i + k      i
         i=0                              q      q
In[2]:= %/.SUMRegeln

                ∞ i   -k        -n

                    q  (q  ; q)  (q  ; q)
         k n \  i         i     1 - k + m
        q    ( ⟩ ------------------------) (q         ; q)
               /             1 - k + m                      k
                   (q; q)  (q         ; q)
               i=0       i                i
Out[2]= -----------------------------------------------------
                               (q; q)
                                     k

\underline{See also}: SUM, ph, ps, SUMph, SUMps, phSUM, psSUM, MinusOne, Ers, PosListe.

SUMSammle

\underline{Description}: Rule that causes all terms of an expression Expr, which involves a SUM[] to be put into the SUM[].

\underline{Usage}: Expr/.SUMSammle.

\underline{Example(s)}:

In[1]:= pq[a,n]/pq[b,m]*(-1)^{∧}n*SUM[1/pq[q,k],{k,0,Infinity}]

∞

            n \  1
        (-1)  ( ⟩ -------) (a; q)

                /   (q; q)         n
                          k
                k=0
Out[1]= ----------------------------
                  (b; q)
                        m
In[2]:= %/.SUMSammle

          ∞  n

              (-1)  (a; q)
         \  n
Out[2]= ⟩ ---------------
         /   (b; q)  (q; q)
                   m       k
         k=0

\underline{See also}:  SUM, SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, pqzus, pqinfzus, Ers,
  PosListe.

Alphabetic List: SUMTausche 155

SUMShift \underline{Description}: Rule that shifts the index in a SUM[].

∑_{k=l}^{n} Expr(k) → ∑_{k=l−m}^{n−m} Expr(k + m).

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SUMShift.

\underline{Example(s)}:

In[1]:= SUM[a[k],{k,3,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=3
In[2]:= %/.SUMShift
shift summation index by:  3
        -3 + N

\

Out[2]= ⟩ a[3 + k]
         /
         k=0
\underline{See also}: SUM, SUMErw1, SUMErw2, SUMTausche, SUMZerl, SUMRegeln, SUMUmkehr, Ers, PosListe.

SUMTausche \underline{Description}: Rule that exchanges summations. You should apply SUMSammle before applying SUMTausche.

∑_{k_{1}=l_{1}}^{n_{1}} ∑_{k_{2}=l_{2}}^{n_{2}} Expr → ∑_{k_{2}=l_{2}}^{n_{2}} ∑_{k_{1}=l_{1}}^{n_{1}} Expr.

\underline{Usage}: Expr/.SUMTausche.

\underline{Example(s)}:

In[1]:= SUM[SUM[Binomialq[n,k+l],{k,0,n1}],{l,0,n2}]

156 Alphabetic List: SUMUmkehr

         n2   n1

\  \  n

Out[1]= ⟩  ⟩
         /    /     k + l
         l=0  k=0          q
In[2]:= %/.SUMTausche
         n1   n2

\  \  n

Out[2]= ⟩  ⟩
         /    /     k + l
         k=0  l=0          q
\underline{See also}: SUM, SUMErw1, SUMErw2, SUMSammle, SUMShift, SUMZerl, SUMRegeln, SUMUmkehr, Ers, PosListe.

SUMUmkehr \underline{Description}: Rule that reverses the order of summation. SUMUmkehr applies to SUM[] as well as ph[].

\underline{Usage}: Expr/.SUMUmkehr.

\underline{Example(s)}:

In[1]:= ph[{a,q^{∧}-n},{b},q,z]
                 -n
Out[1]= ϕ a, q  ; q, z
        2 1
               b
In[2]:= %/.SUMUmkehr
Is n a nonnegative integer?
[y|n]:    y

                            1 - n
                       -n  q
                      q  , ------        1 + n   (a; q)
            n  n             b        b q              n
        (-1)  z ϕ  ; q, --------   -------
                 2 1     1 - n          a z      (b; q)
                        q                              n
                        ------
                          a
Out[2]= ------------------------------------------------
                          (n (1 + n))/2
                         q
In[3]:= ph[{q^{∧}-m,q^{∧}-n},{b},q,z]

              -m   -n
Out[3]= ϕ q  , q  ; q, z
        2 1
                b
In[4]:= %/.SUMUmkehr
Is m a nonnegative integer?
[y|n]:    n
Is n a nonnegative integer?
[y|n]:    y
                            1 - n                      -m
                       -n  q             1 + m + n   (q  ; q)
            n  n      q  , ------     b q                    n
        (-1)  z ϕ  b   ; q, ------------   ---------
                 2 1                       z          (b; q)
                       1 + m - n                            n
                      q
Out[4]= ------------------------------------------------------
                             (n (1 + n))/2
                            q
In[5]:= SUM[pq[q^{∧}-n,k]/pq[q,k],{k,0,Infinity}]

          ∞ -n

             (q  ; q)
         \  k
Out[5]= ⟩ ---------
         /    (q; q)
                    k
         k=0

In[6]:= %/.SUMUmkehr
Is n a nonnegative integer?
[y|n]:    y
           n   k + k n   -n
              q        (q  ; q)
          \  k    -n
        ( ⟩ ------------------) (q  ; q)
          /        (q; q)                 n
                         k
          k=0
Out[6]= -----------------------------------
                      (q; q)
                            n
\underline{See also}: SUM, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, SUMRegeln, Ers, PosListe.

SUMZerl

\underline{Description}: Rule that splits a SUM[] into two parts.

∑_{k=l}^{n} Expr → ∑_{k=l}^{l+m−1} Expr + ∑_{k=l+m}^{n} Expr.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SUMZerl.

\underline{Example(s)}:

In[1]:= SUM[a[k],{k,0,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=0
In[2]:= %/.SUMZerl
bottom-split by:  3
                               N

\

Out[2]= a[0] + a[1] + a[2] + ⟩ a[k]
                              /
                              k=3

Alphabetic List: T2104 159

In[3]:= %%/.SUMZerl
bottom-split by:  M
        -1 + M         N

\  \

Out[3]= ⟩ a[k] + ⟩ a[k]
         /            /
         k=0          k=M
\underline{See also}: SUM, SUMErw1, SUMErw2, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe.

T2101 \underline{Description}: Transformation formula ([1], (1.4.1); Appendix (III.1)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(b, az; q)_{∞}}{(c, z; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}c/b, z \\ az \end{matrix}; q, b]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2102 \underline{Description}: Transformation formula ([1], (1.4.5); Appendix (III.2)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(c/b, bz; q)_{∞}}{(c, z; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}abz/c, b \\ bz \end{matrix}; q, \frac{c}{b}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2103 \underline{Description}: Transformation formula ([1], (1.4.6); Appendix (III.3)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(abz/c; q)_{∞}}{(z; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}c/a, c/b \\ c \end{matrix}; q, \frac{abz}{c} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2104 \underline{Description}: Transformation formula ([1], (1.5.4); Appendix (III.4)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(az; q)_{∞}}{(z; q)_{∞}}_{2}ϕ_{2}[ \begin{matrix}a, c/b \\ c, az \end{matrix}; q, bz]

T2105 \underline{Description}: Transformation formula ([1], (3.2.4); Appendix (III.5)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(abz/c; q)_{∞}}{(bz/c; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, c/b, 0 \\ c, cq/bz \end{matrix}; q, q]

provided the _{3}ϕ_{2} is terminating.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2106 \underline{Description}: Transformation formula ([1], (1.5.6); Appendix (III.6)) in form of a rule.

2ϕ_{1}[ \begin{matrix}b, q^{−n} \\ c \end{matrix}; q, z] ⟶ \frac{b^{n}z^{n}}{q^{n}} \begin{matrix}(c/b; q) & n \\ (c; q) & n \end{matrix}_{3}ϕ_{2}[ \begin{matrix}q^{−n}, q/z, q^{1−n}/c \\ bq^{1−n}/c, 0 \end{matrix} ; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2107 \underline{Description}: Transformation formula ([1], Ex. 1.15(iii); Appendix (III.7)) in form of a rule.

2ϕ_{1}[ \begin{matrix}b, q^{−n} \\ c \end{matrix}; q, z] ⟶ \begin{matrix}(c/b; q) & n \\ (c; q) & n \end{matrix}_{3}ϕ_{2}[ \begin{matrix}q^{−n}, b, bz/cq^{n} \\ bq^{1−n}/c, 0 \end{matrix} ; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2108 \underline{Description}: Transformation formula ([1], Ex. 1.15(ii); Appendix (III.8)) in form of a rule.

2ϕ_{1}[ \begin{matrix}b, q^{−n} \\ c \end{matrix}; q, z] ⟶ \begin{matrix}(c/b; q) & n \\ (c; q) & n \end{matrix} b^{n}_{3}ϕ_{1}[ \begin{matrix}q^{−n}, b, q/z \\ bq^{1−n}/c \end{matrix}; q, \frac{z}{c} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2109 \underline{Description}: Transformation formula ([1], Ex. 2.2, reversed) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ bq/a \end{matrix}; q, t] ⟶ \frac{(b, at; q)_{∞}}{(t, b/a; q)_{∞}}_{4}W_{3}(\frac{t}{q}; \frac{1}{a}; q, b)

T2110 \underline{Description}: Transformation formula ([1], (3.4.7)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ aq/b \end{matrix}; q, \frac{qx}{b^{2}}] ⟶ \frac{(qx/b, aqx^{2}/b^{2}; q)_{∞}}{(aqx/b, qx^{2}/b^{2}; q)_{∞}}_{8}W_{7}(\frac{ax}{b}; x, \sqrt{a}, −\sqrt{a}, \sqrt{aq}, −\sqrt{aq}; q, \frac{qx}{b^{2}})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2111 \underline{Description}: Transformation formula ([1], (3.5.4)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a^{2}, b^{2} \\ a^{2}q^{2}/b^{2} \end{matrix} ; q^{2}, \frac{q^{2}x^{2}}{b^{4}} ] ⟶ \frac{(aq/b^{2}, qx^{2}/b^{2}; q)_{∞}}{(a^{2}q/b^{2}, aqx^{2}/b^{2}; q)_{∞}} \frac{(a^{2}qx^{2}/b^{2}, a^{2}q^{2}x^{2}/b^{4}; q^{2})_{∞}}{(qx^{2}/b^{2}, q^{2}x^{2}/b^{4}; q^{2})_{∞}}_{8}W_{7}(\frac{ax^{2}}{b^{2}} ; a, x, −x, \frac{\sqrt{q}x}{b}, −\frac{\sqrt{q}x}{b}; q, \frac{aq}{b^{2}} )

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2112 \underline{Description}: Transformation formula ([1], Ex. 3.2(i), reversed) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, aq \\ b^{2}q \end{matrix}; q^{2}, z^{2}] ⟶ \frac{(az; q)_{∞}}{(z; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, b, −b \\ b^{2}, az \end{matrix}; q, −z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2161 \underline{Description}: Transformation formula ([1], (3.3.5); Appendix (III.31)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(abz/c, q/c; q)_{∞}}{(az/c, q/a; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}c/a, cq/abz \\ cq/az \end{matrix}; q, \frac{bq}{c} ]

− \frac{(b, q/c, c/a, az/q, q^{2}/az; q)_{∞}}{(c/q, bq/c, q/a, az/c, cq/az; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}aq/c, bq/c \\ q^{2}/c \end{matrix}; q, z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2162 \underline{Description}: Transformation formula ([1], Appendix (III.32)) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z] ⟶ \frac{(b, c/a, az, q/az; q)_{∞}}{(c, b/a, z, q/z; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a, aq/c \\ aq/b \end{matrix}; q, \frac{cq}{abz} ] + \frac{(a, c/b, bz, q/bz; q)_{∞}}{(c, a/b, z, q/z; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}b, bq/c \\ bq/a \end{matrix}; q, \frac{cq}{abz} ]

162 Alphabetic List: T3202

T2163 \underline{Description}: Transformation formula ([1], Ex. 3.8) in form of a rule.

2ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, x] ⟶ \frac{(b, c/a, ax; q)_{∞}}{(b/a, c, x; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, c/b, 0 \\ aq/b, ax \end{matrix}; q, q] + \frac{(a, c/b, bx; q)_{∞}}{(a/b, c, x; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}b, c/a, 0 \\ bq/a, bx \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2201 \underline{Description}: Transformation formula ([1], (1.5.4), reversed; Appendix (III.4), reversed) in form of a rule.

2ϕ_{2}[ \begin{matrix}a, b \\ c, d \end{matrix}; q, \frac{cd}{ab}] ⟶ \frac{(d/a; q)_{∞}}{(d; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a, c/b \\ c \end{matrix}; q, \frac{d}{a}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2202 \underline{Description}: Transformation formula ([1], Ex. 3.2(ii), reversed) in form of a rule.

2ϕ_{2}[ \begin{matrix}a^{2}, b^{2} \\ a^{2}b^{2}q, a^{2}z^{2} \end{matrix} ; q^{2}, a^{2}qz^{2}] ⟶ \frac{(z; q)_{∞}(−a^{2}z; q)_{∞}}{(a^{2}z^{2}; q^{2})_{∞}}_{3}ϕ_{2}[ \begin{matrix}a^{2}, ab, −ab \\ a^{2}b^{2}, −a^{2}z \end{matrix}; q, z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3101 \underline{Description}: Transformation formula ([1], Ex. 1.15(ii), reversed; Appendix (III.8), reversed) in form of a rule.

3ϕ_{1}[ \begin{matrix}b, c, q^{−n} \\ bcz/q^{n} \end{matrix}; q, z] ⟶ b^{n} \begin{matrix}(q/cz; q) & n \\ (q/bcz; q) & n \end{matrix}_{2}ϕ_{1}[ \begin{matrix}q^{−n}, b \\ q/cz \end{matrix}; q, \frac{q}{c} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3201 \underline{Description}: Transformation formula ([1], (3.2.4), reversed; Appendix (III.5), reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, 0 \\ c, d \end{matrix}; q, q] ⟶ \frac{(q/d; q)_{∞}}{(aq/d; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a, c/b \\ c \end{matrix}; q, \frac{bq}{d} ]

provided the _{3}ϕ_{2} is terminating.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3202 \underline{Description}: Transformation formula ([1], (1.5.6), reversed; Appendix (III.6), reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}b, c, q^{−n} \\ d, 0 \end{matrix} ; q, q] ⟶ \frac{b^{n}c^{n}}{d^{n}} \begin{matrix}(q^{1−n}/c; q) & n \\ (q^{1−n}/d; q) & n \end{matrix}_{2}ϕ_{1}[ \begin{matrix}q^{−n}, d/c \\ q^{1−n}/c \end{matrix}; q, \frac{q}{b} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

Alphabetic List: T3208 163

T3203 \underline{Description}: Transformation formula ([1], Ex. 1.15(iii), reversed; Appendix (III.7), reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}b, c, q^{−n} \\ d, 0 \end{matrix} ; q, q] ⟶ \begin{matrix}(bq^{1−n}/d; q) & n \\ (q^{1−n}/d; q) & n \end{matrix}_{2}ϕ_{1}[ \begin{matrix}q^{−n}, b \\ bq^{1−n}/d \end{matrix}; q, \frac{cq}{d} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3204 \underline{Description}: Transformation formula ([1], (3.2.7); Appendix (III.9)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(e/a, de/bc; q)_{∞}}{(e, de/abc; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, d/b, d/c \\ d, de/bc \end{matrix}; q, \frac{e}{a}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3205 \underline{Description}: Transformation formula ([1], (3.2.10); Appendix (III.10)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(b, de/ab, de/bc; q)_{∞}}{(d, e, de/abc; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}d/b, e/b, de/abc \\ de/ab, de/bc \end{matrix}; q, b]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3206 \underline{Description}: Transformation formula ([1], (3.2.3); Appendix (III.11)) in form of a rule.

3ϕ_{2}[ \begin{matrix}b, c, q^{−n} \\ d, e \end{matrix}; q, q] ⟶ \frac{b^{n}c^{n}}{d^{n}} \begin{matrix}(de/bc; q) & n \\ (e; q) & n \end{matrix}_{3}ϕ_{2}[ \begin{matrix}q^{−n}, d/b, d/c \\ d, de/bc \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3207 \underline{Description}: Transformation formula ([1], (3.2.2); Appendix (III.12)) in form of a rule.

3ϕ_{2}[ \begin{matrix}b, c, q^{−n} \\ d, e \end{matrix}; q, q] ⟶ c^{n} \begin{matrix}(e/c; q) & n \\ (e; q) & n \end{matrix}_{3}ϕ_{2}[ \begin{matrix}q^{−n}, c, d/b \\ d, cq^{1−n}/e \end{matrix}; q, \frac{bq}{e} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3208 \underline{Description}: Transformation formula ([1], (3.2.5); Appendix (III.13)) in form of a rule.

