%\input amstex %\documentstyle{amspptbk} %\input sekr \def\poq#1#2{(#1;q)_#2} \def\prodl{\prod\limits} \def\la{\lambda} \def\[{\left[} \def\]{\right]} \def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\smc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} \def\AmS{{\textfontii A}\kern-.1667em\lower.5ex\hbox {\textfontii M}\kern-.125em{\textfontii S}} \def\WolfAA{4} \def\RaVeAA{3} \def\KratAT{2} \def\GaRaAA{1} \magnification1000 \catcode`\@=11 \font@\twelverm=cmr10 scaled\magstep1 \font@\twelveit=cmti10 scaled\magstep1 \font@\twelvebf=cmbx10 scaled\magstep1 \font@\twelvei=cmmi10 scaled\magstep1 \font@\twelvesy=cmsy10 scaled\magstep1 \font@\twelveex=cmex10 scaled\magstep1 \newtoks\twelvepoint@ \def\twelvepoint{\normalbaselineskip15\p@ \abovedisplayskip15\p@ plus3.6\p@ minus10.8\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus3.6\p@ \belowdisplayshortskip8.4\p@ plus3.6\p@ minus4.8\p@ \textonlyfont@\rm\twelverm \textonlyfont@\it\twelveit \textonlyfont@\sl\twelvesl 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\def\tab{\bgroup\tab@true\vtab@false\vst@bfalse\Strich@false% \def\\{\global\hline@@false% \ifhline@\global\hline@false\global\hline@@true\fi\cr} \edef\l@{\the\leftskip}\ialign\bgroup\hskip\l@##\hfil&&##\hfil\cr} \def\endtab{\cr\egroup\egroup} \def\vtab{\vtop\bgroup\vst@bfalse\vtab@true\tab@true\Strich@false% \bgroup\def\\{\cr}\ialign\bgroup&##\hfil\cr} \def\endvtab{\cr\egroup\egroup\egroup} \def\stab{\D@cke0.5pt\null \bgroup\tab@true\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@ \edef\l@{\the\leftskip}\ialign \bgroup\hskip\l@##\hfil&&##\hfil\crcr} \def\endstab{\crcr\egroup \egroup} \newif\ifvst@b\vst@bfalse \def\vstab{\D@cke0.5pt\null \vtop\bgroup\tab@true\vtab@false\vst@btrue\Strich@true\bgroup\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@\bgroup} \def\endvstab{\crcr\egroup\egroup \egroup\tab@false\Strich@false} \newdimen\htstrut@ \htstrut@8.5\p@ \newdimen\htStrut@ \htStrut@12\p@ \newdimen\dpstrut@ \dpstrut@3.5\p@ \newdimen\dpStrut@ \dpStrut@3.5\p@ \def\openup{\afterassignment\@penup\dimen@=} \def\@penup{\advance\lineskip\dimen@ \advance\baselineskip\dimen@ \advance\lineskiplimit\dimen@ \divide\dimen@ by2 \advance\htstrut@\dimen@ \advance\htStrut@\dimen@ \advance\dpstrut@\dimen@ \advance\dpStrut@\dimen@} \def\Let@@{\relax\iffalse{\fi% \def\\{\global\hline@@false% \ifhline@\global\hline@false\global\hline@@true\fi\cr}% \iffalse}\fi} \def\matrix{\null\,\vcenter\bgroup \tab@false\vtab@false\vst@bfalse\Strich@false\Let@@\vspace@ \normalbaselines\openup\spreadmlines@\ialign \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr \Mathstrut@\crcr\noalign{\kern-\baselineskip}} \def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup \egroup\,} \def\smatrix{\D@cke0.5pt\null\, \vcenter\bgroup\tab@false\vtab@false\vst@bfalse\Strich@true\Let@@\vspace@ \normalbaselines\offinterlineskip \openup\spreadmlines@\ialign \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr} \def\endsmatrix{\crcr\egroup \egroup\,\Strich@false} \newdimen\D@cke \def\Dicke#1{\global\D@cke#1} \newtoks\tabs@\tabs@{&} \newif\ifStrich@\Strich@false \newif\iff@rst \def\Stricherr@{\iftab@\ifvtab@\errmessage{\noexpand\s not allowed here. Use \noexpand\vstab!}% \else\errmessage{\noexpand\s not allowed here. Use \noexpand\stab!}% \fi\else\errmessage{\noexpand\s not allowed here. Use \noexpand\smatrix!