3ϕ_{2}[ \begin{matrix}b, c, q^{−n} \\ d, e \end{matrix}; q, \frac{deq^{n}}{bc} ] ⟶ \begin{matrix}(e/c; q) & n \\ (e; q) & n \end{matrix}_{3}ϕ_{2}[ \begin{matrix}q^{−n}, c, d/b \\ d, cq^{1−n}/e \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3209 \underline{Description}: Transformation formula ([1], Ex. 2.26; Appendix (III.14)) in form of a rule.

3ϕ_{2}[ \begin{matrix}q^{−n}, b, c \\ q^{1−n}/b, q^{1−n}/c \end{matrix}; q, \frac{q^{1−n}z}{bc} ] ⟶ \frac{(z/q^{n}; q)_{∞}}{(z; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}q^{−n/2}, −q^{−n/2}, q^{1/2−n/2}, −q^{1/2−n/2}, q^{1−n}/bc \\ q^{1−n}/b, q^{1−n}/c, z/q^{n}, q/z \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3210 \underline{Description}: Transformation formula ([1], (3.2.6)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, q^{−n} \\ e, d \end{matrix}; q, \frac{deq^{n}}{ab} ] ⟶ (\frac{deq^{n−1}}{a} )^{n} \frac{(aq^{1−n}/d, aq^{1−n}/e; q)_{n}}{(d, e; q)_{n}}_{3}ϕ_{2}[ \begin{matrix}a, abq^{1−n}/de, q^{−n} \\ aq^{1−n}/e, aq^{1−n}/d \end{matrix}; q, \frac{q}{b} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3211 \underline{Description}: Transformation formula ([1], Ex. 3.1) in form of a rule.

3ϕ_{2}[ \begin{matrix}a^{2}, b^{2}, z \\ ab\sqrt{q}, −ab\sqrt{q} \end{matrix}; q, q] ⟶ _{3}ϕ_{2}[ \begin{matrix}a^{2}, b^{2}, z^{2} \\ a^{2}b^{2}q, 0 \end{matrix} ; q^{2}, q^{2}]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3212 \underline{Description}: Transformation formula ([1], Ex. 3.1, reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}a^{2}, b^{2}, z^{2} \\ a^{2}b^{2}q, 0 \end{matrix} ; q^{2}, q^{2}] ⟶ _{3}ϕ_{2}[ \begin{matrix}a^{2}, b^{2}, z \\ ab\sqrt{q}, −ab\sqrt{q} \end{matrix}; q, q]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3213 \underline{Description}: Transformation formula ([1], Ex. 3.2(i)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, −b \\ b^{2}, az \end{matrix}; q, −z] ⟶ \frac{(z; q)_{∞}}{(az; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a, aq \\ b^{2}q \end{matrix}; q^{2}, z^{2}]

T3214 \underline{Description}: Transformation formula ([1], Ex. 3.2(ii)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, −b \\ b^{2}, −az \end{matrix}; q, z] ⟶ \frac{(az^{2}; q^{2})_{∞}}{(z; q)_{∞}(−az; q)_{∞}}_{2}ϕ_{2}[ \begin{matrix}a, b^{2}/a \\ b^{2}q, az^{2} \end{matrix} ; q^{2}, aqz^{2}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3215 \underline{Description}: Transformation formula ([1], Ex. 3.3) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, q/a, z \\ c, −q \end{matrix}; q, q] ⟶ \frac{(−1, −qz/c; q)_{∞}}{(−q/c, −z; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}c/a, ac/q, z^{2} \\ c^{2}, 0 \end{matrix} ; q^{2}, q^{2}]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3216 \underline{Description}: Transformation formula ([1], Ex. 3.3, reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}c, a^{2}/cq, z^{2} \\ a^{2}, 0 \end{matrix} ; q^{2}, q^{2}] ⟶ \frac{(−q/a, −z; q)_{∞}}{(−1, −qz/a; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a/c, cq/a, z \\ a, −q \end{matrix}; q, q]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3217 \underline{Description}: Transformation formula ([1], (3.2.11)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(de/ab, de/ac; q)_{∞}}{(de/a, de/abc; q)_{∞}}_{7}ϕ_{7}[ \begin{matrix}de/aq, \sqrt{deq}/\sqrt{a}, −\sqrt{deq}/\sqrt{a}, e/a, d/a, b, c \\ \sqrt{de}/\sqrt{aq}, −\sqrt{de}/\sqrt{aq}, d, e, de/ab, de/ac, 0 \end{matrix} ; q, \frac{de}{bc}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3261 \underline{Description}: Transformation formula ([1], (3.3.3); Appendix (III.33)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(e/b, e/c, cq/a, q/d; q)_{∞}}{(e, cq/d, q/a, e/bc; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}c, d/a, cq/e \\ cq/a, bcq/e \end{matrix}; q, \frac{bq}{d} ]

− \frac{(q/d, eq/d, b, c, d/a, de/bcq, bcq^{2}/de; q)_{∞}}{(d/q, e, bq/d, cq/d, q/a, e/bc, bcq/e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}aq/d, bq/d, cq/d \\ q^{2}/d, eq/d \end{matrix}; q, \frac{de}{abc}]

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(e/b, e/c; q)_{∞}}{(e, e/bc; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}d/a, b, c \\ d, bcq/e \end{matrix}; q, q] + \frac{(d/a, b, c, de/bc; q)_{∞}}{(d, e, bc/e, de/abc; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}e/b, e/c, de/abc \\ de/bc, eq/bc \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3263 \underline{Description}: Transformation formula ([1], (3.3.3), reversed; Appendix (III.33)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(dq/bc, e/c, a, b, abq/de, de/ab; q)_{∞}}{(aq/e, q/c, d, ab/d, de/abc, e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}q/b, de/abc, d/b \\ dq/ab, dq/bc \end{matrix}; q, \frac{bq}{e} ]

+ \frac{(aq/c, d/b, d/a, q/e; q)_{∞}}{(aq/e, q/c, d, d/ab; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}aq/d, e/c, a \\ abq/d, aq/c \end{matrix}; q, \frac{bq}{e} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3264 \underline{Description}: Transformation formula ([1], (3.3.1), reversed; Appendix (III.34)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, q] ⟶ \frac{(bcq/e, q/e; q)_{∞}}{(cq/e, bq/e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}d/a, b, c \\ d, bcq/e \end{matrix}; q, \frac{aq}{e} ] − \frac{(q/e, a, b, c, dq/e; q)_{∞}}{(cq/e, bq/e, d, e/q, aq/e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}cq/e, bq/e, aq/e \\ dq/e, q^{2}/e \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3265 \underline{Description}: Transformation formula ([1], (3.4.1); Appendix (III.35)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ aq/b, aq/c \end{matrix}; q, \frac{aqx}{bc}] ⟶ \frac{(ax; q)_{∞}}{(x; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{a}, −\sqrt{a}, \sqrt{a}\sqrt{q}, −\sqrt{a}\sqrt{q}, aq/bc \\ aq/b, aq/c, ax, q/x \end{matrix}; q, q]

+ \frac{(a, aq/bc, aqx/b, aqx/c; q)_{∞}}{(aq/b, aq/c, aqx/bc, 1/x; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{a}x, −\sqrt{a}x, \sqrt{a}\sqrt{q}x, −\sqrt{a}\sqrt{q}x, aqx/bc \\ aqx/b, aqx/c, qx, ax^{2} \end{matrix} ; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3266 \underline{Description}: Transformation formula ([1], (3.5.2)) in form of a rule.

3ϕ_{2}[ \begin{matrix}a^{2}, x^{2}, y^{2} \\ a^{2}b^{2}, b^{2}x^{2}y^{2} \end{matrix} ; q^{2}, b^{2}q] ⟶ \frac{(b^{2}; q^{2})_{∞}}{(a^{2}b^{2}; q^{2})_{∞}} \frac{(a, −ab^{2}; q)_{∞}}{(−1, b^{2}; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}−a, −bx, bx, −by, by \\ −q, −ab^{2}, −bxy, bxy \end{matrix}; q, q]

+ \frac{(b^{2}; q^{2})_{∞}}{(a^{2}b^{2}; q^{2})_{∞}} \frac{(−a, ab^{2}; q)_{∞}}{(−1, b^{2}; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}a, bx, −bx, by, −by \\ −q, ab^{2}, bxy, −bxy \end{matrix}; q, q]

T3267 \underline{Description}: Transformation formula ([1], Ex. 3.6) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, q] ⟶ \frac{(q/e, abq/e, acq/e, d/a; q)_{∞}}{(d, aq/e, bq/e, cq/e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, aq/e, abcq/de \\ abq/e, acq/e \end{matrix}; q, \frac{d}{a}]

− \frac{(q/e, a, b, c, dq/e; q)_{∞}}{(e/q, aq/e, bq/e, cq/e, d; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}aq/e, bq/e, cq/e \\ q^{2}/e, dq/e \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3268 \underline{Description}: Transformation formula ([1], Ex. 3.6, reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; q, \frac{de}{abc}] ⟶ \frac{(de/bc, b, d/a, e/a; q)_{∞}}{(b/a, d, e, de/abc; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, d/b, e/b \\ de/bc, aq/b \end{matrix}; q, q]+ \frac{(a, d/b, e/b, de/ac; q)_{∞}}{(d, e, de/abc, a/b; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}b, d/a, e/a \\ bq/a, de/ac \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3269 \underline{Description}: Transformation formula ([1], Ex. 3.8, reversed) in form of a rule.

3ϕ_{2}[ \begin{matrix}a, b, 0 \\ c, x \end{matrix}; q, q] ⟶ \frac{(q/c, abq/c, x/a; q)_{∞}}{(aq/c, bq/c, x; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a, aq/c \\ abq/c \end{matrix}; q, \frac{x}{a}] − \frac{(a, b, qx/c, q/c; q)_{∞}}{(c/q, aq/c, bq/c, x; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}aq/c, bq/c, 0 \\ q^{2}/c, qx/c \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4201 \underline{Description}: Transformation formula ([1], Ex. 3.4, reversed) in form of a rule.

4ϕ_{2}[ \begin{matrix}a, b, −b, q^{−n} \\ aq^{1−n}/d, b^{2} \end{matrix} ; q, −\frac{q}{d}] ⟶ a^{−n} \frac{(d; q)_{n}}{(d/a; q)_{n}}_{4}ϕ_{3}[ \begin{matrix}a, aq, q^{1−n}, q^{−n} \\ d, dq, b^{2}q \end{matrix}; q^{2}, q^{2}]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4301 \underline{Description}: Transformation formula ([1], (2.10.4); Appendix (III.15)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, q^{−n} \\ e, f, abcq^{1−n}/ef \end{matrix} ; q, q] ⟶ a^{n} \begin{matrix}(e/a, f/a; q) & n \\ (e, f; q) & n \end{matrix}_{4}ϕ_{3}[ \begin{matrix}q^{−n}, a, acq^{1−n}/ef, abq^{1−n}/ef \\ abcq^{1−n}/ef, aq^{1−n}/e, aq^{1−n}/f \end{matrix} ; q, q]

4ϕ_{3}[ \begin{matrix}a, b, c, q^{−n} \\ e, f, abcq^{1−n}/ef \end{matrix} ; q, q] ⟶ \begin{matrix}(a, ef/ab, ef/ac; q) & n \\ (e, f, ef/abc; q) & n \end{matrix}_{4}ϕ_{3}[ \begin{matrix}q^{−n}, e/a, f/a, ef/abc \\ ef/ab, ef/ac, q^{1−n}/a \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4303 \underline{Description}: Transformation formula ([1], (2.5.1), reversed; Appendix (III.19)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, q^{−n} \\ e, f, abcq^{1−n}/ef \end{matrix} ; q, q] ⟶ \begin{matrix}(ef/ab, ef/ac; q) & n \\ (ef/a, ef/abc; q) & n \end{matrix}_{8}W_{7}(\frac{ef}{aq}; \frac{f}{a} , \frac{e}{a}, b, c, q^{−n}; q, \frac{efq^{n}}{bc})

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4304 \underline{Description}: Transformation formula ([1], (2.10.7); Appendix (III.20)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, q^{−n} \\ d, e, abcq^{1−n}/de \end{matrix}; q, q] ⟶ \frac{(deq^{n}/c, deq^{n}/b, deq^{n}/a, deq^{n}/abc; q)_{∞}}{(deq^{n}/bc, deq^{n}/ac, deq^{n}/ab, deq^{n}; q)_{∞}}_{8}W_{7}(deq^{−1+n}; a, b, c, dq^{n}, eq^{n}; q, \frac{de}{abc})

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4305 \underline{Description}: Transformation formula ([1], (3.10.13); Appendix (III.21)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a^{2}, b^{2}, c, d \\ ab\sqrt{q}, −ab\sqrt{q}, −cd \end{matrix}; q, q] ⟶ _{4}ϕ_{3}[ \begin{matrix}a^{2}, b^{2}, c^{2}, d^{2} \\ a^{2}b^{2}q, −cd, −cdq \end{matrix}; q^{2}, q^{2}]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4306 \underline{Description}: Transformation formula ([1], (3.10.13), reversed; Appendix (III.21), reversed) in form of a rule.

4ϕ_{3}[ \begin{matrix}a^{2}, b^{2}, c^{2}, d^{2} \\ a^{2}b^{2}q, −cd, −cdq \end{matrix}; q^{2}, q^{2}] ⟶ _{4}ϕ_{3}[ \begin{matrix}a^{2}, b^{2}, c, d \\ ab\sqrt{q}, −ab\sqrt{q}, −cd \end{matrix}; q, q]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

4ϕ_{3}[ \begin{matrix}a, b, abz, ab/z \\ ab\sqrt{q}, −ab\sqrt{q}, −ab \end{matrix}; q, q]^{2} ⟶ _{5}ϕ_{4}[ \begin{matrix}a^{2}, b^{2}, ab, abz, ab/z \\ ab\sqrt{q}, −ab\sqrt{q}, −ab, a^{2}b^{2} \end{matrix} ; q, q]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4308 \underline{Description}: Transformation formula ([1], Ex. 2.2) in form of a rule.

4W_{3}(a; b; q, t) ⟶ \frac{(aq, bt; q)_{∞}}{(t, aq/b; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}1/b, t \\ bqt \end{matrix}; q, aq]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4309 \underline{Description}: Transformation formula ([1], Ex. 2.13(i)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, d \\ aq/b, aq/c, aq/d \end{matrix}; q, \frac{a^{2}q^{3}}{b^{2}c^{2}d^{2}} ]

⟶ \frac{(aq^{2}/bcd, a^{3}q^{3}/b^{2}c^{2}d^{2}; q)_{∞}}{(a^{2}q^{2}/bcd, a^{2}q^{3}/b^{2}c^{2}d^{2}; q)_{∞}}_{10}W_{9}(\frac{a^{2}q}{bcd}; \sqrt{a}, −\sqrt{a}, \sqrt{aq}, −\sqrt{aq}, \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc}; q, \frac{aq^{2}}{bcd})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4310 \underline{Description}: Transformation formula ([1], Ex. 2.13(ii)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, d \\ aq/b, aq/c, aq/d \end{matrix}; q, −\frac{aq^{2}}{bcd} ] ⟶ \frac{(aq, −q, a^{3/2}q^{2}/bcd, −a^{3/2}q^{2}/bcd; q)_{∞}}{(a^{2}q^{2}/bcd, −aq^{2}/bcd, \sqrt{a}q, −\sqrt{a}q; q)_{∞}}_{8}W_{7}(\frac{a^{2}q}{bcd}; \sqrt{a}, −\sqrt{a}, \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc}; q, −q)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4311 \underline{Description}: Transformation formula ([1], (3.4.8)) in form of a rule.

4W_{3}(a; b; q, \frac{x}{b^{2}q}) ⟶ \frac{(ax^{2}/b^{2}, x/bq; q)_{∞}}{(aqx/b, x^{2}/b^{2}q; q)_{∞}}_{8}W_{7}(\frac{ax}{b}; \sqrt{aq}, −\sqrt{aq}, \sqrt{a}q, −\sqrt{a}q, x; q, \frac{x}{b^{2}q})

170 Alphabetic List: T5401

T4312 \underline{Description}: Transformation formula ([1], Ex. 3.4) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, aq, q^{1−n}, q^{−n} \\ d, dq, b^{2}q \end{matrix}; q^{2}, q^{2}] ⟶ a^{n} \frac{(d/a; q)_{n}}{(d; q)_{n}}_{4}ϕ_{2}[ \begin{matrix}a, b, −b, q^{−n} \\ aq^{1−n}/d, b^{2} \end{matrix} ; q, −\frac{q}{d}]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4313 \underline{Description}: Transformation formula ([1], Ex. 8.15) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, d \\ bq/a, cq/a, dq/a \end{matrix}; q, \frac{q^{2}}{a^{2}} ] ⟶ \frac{(a/d, bq/d, cq/d, abc/d; q)_{∞}}{(q/d, ab/d, ac/d, bcq/d; q)_{∞}}

×_{12}W_{11} (\frac{bc}{d}; \begin{matrix}√ & \overline{bcq} \\ √ & \overline{ad} \end{matrix} , −\begin{matrix}√ & \overline{bcq} \\ √ & \overline{ad} \end{matrix} , \frac{\sqrt{bc}q}{\sqrt{ad}} , −\frac{\sqrt{bc}q}{\sqrt{ad}} , \frac{ab}{d}, \frac{ac}{d}, a, b, c; q, \frac{q}{a})

provided at least one of a, b, c is of the form q^{−}n, n = 0, 1, 2, . . ..

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4361 \underline{Description}: Transformation formula ([1], Ex. 3.16) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, −\sqrt{a}q, b, c \\ −\sqrt{a}, aq/b, aq/c \end{matrix}; q, x] ⟶ (1 − \frac{bcx}{\sqrt{a}q} ) \frac{(bcx; q)_{∞}}{(bcx/aq; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{a}, −\sqrt{a}q, \sqrt{aq}, −\sqrt{aq}, aq/bc \\ aq/b, aq/c, aq^{2}/bcx, bcx \end{matrix}; q, q]

+ (1 − \sqrt{a}) \frac{(aq, aq/bc, cx, bx; q)_{∞}}{(aq/b, aq/c, x, aq/bcx; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}bcx/\sqrt{a}q, −bcx/\sqrt{a}, bcx/\sqrt{aq}, −bcx/\sqrt{aq}, x \\ cx, bx, bcx/a, b^{2}c^{2}x^{2}/aq \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4362 \underline{Description}: Transformation formula ([1], (2.10.10), reversed; Appendix (III.36)) in form of a rule.

4ϕ_{3}[ \begin{matrix}a, b, c, d \\ e, f, abcdq/ef \end{matrix} ; q, q] ⟶ \frac{(ef/ab, ef/ac, ef/ad, ef/abcd; q)_{∞}}{(ef/a, ef/abc, ef/abd, ef/acd; q)_{∞}}_{8}W_{7}(\frac{ef}{aq}; \frac{f}{a} , \frac{e}{a}, b, c, d; q, \frac{ef}{bcd})

− \frac{(ef/abcd, a, b, c, d, e^{2}f/abcd, ef^{2}/abcd; q)_{∞}}{(ef/abc, ef/abd, ef/acd, e, f, ef/bcd, abcd/ef; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}ef/abc, ef/abd, ef/acd, ef/bcd \\ e^{2}f/abcd, ef^{2}/abcd, efq/abcd \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5401 \underline{Description}: Transformation formula ([1], (2.8.3); Appendix (III.25)) in form of a rule.

5ϕ_{4}[ \begin{matrix}a, b, c, d, q^{−n} \\ aq/b, aq/c, aq/d, b^{2}c^{2}d^{2}q^{−2−n}/a^{2} \end{matrix} ; q, q] ⟶ \begin{matrix}(aq^{2}/bcd, a^{3}q^{3}/b^{2}c^{2}d^{2}; q) & n \\ (a^{2}q^{2}/bcd, a^{2}q^{3}/b^{2}c^{2}d^{2}; q) & n \end{matrix}

12W_{11}(\frac{a^{2}q}{bcd}; \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc} , \sqrt{a}, −\sqrt{a}, \sqrt{a}\sqrt{q}, −\sqrt{a}\sqrt{q}, \begin{matrix}a^{3} & q^{3+n} \\ b^{2} & c^{2} & d^{2} \end{matrix} , q^{−n}; q, q)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

5ϕ_{4}[ \begin{matrix}q^{−n}, b, c, d, e \\ q^{1−n}/b, q^{1−n}/c, q^{1−n}/d, b^{2}c^{2}d^{2}eq^{−2+2n} \end{matrix}; q, q] ⟶ \begin{matrix}(q^{2−2n}/bcde, q^{3−3n}/b^{2}c^{2}d^{2}; q) & n \\ (q^{2−2n}/bcd, q^{3−3n}/b^{2}c^{2}d^{2}e; q) & n \end{matrix}

12W_{11}(\frac{q^{1−2n}}{bcd}; \frac{q^{1−n}}{cd} , \frac{q^{1−n}}{bd} , \frac{q^{1−n}}{bc} , q^{\frac{−n}{2}} , −q^{\frac{−n}{2}} , q^{\frac{1}{2} − \frac{n}{2}} , −q^{\frac{1}{2} − \frac{n}{2}} , e, \frac{q^{3−3n}}{b^{2}c^{2}d^{2}e}; q, q)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5403 \underline{Description}: Transformation formula ([1], (3.10.4), reversed) in form of a rule.

5ϕ_{4}[ \begin{matrix}x^{2}, y^{2}, a, aq, q^{−2n} \\ a^{2}, ab, abq, q^{2−2n}x^{2}y^{2}/a^{2}b^{2} \end{matrix} ; q^{2}, q^{2}]

⟶ \frac{(a^{2}b^{2}/x^{2}, a^{2}b^{2}/y^{2}; q^{2})_{n}}{(a^{2}b^{2}, a^{2}b^{2}/x^{2}y^{2}; q^{2})_{n}}_{10}W_{9}(−\frac{ab}{q}; b, x, −x, y, −y, −q^{−n}, q^{−n}; q, \begin{matrix}a^{3} & b^{2} & q^{2n} \\ x^{2} & y^{2} \end{matrix} )

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5404 \underline{Description}: Transformation formula ([1], Ex. 2.26, Appendix (III.14), reversed) in form of a rule.

5ϕ_{4}[ \begin{matrix}q^{1−n}/bc, −q^{1/2−n/2}, q^{1/2−n/2}, −q^{−n/2}, q^{−n/2} \\ q^{1−n}/b, q^{1−n}/c, z/q^{n}, q/z \end{matrix}; q, q] ⟶ \frac{(z; q)_{∞}}{(z/q^{n}; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}q^{−n}, b, c \\ q^{1−n}/b, q^{1−n}/c \end{matrix}; q, \frac{q^{1−n}z}{bc} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5405 \underline{Description}: Transformation formula ([1], (8.8.3), Appendix (III.22), reversed) in form of a rule.

5ϕ_{4}[ \begin{matrix}a^{2}, b^{2}, ab, z, a^{2}b^{2}/z \\ ab\sqrt{q}, −ab\sqrt{q}, −ab, a^{2}b^{2} \end{matrix} ; q, q] ⟶ _{4}ϕ_{3}[ \begin{matrix}a, b, z, a^{2}b^{2}/z \\ ab\sqrt{q}, −ab\sqrt{q}, −ab \end{matrix}; q, q]^{2}

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5461 \underline{Description}: Transformation formula ([1], (3.4.4)) in form of a rule.

5W_{4}(a; b, c; q, \frac{\sqrt{aq}x}{bc}) ⟶ (1 − x^{2}) \frac{(\sqrt{a}q^{3/2}x; q)_{∞}}{(x/\sqrt{aq}; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{aq}, −\sqrt{aq}, \sqrt{a}q, −\sqrt{a}q, aq/bc \\ aq/b, aq/c, \sqrt{a}q^{3/2}x, \sqrt{a}q^{3/2}/x \end{matrix}; q, q]

+ \frac{(aq, aq/bc, \sqrt{aq}x/b, \sqrt{aq}x/c; q)_{∞}}{(aq/b, aq/c, \sqrt{aq}x/bc, \sqrt{aq}/x; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}x, −x, \sqrt{q}x, −\sqrt{q}x, \sqrt{aq}x/bc \\ \sqrt{aq}x/b, \sqrt{aq}x/c, \sqrt{q}x/\sqrt{a}, qx^{2} \end{matrix} ; q, q]

T5462 \underline{Description}: Transformation formula ([1], (3.4.4), reversed, first form) in form of a rule.

5ϕ_{4}[ \begin{matrix}\sqrt{aq}, −\sqrt{aq}, \sqrt{a}q, −\sqrt{a}q, bc/aq \\ b, c, \sqrt{a}q^{3/2}x, \sqrt{a}q^{3/2}/x \end{matrix}; q, q] ⟶ \frac{1}{1 − x^{2}} \frac{(x/\sqrt{aq}; q)_{∞}}{(\sqrt{a}q^{3/2}x; q)_{∞}}_{5}W_{4}(a; \frac{aq}{b} , \frac{aq}{c}; q, \frac{bcx}{(aq)^{\frac{3}{2}}} )

− \frac{1}{1 − x^{2}} \frac{(x/\sqrt{aq}, aq, bc/aq, bx/\sqrt{aq}, cx/\sqrt{aq}; q)_{∞}}{(\sqrt{a}q^{3/2}x, b, c, bcx/(aq)^{3/2}, \sqrt{aq}/x; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}x, −x, \sqrt{q}x, −\sqrt{q}x, bcx/(aq)^{3/2} \\ bx/\sqrt{aq}, cx/\sqrt{aq}, \sqrt{q}x/\sqrt{a}, qx^{2} \end{matrix} ; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5463 \underline{Description}: Transformation formula ([1], (3.4.4), reversed, second form) in form of a rule.

5ϕ_{4}[ \begin{matrix}x, −x, \sqrt{q}x, −\sqrt{q}x, bc/\sqrt{aq}x \\ b, c, \sqrt{q}x/\sqrt{a}, qx^{2} \end{matrix} ; q, q]

⟶ \frac{(b\sqrt{aq}/x, c\sqrt{aq}/x, bc/\sqrt{aq}x, \sqrt{aq}/x; q)_{∞}}{(aq, bc/x^{2}, b, c; q)_{∞}}_{5}W_{4}(a; \frac{\sqrt{aq}x}{b}, \frac{\sqrt{aq}x}{c}; q, \frac{bc}{\sqrt{aq}x})

− (1 − x^{2}) \frac{(\sqrt{a}q^{3/2}x, b\sqrt{aq}/x, c\sqrt{aq}/x, bc/\sqrt{aq}x, \sqrt{aq}/x; q)_{∞}}{(x/\sqrt{aq}, aq, bc/x^{2}, b, c; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{aq}, −\sqrt{aq}, \sqrt{a}q, −\sqrt{a}q, bc/x^{2} \\ b\sqrt{aq}/x, c\sqrt{aq}/x, \sqrt{a}q^{3/2}x, \sqrt{a}q^{3/2}/x \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5464 \underline{Description}: Transformation formula ([1], (3.5.2), reversed) in form of a rule.

5ϕ_{4}[ \begin{matrix}a, x, −x, y, −y \\ −q, ab^{2}, xy/b, −xy/b \end{matrix}; q, q]

⟶ \frac{(a^{2}b^{2}; q^{2})_{∞}}{(b^{2}; q^{2})_{∞}} \frac{(−1, b^{2}; q)_{∞}}{(−a, ab^{2}; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a^{2}, x^{2}/b^{2}, y^{2}/b^{2} \\ a^{2}b^{2}, x^{2}y^{2}/b^{2} \end{matrix} ; q^{2}, b^{2}q] − \frac{(a, −ab^{2}; q)_{∞}}{(−a, ab^{2}; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}−a, −x, x, −y, y \\ −q, −ab^{2}, −xy/b, xy/b \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5465 \underline{Description}: Transformation formula ([1], (3.5.7), reversed) in form of a rule.

5ϕ_{4}[ \begin{matrix}\sqrt{a}, −\sqrt{a}, c, d, e \\ \sqrt{ab}, −\sqrt{ab}, a, cdeq/ab \end{matrix}; q, q] ⟶ −\frac{(abq, a^{3}b^{3}/c^{2}d^{2}e^{2}; q^{2})_{∞}}{(aq, a^{3}b^{2}/c^{2}d^{2}e^{2}; q^{2})_{∞}} \frac{(c, d, e, a^{2}b/cde, ab/cde; q)_{∞}}{(cde/ab, ab, ab/cd, ab/ce, ab/de; q)_{∞}}

5ϕ_{4}[ \begin{matrix}a^{3/2}b/cde, −a^{3/2}b/cde, ab/cd, ab/ce, ab/de \\ (ab)^{3/2}/cde, −(ab)^{3/2}/cde, a^{2}b/cde, abq/cde \end{matrix}; q, q]

+ \frac{(ab/c, ab/d, ab/e, ab/cde; q)_{∞}}{(ab, ab/cd, ab/ce, ab/de; q)_{∞}}_{10}W_{9}(\frac{ab}{q}; b, c, cq, d, dq, e, eq; q^{2}, \begin{matrix}a^{3} & b^{2} \\ c^{2} & d^{2} & e^{2} \end{matrix} )

T5466 \underline{Description}: Transformation formula ([1], Ex. 3.16, reversed, first form) in form of a rule.

5ϕ_{4}[ \begin{matrix}a, −aq, a\sqrt{q}, −a\sqrt{q}, bc/a^{2}q \\ b, c, x, a^{2}q^{2}/x \end{matrix}; q, q] ⟶ \frac{1}{(1 − \frac{aq}{x})} \frac{(q/x; q)_{∞}}{(a^{2}q^{2}/x; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}a^{2}, −aq, a^{2}q/b, a^{2}q/c \\ −a, b, c \end{matrix}; q, \frac{bc}{a^{2}x}]

− \frac{(1 − a)}{(1 − \frac{aq}{x})} \frac{(q/x, a^{2}q, bc/a^{2}q, bq/x, cq/x; q)_{∞}}{(a^{2}q^{2}/x, b, c, bc/a^{2}x, x/q; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}aq/x, −aq^{2}/x, aq^{3/2}/x, −aq^{3/2}/x, bc/a^{2}x \\ bq/x, cq/x, q^{2}/x, a^{2}q^{3}/x^{2} \end{matrix} ; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5467 \underline{Description}: Transformation formula ([1], Ex. 3.16, reversed, second form) in form of a rule.

5ϕ_{4}[ \begin{matrix}a, −aq, a\sqrt{q}, −a\sqrt{q}, bcx/a^{2}q^{2} \\ b, c, x, a^{2}q \end{matrix}; q, q]

⟶ \frac{1}{(1 − \frac{aq}{x})} \frac{(bq/x, cq/x, bcx/a^{2}q^{2}, q/x; q)_{∞}}{(a^{2}q^{3}/x^{2}, bc/a^{2}q, b, c; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}a^{2}q^{2}/x^{2}, −aq^{2}/x, a^{2}q^{2}/bx, a^{2}q^{2}/cx \\ −aq/x, bq/x, cq/x \end{matrix}; q, \frac{bcx}{a^{2}q^{2}} ]

− \frac{(1 − a)}{(1 − \frac{aq}{x})} \frac{(a^{2}q^{2}/x, bq/x, cq/x, bcx/a^{2}q^{2}, q/x; q)_{∞}}{(x/q, a^{2}q^{3}/x^{2}, bc/a^{2}q, b, c; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}aq/x, −aq^{2}/x, aq^{3/2}/x, −aq^{3/2}/x, bc/a^{2}q \\ bq/x, cq/x, q^{2}/x, a^{2}q^{2}/x \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5468 \underline{Description}: Transformation formula ([1], Ex. 2.25) in form of a rule.

5ϕ_{4}[ \begin{matrix}a, b, c, d, e \\ aq/b, aq/c, aq/d, b^{2}c^{2}d^{2}e/a^{2}q^{2} \end{matrix} ; q, q]

⟶ −\frac{(a, b, c, d, e, a^{2}q^{3}/b^{2}c^{2}d^{2}e, a^{3}q^{4}/b^{3}c^{2}d^{2}e, a^{3}q^{4}/b^{2}c^{3}d^{2}e, a^{3}q^{4}/b^{2}c^{2}d^{3}e; q)_{∞}}{(aq/b, aq/c, aq/d, b^{2}c^{2}d^{2}e/a^{2}q^{3}, a^{3}q^{3}/b^{2}c^{2}d^{2}e, a^{2}q^{3}/bc^{2}d^{2}e, a^{2}q^{3}/b^{2}cd^{2}e, a^{2}q^{3}/b^{2}c^{2}de, a^{2}q^{3}/b^{2}c^{2}d^{2}; q)_{∞}}

5ϕ_{4}[ \begin{matrix}a^{2}q^{3}/b^{2}c^{2}d^{2}, a^{3}q^{3}/b^{2}c^{2}d^{2}e, a^{2}q^{3}/bc^{2}d^{2}e, a^{2}q^{3}/b^{2}cd^{2}e, a^{2}q^{3}/b^{2}c^{2}de \\ a^{2}q^{4}/b^{2}c^{2}d^{2}e, a^{3}q^{4}/b^{3}c^{2}d^{2}e, a^{3}q^{4}/b^{2}c^{3}d^{2}e, a^{3}q^{4}/b^{2}c^{2}d^{3}e \end{matrix}; q, q]

+ \frac{(aq^{2}/bcd, a^{2}q^{2}/bcde, a^{3}q^{3}/b^{2}c^{2}d^{2}, a^{2}q^{3}/b^{2}c^{2}d^{2}e; q)_{∞}}{(a^{2}q^{2}/bcd, a^{3}q^{3}/b^{2}c^{2}d^{2}e, a^{2}q^{3}/b^{2}c^{2}d^{2}, aq^{2}/bcde; q)_{∞}}

12W_{11}(\frac{a^{2}q}{bcd}; \sqrt{a}, −\sqrt{a}, \sqrt{a}\sqrt{q}, −\sqrt{a}\sqrt{q}, \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc} , e, \frac{a^{3}q^{3}}{b^{2}c^{2}d^{2}e}; q, q)

+ \frac{(a, e, aq/cd, aq/bd, aq/bc, a^{2}q^{3}/b^{2}c^{2}d^{2}e, a^{2}q^{3}/b^{2}cde, a^{2}q^{3}/bc^{2}de, a^{2}q^{3}/bcd^{2}e; q)_{∞}}{(aq/b, aq/c, aq/d, a^{3}q^{3}/b^{2}c^{2}d^{2}e, a^{2}q^{3}/bc^{2}d^{2}e, a^{2}q^{3}/b^{2}cd^{2}e, a^{2}q^{3}/b^{2}c^{2}de, a^{2}q^{3}/b^{2}c^{2}d^{2}, bcde/aq^{2}; q)_{∞}}

\frac{(a^{5}q^{7}/b^{4}c^{4}d^{4}e^{2}; q)_{∞}}{(a^{4}q^{6}/b^{3}c^{3}d^{3}e^{2}; q)_{∞}}_{12}W_{11}(\frac{a^{4}q^{5}}{b^{3}c^{3}d^{3}e^{2}} ; \frac{a^{\frac{3}{2}} q^{2}}{bcde} , −\frac{a^{\frac{3}{2}} q^{2}}{bcde} , \frac{a^{\frac{3}{2}} q^{\frac{5}{2}}}{bcde} , −\frac{a^{\frac{3}{2}} q^{\frac{5}{2}}}{bcde} , \frac{aq^{2}}{bcd} , \frac{a^{3}q^{3}}{b^{2}c^{2}d^{2}e}, \frac{a^{2}q^{3}}{bc^{2}d^{2}e}, \frac{a^{2}q^{3}}{b^{2}cd^{2}e}, \frac{a^{2}q^{3}}{b^{2}c^{2}de}; q, q)

T5469 \underline{Description}: Transformation formula ([1], (3.4.1), reversed; Appendix (III.35)) in form of a rule.

5ϕ_{4}[ \begin{matrix}\sqrt{a}, −\sqrt{a}, \sqrt{a}\sqrt{q}, −\sqrt{a}\sqrt{q}, bc/aq \\ b, c, ax, q/x \end{matrix}; q, q] ⟶ \frac{(x; q)_{∞}}{(ax; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}a, aq/b, aq/c \\ b, c \end{matrix}; q, \frac{bcx}{aq}]

− \frac{(x, a, bc/aq, bx, cx; q)_{∞}}{(ax, b, c, bcx/aq, 1/x; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{a}x, −\sqrt{a}x, \sqrt{a}\sqrt{q}x, −\sqrt{a}\sqrt{q}x, bcx/aq \\ bx, cx, qx, ax^{2} \end{matrix} ; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T6501 \underline{Description}: Transformation formula ([1], Ex. 2.14(ii)) in form of a rule.

6ϕ_{5}[ \begin{matrix}a, \sqrt{a}q, b, c, d, q^{−n} \\ \sqrt{a}, aq/b, aq/c, aq/d, b^{2}c^{2}d^{2}q^{−1−n}/a^{2} \end{matrix} ; q, q]

⟶ \frac{(aq/bcd, a^{3}q^{2}/b^{2}c^{2}d^{2}, −a^{3/2}q^{2}/bcd; q)_{n}}{(a^{2}q^{2}/bcd, a^{2}q^{2}/b^{2}c^{2}d^{2}, −a^{3/2}q/bcd; q)_{n}}_{12}W_{11}(\frac{a^{2}q}{bcd}; \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc} , \sqrt{a}q, −\sqrt{a}, \sqrt{aq}, −\sqrt{aq}, \begin{matrix}a^{3} & q^{2+n} \\ b^{2} & c^{2} & d^{2} \end{matrix} , q^{−n}; q, q)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7601 \underline{Description}: Transformation formula ([1], (2.8.5); Appendix (III.27)) in form of a rule.

7ϕ_{6}[ \begin{matrix}a, \sqrt{a}q, −\sqrt{a}q, b, c, d, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, b^{2}c^{2}d^{2}/a^{2}q^{n} \end{matrix}; q, q] ⟶ \frac{(1 − a^{3}q^{1+2n}/b^{2}c^{2}d^{2})}{(1 − a^{3}q/b^{2}c^{2}d^{2})} \begin{matrix}(a/bcd, a^{3}q/b^{2}c^{2}d^{2}; q) & n \\ (a^{2}q^{2}/bcd, a^{2}q/b^{2}c^{2}d^{2}; q) & n \end{matrix}

12W_{11}(\frac{a^{2}q}{bcd}; \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc} , \sqrt{a}\sqrt{q}, −\sqrt{a}\sqrt{q}, \sqrt{a}q, −\sqrt{a}q, \begin{matrix}a^{3} & q^{1+n} \\ b^{2} & c^{2} & d^{2} \end{matrix} , q^{−n}; q, q)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7701 \underline{Description}: Transformation formula ([1], (3.2.11), reversed) in form of a rule.

7ϕ_{7}[ \begin{matrix}a, \sqrt{a}q, −\sqrt{a}q, b, c, d, e \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, 0 \end{matrix} ; q, \frac{a^{2}q^{2}}{bcde} ] ⟶ \frac{(aq, aq/de; q)_{∞}}{(aq/d, aq/e; q)_{∞}}_{3}ϕ_{2}[ \begin{matrix}aq/bc, d, e \\ aq/b, aq/c \end{matrix}; q, \frac{aq}{de} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8701 \underline{Description}: Transformation formula ([1], (2.10.10) terminated; Appendix (III.17)) in form of a rule.

8W_{7}(a; b, c, d, e, f; q, \frac{a^{2}q^{2}}{bcdef}) ⟶ \frac{(aq, aq/de, aq/df, aq/ef; q)_{∞}}{(aq/d, aq/e, aq/f, aq/def; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}aq/bc, d, e, f \\ aq/b, aq/c, def/a \end{matrix}; q, q] ,

provided the _{8}ϕ_{7} series converges and the _{4}ϕ_{3} series terminates.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

8W_{7}(a; b, c, d, e, q^{−n}; q, \frac{a^{2}q^{2+n}}{bcde}) ⟶ \begin{matrix}(aq, aq/de; q) & n \\ (aq/d, aq/e; q) & n \end{matrix}_{4}ϕ_{3}[ \begin{matrix}aq/bc, d, e, q^{−n} \\ aq/b, aq/c, de/aq^{n} \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8703 \underline{Description}: Transformation formula ([1], (2.10.7), reversed; Appendix (III.20), reversed) in form of a rule.

8W_{7}(a; b, c, d, e, \frac{aq^{1+n}}{e}; q, \frac{aq^{1−n}}{bcd}) ⟶ \frac{(aq/cd, aq/bd, aq/bc, aq; q)_{∞}}{(aq/d, aq/c, aq/b, aq/bcd; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}b, c, d, q^{−n} \\ aq/e, bcd/a, e/q^{n} \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8704 \underline{Description}: Transformation formula ([1], (2.10.1); Appendix (III.23)) in form of a rule.

8W_{7}(a; b, c, d, e, f; q, \frac{a^{2}q^{2}}{bcdef}) ⟶ \frac{(aq, aq/ef, a^{2}q^{2}/bcde, a^{2}q^{2}/bcdf; q)_{∞}}{(aq/e, aq/f, a^{2}q^{2}/bcd, a^{2}q^{2}/bcdef; q)_{∞}}_{8}W_{7}(\frac{a^{2}q}{bcd}; \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc} , e, f; q, \frac{aq}{ef})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8705 \underline{Description}: Transformation formula ([1], (2.10.1), iterated; Appendix (III.24)) in form of a rule.

8W_{7}(a; b, c, d, e, f; q, \frac{a^{2}q^{2}}{bcdef}) ⟶ \frac{(aq, b, a^{2}q^{2}/bdef, a^{2}q^{2}/bcef, a^{2}q^{2}/bcdf, a^{2}q^{2}/bcde; q)_{∞}}{(aq/c, aq/d, aq/e, aq/f, a^{3}q^{3}/b^{2}cdef, a^{2}q^{2}/bcdef; q)_{∞}}

8W_{7}(\frac{a^{3}q^{2}}{b^{2}cdef}; \frac{aq}{bc} , \frac{aq}{bd} , \frac{aq}{be} , \frac{aq}{bf} , \frac{a^{2}q^{2}}{bcdef}; q, b)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8706 \underline{Description}: Transformation formula ([1], Ex. 2.13(ii), reversed) in form of a rule.

8W_{7}(a; b, −b, c, d, \frac{a^{2}q}{b^{2}cd}; q, −q) ⟶ \frac{(aq, −aq/b^{2}, bq, −bq; q)_{∞}}{(b^{2}q, −q, aq/b, −aq/b; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}b^{2}, b^{2}c/a, b^{2}d/a, aq/cd \\ aq/c, aq/d, b^{2}cd/a \end{matrix}; q, −\frac{aq}{b^{2}} ]

T8707 \underline{Description}: Transformation formula ([1], (3.4.7), reversed) in form of a rule.

8W_{7}(b; \sqrt{a}, −\sqrt{a}, \sqrt{aq}, −\sqrt{aq}, x; q, \frac{b^{2}q}{a^{2}x}) ⟶ \frac{(bq, b^{2}q/a^{2}; q)_{∞}}{(bq/a, b^{2}q/a; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a, ax/b \\ bq/x \end{matrix}; q, \frac{b^{2}q}{a^{2}x}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8708 \underline{Description}: Transformation formula ([1], (3.4.8), reversed) in form of a rule.

8W_{7}(b; \sqrt{aq}, −\sqrt{aq}, \sqrt{a}q, −\sqrt{a}q, x; q, \frac{b^{2}}{a^{2}qx}) ⟶ \frac{(bq, b^{2}/a^{2}q; q)_{∞}}{(b^{2}/a, b/aq; q)_{∞}}_{4}W_{3}(a; \frac{ax}{b}; q, \frac{b^{2}}{a^{2}qx})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8709 \underline{Description}: Transformation formula ([1], (3.5.4), reversed) in form of a rule.

8W_{7}(b; x, −x, \begin{matrix}√ & \overline{bq} \\ √ & \overline{a} \end{matrix} , −\begin{matrix}√ & \overline{bq} \\ √ & \overline{a} \end{matrix} , a; q, \frac{bq}{x^{2}} ) ⟶ \frac{(bq/a, b^{2}q^{2}/a^{2}x^{2}; q^{2})_{∞}}{(abq, b^{2}q^{2}/x^{2}; q^{2})_{∞}} \frac{(abq/x^{2}, bq; q)_{∞}}{(bq/x^{2}, bq/a; q)_{∞}}_{2}ϕ_{1}[ \begin{matrix}a^{2}, ax^{2}/b \\ abq^{2}/x^{2} \end{matrix} ; q^{2}, \frac{b^{2}q^{2}}{a^{2}x^{2}} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8710 \underline{Description}: Transformation formula ([1], (3.5.10)) in form of a rule.

8W_{7}(a; b, c, cq, d, dq; q^{2}, \frac{a^{2}q^{2}}{bc^{2}d^{2}} )

⟶ \frac{(aq, aq/bc, aq/cd, −aq/cd, aq/\sqrt{b}d, −aq/\sqrt{b}d; q)_{∞}}{(aq/b, aq/c, aq/d, −aq/d, aq/\sqrt{b}cd, −aq/\sqrt{b}cd; q)_{∞}}_{8}W_{7}(−\frac{a}{d}; c, \sqrt{b}, −\sqrt{b}, \frac{\sqrt{aq}}{d} , −\frac{\sqrt{aq}}{d}; q, \frac{aq}{bc})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8711 \underline{Description}: Transformation formula ([1], (3.5.10), reversed) in form of a rule.

8W_{7}(a; b, −b, d, −d, c; q, \frac{a^{2}q^{2}}{b^{2}cd^{2}} )

⟶ \frac{(a^{2}q^{2}/b^{2}d^{2}, a^{2}q^{2}/cd^{2}, −aq, aq, −aq/bc, aq/bc; q)_{∞}}{(a^{2}q^{2}/d^{2}, a^{2}q^{2}/b^{2}cd^{2}, −aq/c, aq/c, −aq/b, aq/b; q)_{∞}}_{8}W_{7}(\frac{a^{2}q}{d^{2}} ; b^{2}, c, cq, −\frac{aq}{d^{2}} , −\frac{aq^{2}}{d^{2}} ; q^{2}, \frac{a^{2}q^{2}}{b^{2}c^{2}} )

8W_{7}(a; b, c, d, e, f; q, \frac{a^{2}q^{2}}{bcdef}) ⟶ \frac{(aq, aq/de, aq/df, aq/ef; q)_{∞}}{(aq/d, aq/e, aq/f, aq/def; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}aq/bc, d, e, f \\ aq/b, aq/c, def/a \end{matrix}; q, q]

+ \frac{(aq, aq/bc, d, e, f, a^{2}q^{2}/bdef, a^{2}q^{2}/cdef; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, aq/f, a^{2}q^{2}/bcdef, def/aq; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}aq/de, aq/df, aq/ef, a^{2}q^{2}/bcdef \\ a^{2}q^{2}/bdef, a^{2}q^{2}/cdef, aq^{2}/def \end{matrix} ; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8762 \underline{Description}: Transformation formula ([1], (2.11.1); Appendix (III.37)) in form of a rule.

8W_{7}(a; b, c, d, e, f; q, \frac{a^{2}q^{2}}{bcdef})

⟶ −\frac{(aq, b/a, bq/c, bq/d, bq/e, bq/f, d, e, f, aq/bc, bdef/a^{2}, a^{2}q/bdef; q)_{∞}}{(a/b, aq/c, aq/d, aq/e, aq/f, bd/a, be/a, bf/a, def/a, aq/def, q/c, b^{2}q/a; q)_{∞}}

8W_{7}(\frac{b^{2}}{a}; b, \frac{bc}{a}, \frac{bd}{a}, \frac{be}{a}, \frac{bf}{a}; q, \frac{a^{2}q^{2}}{bcdef})

+ \frac{(aq, aq/de, aq/df, aq/ef, eq/c, fq/c, b/a, bef/a; q)_{∞}}{(aq/d, aq/e, aq/f, aq/def, q/c, efq/c, be/a, bf/a; q)_{∞}}

8W_{7}(\frac{ef}{c}; \frac{aq}{bc} , \frac{aq}{cd} , \frac{ef}{a} , e, f; q, \frac{bd}{a})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8763 \underline{Description}: Transformation formula ([1], (2.11.1), reversed; Appendix (III.37)) in form of a rule.

8W_{7}(a; b, c, d, e, f; q, \frac{a^{2}q^{2}}{bcdef})

⟶ \frac{(cef/a, fq/d, eq/d, c/a, aq/ef, aq, aq/bf, aq/be; q)_{∞}}{(efq/d, cf/a, ce/a, q/d, aq/f, aq/e, aq/bef, aq/b; q)_{∞}}_{8}W_{7}(\frac{ef}{d}; \frac{aq}{bd} , \frac{ef}{a} , \frac{aq}{cd} , e, f; q, \frac{bc}{a})

+ \frac{(aq, a^{2}q^{2}/bdef, cq/b, aq^{2}/bde, aq^{2}/bdf, aq/cd, e, f, b, a^{2}q^{2}/bcef, bcef/a^{2}q; q)_{∞}}{(cf/a, ce/a, q/d, aq/f, aq/e, aq/b, bef/aq, aq/d, a^{2}q^{2}/bcdef, aq/c, a^{2}q^{3}/b^{2}def; q)_{∞}}

8W_{7}(\frac{a^{2}q^{2}}{b^{2}def}; \frac{aq}{bd} , \frac{q}{b} , \frac{a^{2}q^{2}}{bcdef} , \frac{aq}{bf} , \frac{aq}{be}; q, \frac{bc}{a})

T8764 \underline{Description}: Transformation formula ([1], Ex. 2.15) in form of a rule.

8W_{7}(a^{2}; ab, ac, ad, ae, af; q, \frac{q^{2}}{abcdef})

⟶ −\frac{a}{b} \frac{(a^{2}q, ac, b/a, c/a, bq/d, bq/e, bq/f, q/bd, q/be, q/bf; q)_{∞}}{(aq/d, aq/e, aq/f, q/ad, q/ae, q/af, b^{2}q, bc, a/b, c/b; q)_{∞}}_{8}W_{7}(b^{2}; ab, bc, bd, be, bf; q, \frac{q^{2}}{abcdef})

− \frac{a}{c} \frac{(a^{2}q, ab, b/a, c/a, cq/d, cq/e, cq/f, q/cd, q/ce, q/cf; q)_{∞}}{(aq/d, aq/e, aq/f, q/ad, q/ae, q/af, c^{2}q, bc, a/c, b/c; q)_{∞}}_{8}W_{7}(c^{2}; ac, bc, cd, ce, cf; q, \frac{q^{2}}{abcdef})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8810 \underline{Description}: Transformation formula ([1], (5.6.1); Appendix (III.38)) in form of a rule.

8ψ_{8}[ \begin{matrix}\sqrt{a}q, −\sqrt{a}q, b, c, d, e, f, g \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq/f, aq/g \end{matrix}; q, \frac{a^{3}q^{2}}{bcdefg} ]

⟶ \frac{(f, f/a, aq, q/a, q, aq/bg, aq/cg, aq/dg, aq/eg, gq/b, gq/c, gq/d, gq/e; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, q/g, aq/g, f/g, fg/a, g^{2}q/a; q)_{∞}}_{8}W_{7}(\frac{g^{2}}{a}; \frac{bg}{a} , \frac{cg}{a} , \frac{dg}{a} , \frac{eg}{a} , \frac{fg}{a}; q, \frac{a^{3}q^{2}}{bcdefg})

+\frac{(g, g/a, aq, q/a, q, aq/bf, aq/cf, aq/df, aq/ef, fq/b, fq/c, fq/d, fq/e; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, q/f, aq/f, g/f, fg/a, f^{2}q/a; q)_{∞}}_{8}W_{7}(\frac{f^{2}}{a}; \frac{bf}{a} , \frac{cf}{a} , \frac{df}{a} , \frac{ef}{a} , \frac{fg}{a}; q, \frac{a^{3}q^{2}}{bcdefg})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10901 \underline{Description}: Transformation formula ([1], (2.9.1); Appendix (III.28)) in form of a rule.

10W_{9}(a; b, c, d, e, f, \frac{a^{3}q^{2+n}}{bcdef} , q^{−n}; q, q)

⟶ \begin{matrix}(aq, aq/ef, a^{2}q^{2}/bcde, a^{2}q^{2}/bcdf; q) & n \\ (aq/e, aq/f, a^{2}q^{2}/bcdef, a^{2}q^{2}/bcd; q) & n \end{matrix}_{10}W_{9}(\frac{a^{2}q}{bcd}; \frac{aq}{cd} , \frac{aq}{bd} , \frac{aq}{bc} , e, f, \frac{a^{3}q^{2+n}}{bcdef} , q^{−n}; q, q)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10902 \underline{Description}: Transformation formula ([1], Ex. 2.13(i), reversed) in form of a rule.

10W_{9}(a; b, −b, b\sqrt{q}, −b\sqrt{q}, c, d, \frac{a^{2}q}{b^{2}cd}; q, \frac{aq}{b^{2}} ) ⟶ \frac{(aq, a^{2}q/b^{4}; q)_{∞}}{(aq/b^{2}, a^{2}q/b^{2}; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}b^{2}, b^{2}c/a, b^{2}d/a, aq/cd \\ aq/c, aq/d, b^{2}cd/a \end{matrix}; q, \frac{a^{2}q}{b^{4}} ]

Alphabetic List: T10907 179

T10903 \underline{Description}: Transformation formula ([1], Ex. 2.19) in form of a rule.

10W_{9}(a; b, c, d, e, f, g, q^{−n}; q, q)

⟶ e^{n} \frac{(aq, aq/ce, aq/de, aq/ef, aq/eg, b; q)_{n}}{(aq/c, aq/d, aq/e, aq/f, aq/g, b/e; q)_{n}}_{10}W_{9}(\frac{e}{bq^{n}}; e, \frac{aq}{bc} , \frac{aq}{bd} , \frac{aq}{bf} , \frac{aq}{bg} , \frac{e}{aq^{n}} , q^{−n}; q, q)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10904 \underline{Description}: Transformation formula ([1], (3.10.4)) in form of a rule.

10W_{9}(a; b, x, −x, y, −y, −q^{−n}, q^{−n}; q, −\frac{a^{3}q^{3+2n}}{bx^{2}y^{2}} )

⟶ \frac{(a^{2}q^{2}, a^{2}q^{2}/x^{2}y^{2}; q^{2})_{n}}{(a^{2}q^{2}/x^{2}, a^{2}q^{2}/y^{2}; q^{2})_{n}}_{5}ϕ_{4}[ \begin{matrix}q^{−2n}, x^{2}, y^{2}, −aq/b, −aq^{2}/b \\ x^{2}y^{2}/a^{2}q^{2n}, a^{2}q^{2}/b^{2}, −aq, −aq^{2} \end{matrix} ; q^{2}, q^{2}]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10905 \underline{Description}: Transformation formula ([1], Ex. 3.21(iii)) in form of a rule.

10ϕ_{9}[ \begin{matrix}a, \sqrt{a}q, −\sqrt{a}q, b, c, a/bc, C/Aq^{n}, 1/BCq^{n}, B/Aq^{n}, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, bcq, 1/Cq^{n}, BC/Aq^{n}, 1/Bq^{n}, 1/Aq^{n} \end{matrix}; q, q]

⟶ \frac{(aq, bq, cq, aq/bc, Aq/B, Aq/C, BCq; q)_{n}}{(Aq, Bq, Cq, Aq/BC, aq/b, aq/c, bcq; q)_{n}}_{10}ϕ_{9}[ \begin{matrix}A, \sqrt{A}q, −\sqrt{A}q, B, C, A/BC, c/aq^{n}, 1/bcq^{n}, b/aq^{n}, q^{−n} \\ \sqrt{A}, −\sqrt{A}, Aq/B, Aq/C, BCq, 1/cq^{n}, bc/aq^{n}, 1/bq^{n}, 1/aq^{n} \end{matrix}; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10906 \underline{Description}: Transformation formula ([3], (7.7), reversed) in form of a rule.

10W_{9}(a^{2}q^{n}; c, d, e, aq^{\frac{1}{2} +n}, −aq^{\frac{1}{2} +n}, \frac{a^{4}q^{1+n}}{cde} , q^{−n}; q, −q^{1+n})

⟶ \frac{(a^{2}q/c, a^{2}q/d, a^{2}q/e, a^{2}q/cde; q)_{n}}{(a^{2}q, a^{2}q/cd, a^{2}q/ce, a^{2}q/de; q)_{n}}_{12}W_{11}(a^{2}; q^{−2n}, c, cq, d, dq, e, eq, \frac{a^{4}q^{1+n}}{cde} , \frac{a^{4}q^{2+n}}{cde}; q^{2}, q^{2})

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10907 \underline{Description}: Transformation formula ([3], (7.8), reversed) in form of a rule.

10W_{9}(\frac{a^{2}}{e}; \frac{a\sqrt{q}}{e} , −\frac{a\sqrt{q}}{e} , c, d, e, \frac{a^{4}q^{1+n}}{cde} , q^{−n}; q, −\frac{q}{e})

⟶ \frac{(a^{2}q/c, a^{2}q/d, a^{2}q/e, a^{2}q/cde; q)_{n}}{(a^{2}q, a^{2}q/cd, a^{2}q/ce, a^{2}q/de; q)_{n}}_{12}W_{11}(a^{2}; e^{2}, c, cq, d, dq, \frac{a^{4}q^{1+n}}{cde} , \frac{a^{4}q^{2+n}}{cde} , q^{1−n}, q^{−n}; q^{2}, q^{2})

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

180 Alphabetic List: T10963

T10961 \underline{Description}: Transformation formula ([1], (2.12.9); Appendix (III.39)) in form of a rule.

10W_{9}(a; b, c, d, e, f, g, \frac{a^{3}q^{2}}{bcdefg}; q, q)

⟶ \frac{(aq, b/a, f, g, a^{3}q^{2}/bcdefg, bq/f, bq/g, b^{2}cdefg/a^{3}q, aq/de, aq/ce, aq/cd; q)_{∞}}{(b^{2}cde/a^{2}, a^{2}q/bcde, aq/c, aq/d, aq/e, aq/f, aq/g, bcdefg/a^{2}q, bc/a, bd/a, be/a; q)_{∞}}

\frac{(bde/a, bce/a, bcd/a; q)_{∞}}{(bf/a, bg/a, a^{2}q^{2}/cdefg; q)_{∞}}_{10}W_{9}(\frac{b^{2}cde}{a^{2}q}; b, \frac{bc}{a}, \frac{bd}{a}, \frac{be}{a}, \frac{bcdef}{a^{2}q} , \frac{bcdeg}{a^{2}q} , \frac{aq}{fg}; q, q)

+ \frac{(aq, b/a, a^{2}q^{2}/cdef, a^{2}q^{2}/cdeg, bfg/a, bcdef/a^{2}q, bcdeg/a^{2}q, aq/fg; q)_{∞}}{(a^{2}q^{2}/cde, bcde/a^{2}q, aq/f, aq/g, bcdefg/a^{2}q, bf/a, bg/a, a^{2}q^{2}/cdefg; q)_{∞}}

10W_{9}(\frac{a^{2}q}{cde}; b, \frac{aq}{de} , \frac{aq}{ce} , \frac{aq}{cd} , f, g, \frac{a^{3}q^{2}}{bcdefg}; q, q)

− \frac{(aq, b/a, c, d, e, f, g, a^{3}q^{2}/bcdefg, bq/c, bq/d, bq/e, bq/f; q)_{∞}}{(b^{2}q/a, a/b, aq/c, aq/d, aq/e, aq/f, aq/g, bcdefg/a^{2}q, bc/a, bd/a, be/a, bf/a; q)_{∞}}

\frac{(bq/g, b^{2}cdefg/a^{3}q; q)_{∞}}{(bg/a, a^{2}q^{2}/cdefg; q)_{∞}}_{10}W_{9}(\frac{b^{2}}{a}; b, \frac{bc}{a}, \frac{bd}{a}, \frac{be}{a}, \frac{bf}{a} , \frac{bg}{a} , \frac{a^{2}q^{2}}{cdefg}; q, q)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10962 \underline{Description}: Transformation formula ([1], (3.5.7)) in form of a rule.

10W_{9}(a; b, c, cq, d, dq, e, eq; q^{2}, \frac{a^{3}q^{3}}{bc^{2}d^{2}e^{2}} ) ⟶ \frac{(aq, aq/cd, aq/ce, aq/de; q)_{∞}}{(aq/c, aq/d, aq/e, aq/cde; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}\sqrt{aq}/\sqrt{b}, −\sqrt{aq}/\sqrt{b}, c, d, e \\ \sqrt{aq}, −\sqrt{aq}, aq/b, cde/a \end{matrix}; q, q]

+ \frac{(aq^{2}, a^{3}q^{3}/c^{2}d^{2}e^{2}; q^{2})_{∞}}{(aq^{2}/b, a^{3}q^{3}/bc^{2}d^{2}e^{2}; q^{2})_{∞}} \frac{(c, d, e, a^{2}q^{2}/bcde; q)_{∞}}{(aq/c, aq/d, aq/e, cde/aq; q)_{∞}}

5ϕ_{4}[ \begin{matrix}(aq)^{3/2}/\sqrt{b}cde, −(aq)^{3/2}/\sqrt{b}cde, aq/cd, aq/ce, aq/de \\ (aq)^{3/2}/cde, −(aq)^{3/2}/cde, a^{2}q^{2}/bcde, aq^{2}/cde \end{matrix}; q, q]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T10963 \underline{Description}: Transformation formula ([1], Ex. 2.30) in form of a rule.

10W_{9}(a; b, c, d, e, f, g, \frac{a^{3}q^{2}}{bcdefg}; q, q)

⟶ −\frac{(bq/f, bq/g, b^{2}cdefg/a^{3}q, aq, c, d, e, f, g, a^{3}q^{2}/bcdefg, b/a, bq/c, bq/d, bq/e; q)_{∞}}{(aq/f, aq/g, bcdefg/a^{2}q, b^{2}q/a, bc/a, bd/a, be/a, bf/a, bg/a, a^{2}q^{2}/cdefg, a/b, aq/c, aq/d, aq/e; q)_{∞}}

10W_{9}(\frac{b^{2}}{a}; b, \frac{bc}{a}, \frac{bd}{a}, \frac{be}{a}, \frac{bf}{a} , \frac{bg}{a} , \frac{a^{2}q^{2}}{cdefg}; q, q)

+ \frac{(beg/a, bfg/a, aq, b/a, a^{3}q^{2}/bcdefg, b^{2}cdefg/a^{3}q, aq/cg, aq/dg, aq/eg, aq/fg, bcg/a, bdg/a; q)_{∞}}{(be/a, bf/a, b^{2}cdefg^{2}/a^{3}q, bg/a, a^{3}q^{2}/bcdefg^{2}, aq/g, aq/c, aq/d, aq/e, aq/f, bc/a, bd/a; q)_{∞}}

10W_{9}(\frac{b^{2}cdefg^{2}}{a^{3}q^{2}} ; b, \frac{bdefg}{a^{2}q} , \frac{bcefg}{a^{2}q} , \frac{bcdfg}{a^{2}q} , \frac{bcdeg}{a^{2}q} , \frac{bg}{a} , g; q, q)

+ \frac{(a^{2}q^{2}/cdfg, a^{2}q^{2}/cdeg, aq, b/a, g, bq/g, bdefg/a^{2}q, bcefg/a^{2}q, bcdfg/a^{2}q, bcdeg/a^{2}q; q)_{∞}}{(be/a, bf/a, a^{3}q^{3}/cdefg^{2}, a^{2}q^{2}/cdefg, bcdefg^{2}/a^{3}q^{2}, bcdefg/a^{2}q, aq/c, aq/d, aq/e, aq/f; q)_{∞}}

\frac{(a^{2}q^{2}/defg, a^{2}q^{2}/cefg; q)_{∞}}{(bc/a, bd/a; q)_{∞}}_{10}W_{9}(\frac{a^{3}q^{2}}{cdefg^{2}} ; b, \frac{aq}{cg} , \frac{aq}{dg} , \frac{aq}{eg} , \frac{aq}{fg} , \frac{a^{2}q^{2}}{cdefg} , \frac{a^{3}q^{2}}{bcdefg}; q, q)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T101010 \underline{Description}: Transformation formula ([1], (5.6.3); Appendix (III.40)) in form of a rule.

10ψ_{10}[ \begin{matrix}\sqrt{a}q, −\sqrt{a}q, b, c, d, e, f, g, h, k \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq/f, aq/g, aq/h, aq/k \end{matrix}; q, \frac{a^{4}q^{3}}{bcdefghk} ]

⟶ \frac{(g, h, g/a, h/a, aq, q/a, q, aq/bk, aq/ck, aq/dk, aq/ek, aq/fk, kq/b, kq/c, kq/d, kq/e, kq/f; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, aq/f, q/b, q/c, q/d, q/e, q/f, hk/a, gk/a, h/k, g/k, q/k, aq/k, k^{2}q/a; q)_{∞}}

×_{10}W_{9}(\frac{k^{2}}{a}; \frac{bk}{a} , \frac{ck}{a} , \frac{dk}{a} , \frac{ek}{a} , \frac{fk}{a} , \frac{hk}{a} , \frac{gk}{a}; q, \frac{a^{4}q^{3}}{bcdefghk})

+ \frac{(g, k, g/a, k/a, aq, q/a, q, aq/bh, aq/ch, aq/dh, aq/eh, aq/fh, hq/b, hq/c, hq/d, hq/e, hq/f; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, aq/f, q/b, q/c, q/d, q/e, q/f, gh/a, hk/a, g/h, k/h, q/h, aq/h, h^{2}q/a; q)_{∞}}

×_{10}W_{9}(\frac{h^{2}}{a}; \frac{bh}{a}, \frac{ch}{a}, \frac{dh}{a}, \frac{eh}{a}, \frac{fh}{a}, \frac{gh}{a}, \frac{hk}{a}; q, \frac{a^{4}q^{3}}{bcdefghk})

+ \frac{(h, k, h/a, k/a, aq, q/a, q, aq/bg, aq/cg, aq/dg, aq/eg, aq/fg, gq/b, gq/c, gq/d, gq/e, gq/f; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, aq/f, q/b, q/c, q/d, q/e, q/f, gh/a, gk/a, h/g, k/g, q/g, aq/g, g^{2}q/a; q)_{∞}}

×_{10}W_{9}(\frac{g^{2}}{a}; \frac{bg}{a} , \frac{cg}{a} , \frac{dg}{a} , \frac{eg}{a} , \frac{fg}{a} , \frac{gh}{a}, \frac{gk}{a}; q, \frac{a^{4}q^{3}}{bcdefghk})

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T121101 \underline{Description}: Transformation formula ([1], (2.8.3), reversed; Appendix (III.25), reversed) in form of a rule.

12W_{11}(a; b, c, d, \frac{a\sqrt{q}}{\sqrt{b}\sqrt{c}\sqrt{d}}, −\frac{a\sqrt{q}}{\sqrt{b}\sqrt{c}\sqrt{d}}, \frac{aq}{\sqrt{b}\sqrt{c}\sqrt{d}}, −\frac{aq}{\sqrt{b}\sqrt{c}\sqrt{d}},

bcdq^{n}, q^{−n}; q, q) ⟶ \begin{matrix}(aq, b^{2}c^{2}d^{2}/a^{2}q; q) & n \\ (bcd/a, bcd; q) & n \end{matrix}_{5}ϕ_{4}[ \begin{matrix}a^{2}q/bcd, aq/cd, aq/bd, aq/bc, q^{−n} \\ aq/b, aq/c, aq/d, a^{2}q^{2−n}/b^{2}c^{2}d^{2} \end{matrix} ; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T121102 \underline{Description}: Transformation formula ([1], (2.8.4), reversed; Appendix (III.26), reversed) in form of a rule.

12W_{11}(\sqrt{b}\sqrt{c}\sqrt{d}q^{− \frac{1}{2} − \frac{n}{2}} ; b, c, d, q^{\frac{−n}{2}} , −q^{\frac{−n}{2}} , q^{\frac{1}{2} − \frac{n}{2}} , −q^{\frac{1}{2} − \frac{n}{2}} , e, \frac{bcd}{e}; q, q)

⟶ \begin{matrix}(\sqrt{b}\sqrt{c}\sqrt{d}q^{1/2−n/2}, bcd/e; q) & n \\ (\sqrt{b}\sqrt{c}\sqrt{d}q^{1/2−n/2}/e, bcd; q) & n \end{matrix}

5ϕ_{4}[ \begin{matrix}q^{−n}, \sqrt{b}q^{1/2−n/2}/\sqrt{c}\sqrt{d}, \sqrt{c}q^{1/2−n/2}/\sqrt{b}\sqrt{d}, \sqrt{d}q^{1/2−n/2}/\sqrt{b}\sqrt{c}, e \\ \sqrt{c}\sqrt{d}q^{1/2−n/2}/\sqrt{b}, \sqrt{b}\sqrt{d}q^{1/2−n/2}/\sqrt{c}, \sqrt{b}\sqrt{c}q^{1/2−n/2}/\sqrt{d}, eq^{1−n}/bcd \end{matrix}; q, q]

T121103 \underline{Description}: Transformation formula ([1], (2.8.5), reversed; Appendix (III.27), reversed) in form of a rule.

12W_{11}(a; b, c, d, \frac{aq}{\sqrt{b}\sqrt{c}\sqrt{d}}, −\frac{aq}{\sqrt{b}\sqrt{c}\sqrt{d}}, \frac{aq^{\frac{3}{2}}}{\sqrt{b}\sqrt{c}\sqrt{d}}, −\frac{aq^{\frac{3}{2}}}{\sqrt{b}\sqrt{c}\sqrt{d}},

bcdq^{−2+n}, q^{−n}; q, q) ⟶ \frac{(1 − bcd/q^{2})}{(1 − bcdq^{−2+2n})} \begin{matrix}(aq, b^{2}c^{2}d^{2}/a^{2}q^{3}; q) & n \\ (bcd/aq^{2}, bcd/q^{2}; q) & n \end{matrix}

7ϕ_{6}[ \begin{matrix}a^{2}q/bcd, aq^{3/2}/\sqrt{b}\sqrt{c}\sqrt{d}, −aq^{3/2}/\sqrt{b}\sqrt{c}\sqrt{d}, aq/cd, aq/bd, aq/bc, q^{−n} \\ a\sqrt{q}/\sqrt{b}\sqrt{c}\sqrt{d}, −a\sqrt{q}/\sqrt{b}\sqrt{c}\sqrt{d}, aq/b, aq/c, aq/d, a^{2}q^{4−n}/b^{2}c^{2}d^{2} \end{matrix} ; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T121104 \underline{Description}: Transformation formula ([1], Ex. 2.14(ii), reversed) in form of a rule.

12W_{11}(a; c, d, \frac{a^{2}q}{bcd} , \sqrt{b}q, −\sqrt{b}, \sqrt{bq}, −\sqrt{bq}, \frac{a^{2}q^{n}}{b} , q^{−n}; q, q)

⟶ \frac{(aq, a^{2}/b^{2}, −a/\sqrt{b}; q)_{n}}{(a/b, a^{2}/b, −aq/\sqrt{b}; q)_{n}}_{6}ϕ_{5}[ \begin{matrix}b, \sqrt{b}q, bc/a, bd/a, aq/cd, q^{−n} \\ \sqrt{b}, aq/c, aq/d, bcd/a, b^{2}q^{1−n}/a^{2} \end{matrix} ; q, q]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T121105 \underline{Description}: Transformation formula ([3], (7.7)) in form of a rule.

12W_{11}(a^{2}; c, cq, d, dq, e, eq, \frac{a^{4}q^{1+n}}{cde} , \frac{a^{4}q^{2+n}}{cde} , q^{−2n}; q^{2}, q^{2})

⟶ \frac{(a^{2}q, a^{2}q/cd, a^{2}q/ce, a^{2}q/de; q)_{n}}{(a^{2}q/c, a^{2}q/d, a^{2}q/e, a^{2}q/cde; q)_{n}}_{10}W_{9}(a^{2}q^{n}; c, d, e, aq^{\frac{1}{2} +n}, −aq^{\frac{1}{2} +n}, \frac{a^{4}q^{1+n}}{cde} , q^{−n}; q, −q^{1+n})

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T121106 \underline{Description}: Transformation formula ([3], (7.8)) in form of a rule.

12W_{11}(a^{2}; e^{2}, c, cq, d, dq, \frac{a^{4}q^{1+n}}{cde} , \frac{a^{4}q^{2+n}}{cde} , q^{1−n}, q^{−n}; q^{2}, q^{2})

⟶ \frac{(a^{2}q, a^{2}q/cd, a^{2}q/ce, a^{2}q/de; q)_{n}}{(a^{2}q/c, a^{2}q/d, a^{2}q/e, a^{2}q/cde; q)_{n}}_{10}W_{9}(\frac{a^{2}}{e}; \frac{a\sqrt{q}}{e} , −\frac{a\sqrt{q}}{e} , c, d, e, \frac{a^{4}q^{1+n}}{cde} , q^{−n}; q, −\frac{q}{e})

T121107 \underline{Description}: Transformation formula ([1], Ex. 8.15, reversed) in form of a rule.

12W_{11} (d; \begin{matrix}√ & \overline{dq} \\ √ & \overline{a} \end{matrix} , −\begin{matrix}√ & \overline{dq} \\ √ & \overline{a} \end{matrix} , \frac{\sqrt{d}q}{\sqrt{a}} , −\frac{\sqrt{d}q}{\sqrt{a}} , \frac{ad}{c}, \frac{ad}{b}, a, b, c; q, \frac{q}{a})

⟶ \frac{(dq/bc, ad/c, ad/b, dq; q)_{∞}}{(ad/bc, dq/c, dq/b, ad; q)_{∞}}_{4}ϕ_{3}[ \begin{matrix}a, b, c, bc/d \\ bq/a, cq/a, bcq/ad \end{matrix}; q, \frac{q^{2}}{a^{2}} ]

provided at least one of a, b, c is of the form q^{−}n, n = 0, 1, 2, . . ..

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T121161 \underline{Description}: Transformation formula ([1], Ex. 2.25, reversed) in form of a rule.

12W_{11}(a; \sqrt{b}, −\sqrt{b}, \sqrt{b}\sqrt{q}, −\sqrt{b}\sqrt{q}, c, d, \frac{a^{2}q}{bcd} , e, \frac{a^{2}q}{be}; q, q)

⟶ \frac{(aq, a^{2}q/be, a^{2}q/b^{2}, aq/be; q)_{∞}}{(aq/b, aq/e, a^{2}q/b, a^{2}q/b^{2}e; q)_{∞}}_{5}ϕ_{4}[ \begin{matrix}b, bc/a, bd/a, aq/cd, e \\ aq/c, aq/d, bcd/a, b^{2}e/a^{2} \end{matrix} ; q, q]

+ \frac{(aq, aq/be, b, bc/a, bd/a, aq/cd, e, a^{3}q^{2}/b^{2}ce, a^{3}q^{2}/b^{2}de, acdq/be; q)_{∞}}{(aq/b, aq/e, a^{2}q/b, aq/c, aq/d, bcd/a, b^{2}e/a^{2}q, acq/be, adq/be, a^{3}q^{2}/b^{2}cde; q)_{∞}}

5ϕ_{4}[ \begin{matrix}a^{2}q/b^{2}, a^{2}q/be, acq/be, adq/be, a^{3}q^{2}/b^{2}cde \\ a^{2}q^{2}/b^{2}e, a^{3}q^{2}/b^{2}ce, a^{3}q^{2}/b^{2}de, acdq/be \end{matrix}; q, q]

− \frac{(aq, aq/be, b, e, c, d, a^{2}q/bcd, a^{2}q^{2}/bce, a^{2}q^{2}/bde, cdq/e, a^{4}q^{3}/b^{3}e^{2}; q)_{∞}}{(aq/b, aq/e, a^{2}q/b, aq/c, aq/d, bcd/a, acq/be, adq/be, a^{3}q^{2}/b^{2}cde, be/aq, a^{3}q^{3}/b^{2}e^{2}; q)_{∞}}

12W_{11}(\frac{a^{3}q^{2}}{b^{2}e^{2}} ; \frac{aq}{\sqrt{b}e}, −\frac{aq}{\sqrt{b}e}, \frac{aq^{\frac{3}{2}}}{\sqrt{b}e}, −\frac{aq^{\frac{3}{2}}}{\sqrt{b}e}, \frac{aq}{b} , \frac{a^{2}q}{be} , \frac{acq}{be} , \frac{adq}{be} , \frac{a^{3}q^{2}}{b^{2}cde}; q, q)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

TeX \underline{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 AMS-TEX. \underline{Usage}: TeX. \underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AMSTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[2]:= TeXForm[ph[{a,b},{c},q,z]]
Out[2]//TeXForm=
{} {2} \phi {1} \! \left [ \matrix \let \over / a, b\\ \let \over /
 c\endmatrix ;q, {\displaystyle z} \right ]

In[3]:= TeX
In[4]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[5]:= TeXForm[ph[{a,b},{c},q,z]]
Out[5]//TeXForm=
{} {2} \phi {1} \! \left [ \matrix { \let \over / a, b\cr \let \over / c}
 ;q, {\displaystyle z} \right ]
\underline{See also}: AmSTeX, AmSLaTeX, LaTeX, TeXMat, TeXphW.

TeXMat

\underline{Description}: Function that writes (to be precise: appends) an expression Expr in InputForm to a file [name].m and the TeXForm of Expr to the file [name].tex. The expressions are numbered automatically. The number can be reset by SchreibeZahl. The string comment is optional. It allows to place the comment comment above the expression and the number in each of the two files.

\underline{Usage}: :  TeXMat[Expr,name,comment].

\underline{Example(s)}:

In[1]:= TeXMat[pq[a,2*n,q^{∧}3],filename]
In[2]:= TeXMat[ph[{a,b},{c},q,z],filename,"A basic hypergeometric  2\phi 1
        series"]
In[3]:= !type filename.m
A[1]:=
pq[a, 2*n, q^{∧}3]
"A basic hypergeometric  2\\phi 1  series"
A[2]:=
ph[{a, b}, {c}, q, z]
In[3]:= !type filename.tex
A[1]:=
({\let \over / a}; {q^{∧}3}) {2\,n}
"A basic hypergeometric  2\\phi 1  series"
A[2]:=
{} {2} \phi {1} \! \left [ \matrix \let \over / a, b\\ \let \over /
  c\endmatrix ;q, {\displaystyle z} \right ]
\underline{See also}: AmSTeX, AmSLaTeX, LaTeX, TeX, TeXphW, SchreibeZahl.

TeXphW

\underline{Description}: Switch that toggles between writing very well-poised basic hypergeometric series in terms of W and in terms of ph, respectively, when written in TeXForm. By default very well-poised hypergeometric series are written in terms of W.

\underline{Usage}: TeXphW.

\underline{Example(s)}:

In[1]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
In[2]:= ph[{a,q*Sqrt[a],-q*Sqrt[a],b,c},{Sqrt[a],-Sqrt[a],a*q/b,a*q/c},q,z]
             a, Sqrt[a] q, -(Sqrt[a] q), b, c
Out[2]= ϕ  a q  a q   ; q, z
        5 4    Sqrt[a], -Sqrt[a], ---, ---
                                   b    c
In[3]:= TeXForm[%]
Out[3]//TeXForm=
{} {5} W {4} ({\displaystyle a; b, c}; q, {\displaystyle z})
In[4]:= TeXphW
In[5]:= hypqAttributes
Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses ph[] for very well-poised basic hypergeometric series.
In[6]:= TeXForm[%2]
Out[6]//TeXForm=
{} {5} \phi {4} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, -\left(
 {\sqrt{a}} q \right) , b, c\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a
 q}\over b}, {{a q}\over c}\endmatrix ;q, {\displaystyle z} \right ]
In[7]:= TeXphW
In[8]:= hypqAttributes

186 Alphabetic List: Tgl2107

Automatic evaluation of pq and ph is inactive.
Automatic cancelling in ph is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses W[] for very well-poised basic hypergeometric series.
\underline{See also}: ph, W, hypqAttributes.

Tgl2101 \underline{Description}: Transformation formula ([1], (1.4.1); Appendix (III.1)) in form of an equation. It is the same transfor-

mation as that in T2101.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2102 \underline{Description}: Transformation formula ([1], (1.4.5); Appendix (III.2)) in form of an equation. It is the same transfor-

mation as that in T2102.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2103 \underline{Description}: Transformation formula ([1], (1.4.6); Appendix (III.3)) in form of an equation. It is the same transfor-

mation as that in T2103.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2104 \underline{Description}: Transformation formula ([1], (1.5.4); Appendix (III.4)) in form of an equation. It is the same transfor-

mation as that in T2104.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2105 \underline{Description}: Transformation formula ([1], (3.2.4), nonterminating; Appendix (III.5)) in form of an equation. It is the

same transformation as that in T2105.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2106 \underline{Description}: Transformation formula ([1], (1.5.6); Appendix (III.6)) in form of an equation. It is the same transfor-

mation as that in T2106.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2107 \underline{Description}: Transformation formula ([1], Ex. 1.15(iii); Appendix (III.7)) in form of an equation. It is the same

transformation as that in T2107.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl2163 187

Tgl2108 \underline{Description}: Transformation formula ([1], Ex. 1.15(ii); Appendix (III.8)) in form of an equation. It is the same

transformation as that in T2108.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2109 \underline{Description}: Transformation formula ([1], Ex. 2.2, reversed) in form of an equation. It is the same transformation as

that in T2109.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2110 \underline{Description}: Transformation formula ([1], (3.4.7)) in form of an equation. It is the same transformation as that in

  T2110.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2111 \underline{Description}: Transformation formula ([1], (3.5.4)) in form of an equation. It is the same transformation as that in

  T2111.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2112 \underline{Description}: Transformation formula ([1], Ex. 3.2(i), reversed) in form of an equation. It is the same transformation

as that in T2112.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2161 \underline{Description}: Transformation formula ([1], (3.3.5); Appendix (III.31)) in form of an equation. It is the same transfor-

mation as that in T2161.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2162 \underline{Description}: Transformation formula ([1], Appendix (III.32)) in form of an equation. It is the same transformation as

that in T2162.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2163 \underline{Description}: Transformation formula ([1], Ex. 3.8) in form of an equation. It is the same transformation as that in

  T2163.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

188 Alphabetic List: Tgl3205

Tgl2201 \underline{Description}: Transformation formula ([1], (1.5.4), reversed; Appendix (III.4), reversed) in form of an equation. It is

the same transformation as that in T2201.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2202 \underline{Description}: Transformation formula ([1], Ex. 3.2(ii), reversed) in form of an equation. It is the same transformation

as that in T2202.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3101 \underline{Description}: Transformation formula ([1], Ex. 1.15(ii), reversed; Appendix (III.8), reversed) in form of an equation.

It is the same transformation as that in T3101.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3201 \underline{Description}: Transformation formula ([1], (3.2.4), reversed; Appendix (III.5), reversed) in form of an equation. It is

the same transformation as that in T3201.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3202 \underline{Description}: Transformation formula ([1], (1.5.6), reversed; Appendix (III.6), reversed) in form of an equation. It is

the same transformation as that in T3202.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3203 \underline{Description}: Transformation formula ([1], Ex. 1.15(iii), reversed; Appendix (III.7), reversed) in form of an equation.

It is the same transformation as that in T3203.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3204 \underline{Description}: Transformation formula ([1], (3.2.7); Appendix (III.9)) in form of an equation. It is the same transfor-

mation as that in T3204.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3205 \underline{Description}: Transformation formula ([1], (3.2.10); Appendix (III.10)) in form of an equation. It is the same transfor-

mation as that in T3205.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl3213 189

Tgl3206 \underline{Description}: Transformation formula ([1], (3.2.3); Appendix (III.11)) in form of an equation. It is the same transfor-

mation as that in T3206.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3207 \underline{Description}: Transformation formula ([1], (3.2.2); Appendix (III.12)) in form of an equation. It is the same transfor-

mation as that in T3207.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3208 \underline{Description}: Transformation formula ([1], (3.2.5); Appendix (III.13)) in form of an equation. It is the same transfor-

mation as that in T3208.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3209 \underline{Description}: Transformation formula ([1], Ex. 2.26; Appendix (III.14)) in form of an equation. It is the same trans-

formation as that in T3209.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3210 \underline{Description}: Transformation formula ([1], (3.2.6)) in form of an equation. It is the same transformation as that in

  T3210.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3211 \underline{Description}: Transformation formula ([1], Ex. 3.1) in form of an equation. It is the same transformation as that in

  T3211.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3212 \underline{Description}: Transformation formula ([1], Ex. 3.1, reversed) in form of an equation. It is the same transformation as

that in T3212.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3213 \underline{Description}: Transformation formula ([1], Ex. 3.2(i)) in form of an equation. It is the same transformation as that in

  T3213.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

190 Alphabetic List: Tgl3266

Tgl3214 \underline{Description}: Transformation formula ([1], Ex. 3.2(ii)) in form of an equation. It is the same transformation as that in

  T3214.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3215 \underline{Description}: Transformation formula ([1], Ex. 3.3) in form of an equation. It is the same transformation as that in

  T3215.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3216 \underline{Description}: Transformation formula ([1], Ex. 3.3, reversed) in form of an equation. It is the same transformation as

that in T3216.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3217 \underline{Description}: Transformation formula ([1], (3.2.11)) in form of an equation. It is the same transformation as that in

  T3217.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3261 \underline{Description}: Transformation formula ([1], (3.3.3); Appendix (III.33)) in form of an equation. It is the same transfor-

mation as that in T3261.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3262 \underline{Description}: Transformation formula ([1], (3.3.1); Appendix (III.34)) in form of an equation. It is the same transfor-

mation as that in T3262.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3265 \underline{Description}: Transformation formula ([1], (3.4.1); Appendix (III.35)) in form of an equation. It is the same transfor-

mation as that in T3265.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3266 \underline{Description}: Transformation formula ([1], (3.5.2)) in form of an equation. It is the same transformation as that in

  T3266.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl4304 191

Tgl3267 \underline{Description}: Transformation formula ([1], Ex. 3.6) in form of an equation. It is the same transformation as that in

  T3267.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3268 \underline{Description}: Transformation formula ([1], Ex. 3.6, reversed) in form of an equation. It is the same transformation as

that in T3268.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3269 \underline{Description}: Transformation formula ([1], Ex. 3.8, reversed) in form of an equation. It is the same transformation as

that in T3269.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4201 \underline{Description}: Transformation formula ([1], Ex. 3.4, reversed) in form of an equation. It is the same transformation as

that in T4201.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4301 \underline{Description}: Transformation formula ([1], (2.10.4); Appendix (III.15)) in form of an equation. It is the same transfor-

mation as that in T4301.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4302 \underline{Description}: Transformation formula ([1], (3.2.9); Appendix (III.16)) in form of an equation. It is the same transfor-

mation as that in T4302.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4303 \underline{Description}: Transformation formula ([1], (2.5.1), reversed; Appendix (III.19)) in form of an equation. It is the same

transformation as that in T4303.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4304 \underline{Description}: Transformation formula ([1], (2.10.7); Appendix (III.20)) in form of an equation. It is the same transfor-

mation as that in T4304.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

192 Alphabetic List: Tgl4312

Tgl4305 \underline{Description}: Transformation formula ([1], (3.10.13); Appendix (III.21)) in form of an equation. It is the same trans-

formation as that in T4305.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4306 \underline{Description}: Transformation formula ([1], (3.10.13), reversed; Appendix (III.21), reversed) in form of an equation. It

is the same transformation as that in T4306.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4307 \underline{Description}: Transformation formula ([1], (8.8.3); Appendix (III.22)) in form of an equation. It is the same transfor-

mation as that in T4307.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4308 \underline{Description}: Transformation formula ([1], Ex. 2.2) in form of an equation. It is the same transformation as that in

  T4308.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4309 \underline{Description}: Transformation formula ([1], Ex. 2.13(i)) in form of an equation. It is the same transformation as that

  in T4309.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4310 \underline{Description}: Transformation formula ([1], Ex. 2.13(ii)) in form of an equation. It is the same transformation as that

  in T4310.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4311 \underline{Description}: Transformation formula ([1], (3.4.8)) in form of an equation. It is the same transformation as that in

  T4311.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4312 \underline{Description}: Transformation formula ([1], Ex. 3.4) in form of an equation. It is the same transformation as that in

  T4312.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl5461 193

Tgl4313 \underline{Description}: Transformation formula ([1], Ex. 8.15) in form of an equation. It is the same transformation as that in

  T4313.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4361 \underline{Description}: Transformation formula ([1], Ex. 3.16) in form of an equation. It is the same transformation as that in

  T4361.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5401 \underline{Description}: Transformation formula ([1], (2.8.3); Appendix (III.25)) in form of an equation. It is the same transfor-

mation as that in T5401.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5402 \underline{Description}: Transformation formula ([1], (2.8.4); Appendix (III.26)) in form of an equation. It is the same transfor-

mation as that in T5402.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5403 \underline{Description}: Transformation formula ([1], (3.10.4), reversed) in form of an equation. It is the same transformation as

that in T5403.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5404 \underline{Description}: Transformation formula ([1], Ex. 2.26, Appendix (III.14), reversed) in form of an equation. It is the same

transformation as that in T5404.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5405 \underline{Description}: Transformation formula ([1], (8.8.3), Appendix (III.22), reversed) in form of an equation. It is the same

transformation as that in T5405.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5461 \underline{Description}: Transformation formula ([1], (3.4.4)) in form of an equation. It is the same transformation as that in

  T5461.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

194 Alphabetic List: Tgl6501

Tgl5462 \underline{Description}: Transformation formula ([1], (3.4.4), reversed, first form) in form of an equation. It is the same transfor-

mation as that in T5462.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5463 \underline{Description}: Transformation formula ([1], (3.4.4), reversed, second form) in form of an equation. It is the same

transformation as that in T5463.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5464 \underline{Description}: Transformation formula ([1], (3.5.2), reversed) in form of an equation. It is the same transformation as

that in T5464.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5465 \underline{Description}: Transformation formula ([1], (3.5.7), reversed) in form of an equation. It is the same transformation as

that in T5465.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5466 \underline{Description}: Transformation formula ([1], Ex. 3.16, reversed, first form) in form of an equation. It is the same

transformation as that in T5466.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5467 \underline{Description}: Transformation formula ([1], Ex. 3.16, reversed, second form) in form of an equation. It is the same

transformation as that in T5467.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5468 \underline{Description}: Transformation formula ([1], Ex. 2.25) in form of an equation. It is the same transformation as that in

  T5468.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl6501 \underline{Description}: Transformation formula ([1], Ex. 2.14(ii)) in form of an equation. It is the same transformation as that

  in T6501.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl8706 195

Tgl7601 \underline{Description}: Transformation formula ([1], (2.8.5); Appendix (III.27)) in form of an equation. It is the same transfor-

mation as that in T7601.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7701 \underline{Description}: Transformation formula ([1], (3.2.11), reversed) in form of an equation. It is the same transformation as

that in T7701.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8701 \underline{Description}: Transformation formula ([1], (2.10.10), terminated; Appendix (III.17)) in form of an equation. It is the

same transformation as that in T8701.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8702 \underline{Description}: Transformation formula ([1], (2.5.1); Appendix (III.18)) in form of an equation. It is the same transfor-

mation as that in T8702.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8703 \underline{Description}: Transformation formula ([1], (2.10.7), reversed; Appendix (III.20), reversed) in form of an equation. It

is the same transformation as that in T8703.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8704 \underline{Description}: Transformation formula ([1], (2.10.1); Appendix (III.23)) in form of an equation. It is the same transfor-

mation as that in T8704.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8705 \underline{Description}: Transformation formula ([1], (2.10.1), iterated; Appendix (III.24)) in form of an equation. It is the same

transformation as that in T8705.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8706 \underline{Description}: Transformation formula ([1], Ex. 2.13(ii), reversed) in form of an equation. It is the same transformation

as that in T8706.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

196 Alphabetic List: Tgl8763

Tgl8707 \underline{Description}: Transformation formula ([1], (3.4.7), reversed) in form of an equation. It is the same transformation as

that in T8707.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8708 \underline{Description}: Transformation formula ([1], (3.4.8), reversed) in form of an equation. It is the same transformation as

that in T8708.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8709 \underline{Description}: Transformation formula ([1], (3.5.4), reversed) in form of an equation. It is the same transformation as

that in T8709.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8710 \underline{Description}: Transformation formula ([1], (3.5.10)) in form of an equation. It is the same transformation as that in

  T8710.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8711 \underline{Description}: Transformation formula ([1], (3.5.10), reversed) in form of an equation. It is the same transformation as

that in T8711.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8761 \underline{Description}: Transformation formula ([1], (2.10.10); Appendix (III.36)) in form of an equation. It is the same trans-

formation as that in T8761.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8762 \underline{Description}: Transformation formula ([1], (2.11.1); Appendix (III.37)) in form of an equation. It is the same transfor-

mation as that in T8762.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8763 \underline{Description}: Transformation formula ([1], (2.11.1), reversed; Appendix (III.37)) in form of an equation. It is the same

transformation as that in T8763.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl10906 197

Tgl8764 \underline{Description}: Transformation formula ([1], Ex. 2.15) in form of an equation. It is the same transformation as that in

  T8764.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8810 \underline{Description}: Transformation formula ([1], (5.6.1); Appendix (III.38)) in form of an equation. It is the same transfor-

mation as that in T8810.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10901 \underline{Description}: Transformation formula ([1], (2.9.1); Appendix (III.28)) in form of an equation. It is the same transfor-

mation as that in T10901.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10902 \underline{Description}: Transformation formula ([1], Ex. 2.13(i), reversed) in form of an equation. It is the same transformation

as that in T10902.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10903 \underline{Description}: Transformation formula ([1], Ex. 2.19) in form of an equation. It is the same transformation as that in

  T10903.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10904 \underline{Description}: Transformation formula ([1], (3.10.4)) in form of an equation. It is the same transformation as that in

  T10904.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10905 \underline{Description}: Transformation formula ([1], Ex. 3.21(iii)) in form of an equation. It is the same transformation as that

  in T10905.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10906 \underline{Description}: Transformation formula ([3], (7.7), reversed) in form of an equation. It is the same transformation as

that in T10906.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

198 Alphabetic List: Tgl121103

Tgl10907 \underline{Description}: Transformation formula ([3], (7.8), reversed) in form of an equation. It is the same transformation as

that in T10907.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10961 \underline{Description}: Transformation formula ([1], (2.12.9); Appendix (III.39)) in form of an equation. It is the same transfor-

mation as that in T10961.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10962 \underline{Description}: Transformation formula ([1], (3.5.7)) in form of an equation. It is the same transformation as that in

  T10962.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl10963 \underline{Description}: Transformation formula ([1], Ex. 2.30) in form of an equation. It is the same transformation as that in

  T10963.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl101010 \underline{Description}: Transformation formula ([1], (5.6.3); Appendix (III.40)) in form of an equation. It is the same transfor-

mation as that in T101010.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121101 \underline{Description}: Transformation formula ([1], (2.8.3), reversed; Appendix (III.25), reversed) in form of an equation. It is

the same transformation as that in T121101.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121102 \underline{Description}: Transformation formula ([1], (2.8.4), reversed; Appendix (III.26), reversed) in form of an equation. It is

the same transformation as that in T121102.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121103 \underline{Description}: Transformation formula ([1], (2.8.5), reversed; Appendix (III.27), reversed) in form of an equation. It is

the same transformation as that in T121103.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tli2101 199

Tgl121104 \underline{Description}: Transformation formula ([1], Ex. 2.14(ii), reversed) in form of an equation. It is the same transformation

as that in T121104.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121105 \underline{Description}: Transformation formula ([3], (7.7)) in form of an equation. It is the same transformation as that in

  T121105.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121106 \underline{Description}: Transformation formula ([3], (7.8)) in form of an equation. It is the same transformation as that in

  T121106.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121107 \underline{Description}: Transformation formula ([1], Ex. 8.15, reversed) in form of an equation. It is the same transformation as

  that in T121107.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl121161 \underline{Description}: Transformation formula ([1], Ex. 2.25, reversed) in form of an equation. It is the same transformation as

  that in T121161.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tli2101 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.4.1);

Appendix (III.1)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2101.

\underline{Example(s)}:

In[1]:= ph[{a,b},{c},q,z]
             a, b
Out[1]= ϕ  ; q, z
        2 1   c
In[2]:= %/.Tli2101

                                 (a, b z; q)       c
                 a, b  ∞  -, z
Out[2]= {1, ϕ  ; q, z  , {{------------ ϕ a   ; q, a },
            2 1   c               (c, z; q)   2 1
                                           ∞  b z
                                   (b, a z; q)       c
                                              ∞  -, z
⟩  {phPerm[2, 1, u], T2101}}, {{------------ ϕ b   ; q, b }, T2101}}
                                    (c, z; q)   2 1
                                             ∞  a z

The first entry in this list counts the number of iterations of Tli-rules (cf. Tli2104), the second entry displays the original expression to which the Tli-rules are applied. The subsequent entries of the list always display an expression together with the sequence of rules that have to be applied to obtain this expression from the original expression. For

instance, the above list says that the number of iterations is 1, the original expression is _{2}ϕ_{1}[ \begin{matrix}a, b \\ c \end{matrix}; q, z], and (if the

parameters of the original series are permuted) by the application of T2101 two different expressions can be obtained from the original series. The first of them is obtained by first permuting the upper parameters by phPerm[2,1,u] and then applying T2101.

In[3]:= %1/.phPerm[2,1,u]/.T2101
        (a, b z; q)       c
                   ∞  -, z
Out[3]= ------------ ϕ a   ; q, a
         (c, z; q)   2 1
                  ∞  b z

For an example of how to iterate Tli-rules see Tli2104.

\underline{See also}: Tli2104, TransListe, Gleichung.

Tli2102

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.4.5); Appendix (III.2)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2102.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2103

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.4.6); Appendix (III.3)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2103.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli2104 201

Tli2104 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.5.4);

Appendix (III.4)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2104.

\underline{Example(s)}: Here we demonstrate the iterated application of Tli-rules. We continue the Tli2101-example. For information of how to read the resulting listing confer Tli2101.

In[4]:= %2/.Tli2104
                                  (a z; q)        c
                 a, b  ∞  -, a
Out[4]= {2, ϕ  ; q, z  , {{{--------- ϕ  b    ; q, b z }, T2101},
            2 1   c                (z; q)   2 2
                                         ∞  a z, c
                 (b z; q)        c
                         ∞  -, b
⟩  T2104}, {{{--------- ϕ  a    ; q, a z }, {phPerm[2, 1, u], T2101}},
                  (z; q)   2 2
                        ∞  b z, c
                 (a z, b z; q)          a b z
                              ∞  z, -----
⟩  T2104}, {{{-------------- ϕ  c  ; q, c },
                   (c, z; q)    2 2
                            ∞  b z, a z
⟩  {phPerm[2, 1, u], T2101}}, {phPerm[2, 1, u], T2104}},
        (b z, a z; q)          a b z
                     ∞  z, -----
⟩  {{{-------------- ϕ  c  ; q, c }, T2101},
          (c, z; q)    2 2
                   ∞  a z, b z
⟩  {phPerm[2, 1, u], T2104}}}
In[5]:= %1/.phPerm[2,1,u]/.T2101/.phPerm[2,1,u]/.T2104
        (a z, b z; q)          a b z
                     ∞  z, -----
Out[5]= -------------- ϕ  c  ; q, c
          (c, z; q)    2 2
                   ∞  b z, a z
\underline{See also}: Tli2101, TransListe, Gleichung.

202 Alphabetic List: Tli2111

Tli2105 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.4),

nonterminating; Appendix (III.5)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2105.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2106 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.5.6);

Appendix (III.6)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2106.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2107 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 1.15(iii); Appendix (III.7)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2107.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2108 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 1.15(ii); Appendix (III.8)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2108.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2109 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 2.2,

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2109.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2110 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.7))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T2110.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2111 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.5.4))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T2111.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli3101 203

Tli2112 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.2(i), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2112.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2161 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.3.5);

Appendix (III.31)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2161.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2162 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Appendix (III.32)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2162.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2163 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.8)

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T2163.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2201 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.5.4),

reversed; Appendix (III.4), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2201.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli2202 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.2(ii), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T2202.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3101 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 1.15(ii), reversed; Appendix (III.8), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3101.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

204 Alphabetic List: Tli3207

Tli3201 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.4),

reversed; Appendix (III.5), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3201.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3202 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (1.5.6),

reversed; Appendix (III.6), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3202.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3203 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 1.15(iii), reversed; Appendix (III.7), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3203.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3204 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.7);

Appendix (III.9)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3204.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3205 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.2.10); Appendix (III.10)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3205.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3206 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.3);

Appendix (III.11)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3206.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3207 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.2);

Appendix (III.12)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3207.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli3214 205

Tli3208 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.5);

Appendix (III.13)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3208.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3209 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.26; Appendix (III.14)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3209.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3210 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.6))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T3210.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3211 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.1)

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T3211.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3212 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.1,

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3212.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3213 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.2(i)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3213.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3214 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.2(ii)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3214.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

206 Alphabetic List: Tli3266

Tli3215 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.3)

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T3215.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3216 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.3,

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3216.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3217 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.2.11)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3217.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3261 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.3.3);

Appendix (III.33)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3261.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3262 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.3.1);

Appendix (III.34)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3262.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3265 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.1);

Appendix (III.35)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3265.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3266 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.5.2))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T3266.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli4303 207

Tli3267 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.6)

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T3267.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3268 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.6,

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3268.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli3269 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.8,

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T3269.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4201 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.4,

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4201.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4301 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.4); Appendix (III.15)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4301.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4302 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.2.9);

Appendix (III.16)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4302.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4303 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.5.1), reversed; Appendix (III.19)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4303.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

208 Alphabetic List: Tli4310

Tli4304 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.7); Appendix (III.20)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4304.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4305 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.10.13); Appendix (III.21)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4305.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4306 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.10.13), reversed; Appendix (III.21), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4305.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4307 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (8.8.3);

Appendix (III.22)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4306.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4308 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 2.2)

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T4308.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4309 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.13(i)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4309.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4310 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.13(ii)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4310.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli5403 209

Tli4311 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.8))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T4311.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4312 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 3.4)

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T4312.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4313 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 8.15) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4313.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli4361 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.16) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T4361.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5401 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.8.3);

Appendix (III.25)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5401.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5402 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.8.4);

Appendix (III.26)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5402.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5403 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.10.4), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5403.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

210 Alphabetic List: Tli5465

Tli5404 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.26, Appendix (III.14), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5404.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5405 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(8.8.3), Appendix (III.22), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5405.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5461 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.4))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T5461.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5462 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.4),

reversed, first form) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5462.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5463 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.4.4), reversed, second form) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5463.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5464 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.5.2),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5464.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5465 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.5.7),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5465.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli8701 211

Tli5466 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.16, reversed, first form) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5466.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5467 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.16, reversed, second form) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5467.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli5468 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.25) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T5468.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli6501 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.14(ii)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T6501.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli7601 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.8.5);

Appendix (III.27)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T7601.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli7701 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.2.11), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T7701.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8701 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.10), terminated; Appendix (III.17)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8701.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

212 Alphabetic List: Tli8708

Tli8702 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.5.1);

Appendix (III.18)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8702.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8703 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.7), reversed; Appendix (III.20), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8703.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8704 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.1); Appendix (III.23)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8704.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8705 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.1), iterated; Appendix (III.24)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8705.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8706 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.13(ii), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8706.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8707 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.7),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8707.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8708 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.4.8),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8708.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli8810 213

Tli8709 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.5.4),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8709.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8710 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.5.10)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8710.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8711 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.5.10), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8711.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8761 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.10.10); Appendix (III.36)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8761.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8762 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.11.1); Appendix (III.37)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8762.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8764 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.15) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8764.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli8810 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (5.6.1);

Appendix (III.38)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T8810.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

214 Alphabetic List: Tli10907

Tli10901 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.9.1);

Appendix (III.28)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10901.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10902 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.13(i), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10902.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10903 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.19) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10903.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10904 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(3.10.4)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10904.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10905 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 3.21(iii)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10905.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10906 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([3], (7.7),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10906.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10907 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([3], (7.8),

reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10907.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli121103 215

Tli10961 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

(2.12.9); Appendix (III.39)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10961.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10962 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (3.5.7))

together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T10962.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli10963 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 2.30) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T10963.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli101010 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (5.6.3);

Appendix (III.40)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T101010.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121101 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.8.3),

reversed; Appendix (III.25), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T121101.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121102 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.8.4),

reversed; Appendix (III.26), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T121102.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121103 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], (2.8.5),

reversed; Appendix (III.27), reversed). of an equation together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T121103.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121104

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 2.14(ii), reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T121104.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121105

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([3], (7.7)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T121105.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121106

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([3], (7.8)) together with information of how to get them. Also iterated use is possible. The transformation is the same as that

  in T121106.
\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121107

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 8.15, reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T121107.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

Tli121161

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 2.25, reversed) together with information of how to get them. Also iterated use is possible. The transformation is the same as that in T121161.

\underline{See also}: Tli2101, Tli2104, TransListe, Gleichung.

TListe

\underline{Description}: Rule that gives for a basic hypergeometric series a list of applicable transformation formulas. Each entry

  of this list has the format {ArgumentPermutations,T⟨number⟩}, where ArgumentPermutations is a sequence of

reorderings of the parameters of the basic hypergeometric series (given in terms of phPerm and phTausche) and T⟨number⟩ is the name of the transformation in form of a rule which can be applied subsequently. You should be aware that TListe automatically applies phOrdne before checking which transformation could be applied.

\underline{Important Note}: If the value returned by TListe is the empty set this does 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 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.

\underline{Usage}: Expr/.TListe.

\underline{Example(s)}:

             a, b
Out[1]= ϕ  ; q, z
        2 1   c

In[2]:= %/.TListe
Be sure to apply "phOrdne" before using the following information!
Out[2]= {{T2101}, {T2102}, {T2103}, {T2104}, {T2161}, {T2162}}
In[3]:= ph[{d,b^{∧}2,q^{∧}-n,a^{∧}2},{-a*b*Sqrt[q],a*b*Sqrt[q],-q^{∧}-n*d},q,q]
                           2   -n   2
                       d, b , q  , a
Out[3]= ϕ  d  ; q, q
        4 3  -(a b Sqrt[q]), a b Sqrt[q], -(--)
                                             n
                                            q
In[4]:= %/.TListe
Is n a nonnegative integer?
[y|n]:    y
Be sure to apply "phOrdne" before using the following information!
Out[4]= {{T4301}, {T4302}, {T4303}, {T4304},
⟩  {phPerm[2,3,1,4,u], phPerm[3,1,2,l], T4305}}
In[5]:= %3/.phOrdne/.phPerm[2,3,1,4,u]/.phPerm[3,1,2,l]/.T4305
Is n a nonnegative integer?
[y|n]:    y
                   2   2   2   -2 n
                  b , a , d , q
                                           2   2
Out[5]= ϕ  2  2      d         1 - n ; q , q
        4 3  a  b  q, -(--), -(d q     )
                         n
                        q

Now we consider two examples illustrating the note above. Though none of the implemented transformations can be applied, both series can be transformed, the first by a limiting case of Watson’s transformation, the second by a number of specialized very well-poised _{8}ϕ_{7} transformations. These facts are also observed by using this package.

In[6]:= ph[{a,q*Sqrt[a],-q*Sqrt[a],b,c,d,e},{Sqrt[a],-Sqrt[a],
        a*q/b,a*q/c,a*q/d,a*q/e,0},q,a^{∧}2*q^{∧}2/(b*c*d*e)]
              a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e        2  2
                                                           a  q
Out[6]= ϕ  a q  a q  a q  a q   ; q, -------
        7 7  Sqrt[a], -Sqrt[a], ---, ---, ---, ---, 0     b c d e
                                 b    c    d    e
In[7]:= %/.TListe
Out[7]= {}
In[8]:= Tgl8702
Do you want to set values for the equation?  [y|n]:  n
Do you want to set a value for q in the equation?  [y|n]:  n
                                                        -n        2  2 + n
               a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e, q         a  q
Out[8]= ϕ  ; q, ---------
        8 7                     a q  a q  a q  a q     1 + n      b c d e
             Sqrt[a], -Sqrt[a], ---, ---, ---, ---, a q
                                 b    c    d    e
                 a q         -n               a q
                 ---, d, e, q           (a q, ---; q)
                 b c                          d e    n
⟩  == ϕ  ; q, q   --------------
            4 3  a q  a q  d e           a q  a q
                 ---, ---, ----         (---, ---; q)
                  b    c      n           d    e     n
                           a q
In[9]:= Limes[%,n-⟩Infinity]
              a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e        2  2
                                                           a  q
Out[9]= ϕ  a q  a q  a q  a q   ; q, -------   ==
        7 7  Sqrt[a], -Sqrt[a], ---, ---, ---, ---, 0     b c d e
                                 b    c    d    e

                            a q
                            ---, d, e
               a q          b c           a q
     (a q;q)  (---;q) ϕ  ; q, ---
            ∞ d e ∞ 3 2  a q  a q      d e
                            ---, ---
                             b    c
⟩  ------------------------------------------
                  a q      a q
                 (---;q)  (---;q)
                   d ∞ e ∞
In[10]:= ph[{q*Sqrt[a],-q*Sqrt[a],q,c,d,e,f},{Sqrt[a],-Sqrt[a],a*q/c,
         a*q/d,a*q/e,a*q/f},q,a^{∧}2*q^{∧}2/(q*c*d*e*f)]
              Sqrt[a] q, -(Sqrt[a] q), q, c, d, e, f       2
                                                          a  q
Out[10]= ϕ  a q  a q  a q  a q ; q, -------
         7 6  Sqrt[a], -Sqrt[a], ---, ---, ---, ---      c d e f
                                  c    d    e    f
In[11]:= %/.TListe
Out[11]= {}
In[12]:= %%/.phEinf
Add the parameter:  a
              a, Sqrt[a] q, -(Sqrt[a] q), q, c, d, e, f       2
                                                             a  q
Out[12]= ϕ  a q  a q  a q  a q ; q, -------
         8 7  a, Sqrt[a], -Sqrt[a], ---, ---, ---, ---      c d e f
                                     c    d    e    f
In[13]:= %/.TListe
Be sure to apply "phOrdne" before using the following information!
Out[13]= {{T8701}, {T8704}, {T8705}, {T8761}, {T8762}}

\underline{See also}: SListe, phPerm, phTausche, TransListe.

trans

\underline{Description}: (a; q)_{n} → (q^{1−n}/a; q)_{n}(−a)^{n}q^{\binom{n}{2}}.

\underline{Usage}: Expr/.trans.

220 Alphabetic List: W

\underline{Example(s)}:

In[1]:= pq[a,n,q^{∧}2]
             2
Out[1]= (a; q )
               n
In[2]:= %/.trans
                        2   2 - 2 n
            n  n  -n + n   q          2
Out[2]= (-1)  a  q        (--------; q )
                              a         n
\underline{See also}: Ers, PosListe, ManipulationsListe.

TransListe \underline{Description}: List of all transformation formulas.

\underline{Usage}: TransListe.
\underline{See also}: TransListe$gl, Liste.

TransListe$gl \underline{Description}: List of all transformation formulas.

\underline{Usage}: TransListe$gl.
\underline{See also}: TransListe.

W \underline{Description}: W[a,List,q,z] is the very well-poised basic hypergeometric series.

\underline{Usage}: W[a,List,q,z].

\underline{Example(s)}:

In[1]:= W[a,{b,c},q,z]
             a, Sqrt[a] q, -(Sqrt[a] q), b, c
Out[1]= ϕ  a q  a q   ; q, z
        5 4    Sqrt[a], -Sqrt[a], ---, ---
                                   b    c
\underline{See also}: ph, TeXphW, PQ, phFormat.

Alphabetic List: zus1 221

zerl

\underline{Description}: \begin{matrix}(a; q)_{n} → (a; q)_{m}(aq^{m}; q)_{n−m}, \\ (a; q)_{∞} → (a; q)_{m}(aq^{m}; q)_{∞}. \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.zerl.

\underline{Example(s)}:

In[1]:= pq[a,2*n]
Out[1]= (a; q)
              2 n
In[2]:= %/.zerl
bottom-split by:  m+n
                        m + n
Out[2]= (a; q)      (a q     ; q)
              m + n              -m + n
\underline{See also}: Ers, PosListe, ManipulationsListe.

zus1

\underline{Description}: \begin{matrix}(a; q)_{n}(aq^{n}; q)_{m} → (a; q)_{n+m}, \\ (a; q)_{n}(aq^{n}; q)_{∞} → (a; q)_{∞}. \end{matrix}

\underline{Usage}: Expr/.zus1.

\underline{Example(s)}:

In[1]:= pq[a,2*n]*pq[a*q^{∧}(2*n),m-n]
                      2 n
Out[1]= (a; q)    (a q   ; q)
              2 n            m - n
In[2]:= %/.zus1
Out[2]= (a; q)
              m + n
In[3]:= pq[a,2*n]*pqinf[a*q^{∧}(2*n)]
            2 n
Out[3]= (a q   ;q)  (a; q)
                  ∞  2 n
In[4]:= %/.zus1

Out[4]= (a;q)

∞

\underline{See also}: zus2, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

zus2

\underline{Description}: \begin{matrix}(a; q)_{n}/(a; q)_{m} → (aq^{m}; q)_{n−m}, \\ (a; q)_{∞}/(a; q)_{m} → (aq^{m}; q)_{∞}. \end{matrix}

\underline{Usage}: Expr/.zus2.

\underline{Example(s)}:

In[1]:= pq[a,m]/pq[a,n]*pq[b,m+n]
        (a; q)  (b; q)
              m       m + n
Out[1]= -------------------
              (a; q)
                    n
In[2]:= %/.zus2
                        n
Out[2]= (b; q)      (a q ; q)
              m + n          m - n
In[3]:= pq[a,m]/pqinf[a]*pq[b,m+n]
        (a; q)  (b; q)
              m       m + n
Out[3]= -------------------
              (a;q)

∞

In[4]:= %/.zus2
        (b; q)
              m + n
Out[4]= -----------
             m
         (a q ;q)

∞

\underline{See also}: zus1, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

zus3

\underline{Description}: (a; q)_{n}/(b; q)_{m} → (a; q)_{n−m},

provided aq^{n} = bq^{m}, and

(a; q)_{∞}/(aq^{n}; q)_{∞} → (a; q)_{n}.

\underline{Usage}: Expr/.zus3.

\underline{Example(s)}:

In[1]:= pq[a*q^{∧}m,n]/pq[a*q^{∧}n,m]
           m
       (a q ; q)
                n
Out[1]= ----------
           n
       (a q ; q)
                m
In[2]:= %/.zus3
           m
Out[2]= (a q ; q)
                -m + n
In[3]:= pqinf[a*q^{∧}n]/pqinf[a*q]
           n
       (a q ;q)

∞

Out[3]= ---------
       (a q;q)

∞

In[4]:= %/.zus3
             1
Out[4]= --------------
       (a q; q)
               -1 + n
\underline{See also}: zus1, zus2, erw1, erw2, Ers, PosListe, ManipulationsListe.

References

1. G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990.

2. C. Krattenthaler, HYP and HYPQ — Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series.

3. M. Rahman and A. Verma, Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335 (1993), 277–302.

4. S. Wolfram, MATHEMATICA — A system for doing mathematics by computer, second edition, Addison–Wesley, New York, 1991.