}\fi} \def\format{\ifvst@b\else\crcr\fi\egroup\iffalse{\fi\ifnum`}=0 \fi\format@} \def\format@#1\\{\def\preamble@{#1}% \def\Str@chfehlt##1{\ifx##1\s\Stricherr@\fi\ifx##1\\\let\Next\relax% \else\let\Next\Str@chfehlt\fi\Next}% \def\c{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}% \def\r{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi}% \def\l{\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi\iftab@\else$\m@th\fi\the\hashtoks@\iftab@\else$\fi\hfil}% \def\s{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@% \fi\the\tabs@\ }% \def\sa{\ifStrich@\vrule width\D@cke\the\hashtoks@% \the\tabs@\ % \fi}% \def\se{\ifStrich@\ \the\tabs@\vrule width\D@cke\the\hashtoks@\fi}% \def\cd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$\hfil}% \def\rd{\hfil\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$}% \def\ld{\noexpand\ifhline@@\hbox{\vrule height\htStrut@% depth\dpstrut@ width\z@}\noexpand\fi% \ifStrich@\hbox{\vrule height\htstrut@ depth\dpstrut@ width\z@}% \fi$\dsize\m@th\the\hashtoks@$\hfil}% \ifStrich@\else\Str@chfehlt#1\\\fi% \setbox\z@\hbox{\xdef\Preamble@{\preamble@}}\ifnum`{=0 \fi\iffalse}\fi \ialign\bgroup\span\Preamble@\crcr} \newif\ifhline@\hline@false \newif\ifhline@@\hline@@false \def\hlinefor#1{\multispan@{\strip@#1 }\leaders\hrule height\D@cke\hfill% \global\hline@true\ignorespaces} \def\Item "#1"{\par\noindent\hangindent2\parindent% \hangafter1\setbox0\hbox{\rm#1\enspace}\ifdim\wd0>2\parindent% \box0\else\hbox to 2\parindent{\rm#1\hfil}\fi\ignorespaces} \def\ITEM #1"#2"{\par\noindent\hangafter1\hangindent#1% \setbox0\hbox{\rm#2\enspace}\ifdim\wd0>#1% \box0\else\hbox to 0pt{\rm#2\hss}\hskip#1\fi\ignorespaces} \def\item"#1"{\par\noindent\hang% \setbox0=\hbox{\rm#1\enspace}\ifdim\wd0>\the\parindent% \box0\else\hbox to \parindent{\rm#1\hfil}\enspace\fi\ignorespaces} \let\plainitem@\item \catcode`\@=13 \catcode`\@=11 \def\leftheadline{\rlap{\folio}\hfill \iftrue\botmark\fi \hfill} \runheads@true \catcode`\@=13 \hsize18cm \vsize22.8cm \hoffset-1cm \def\Ai{(A)} \def\Bi{(B)} \def\qAi{(qA)} \def\qBi{(qB)} \def\Aiq{(A/q)} \def\Biq{(B/q)} \def\ai{A_i} \def\bi{B_i} {\obeylines \gdef\MATH{\begingroup\parindent0pt\parskip0pt plus 0.25pt\obeylines% \def^^M{\penalty10000\par\penalty10000% \vrule height10pt depth2pt width0pt\leavevmode}% \obeyspaces\tt}% } \def\goodbreakpoint{\par\penalty-5000% \vrule height10pt depth2pt width0pt\leavevmode} \def\endMATH{\endgroup} \def\MATHphi{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\phi$\hss}\hss}} \def\MATHGamma{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\Gamma$\hss}\hss}} \def\MATHpi{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\pi$\hss}\hss}} \def\MATHinfty{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\infty$\hss}\hss}} \def\MATHhStrich{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\vrule height4.5pt depth-3.5pt width5.24995pt}\hss}} \def\MATHluEck{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hskip2.12497pt \vrule height4.5pt depth1pt width1pt \vrule height4.5pt depth-3.5pt width2.12498pt}\hss}} \def\MATHruEck{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{% \vrule height4.5pt depth-3.5pt width2.12497pt \vrule height4.5pt depth1pt width1pt \hskip2.12498pt}\hss}} \def\MATHloEck{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hskip2.12497pt \vrule height9pt depth-3.5pt width1pt \vrule height4.5pt depth-3.5pt width2.12498pt}\hss}} \def\MATHroEck{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{% \vrule height4.5pt depth-3.5pt width2.12497pt \vrule height9pt depth-3.5pt width1pt \hskip2.12498pt}\hss}} \def\MATHvStrich{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hskip2.12497pt \vtop to 0pt{\hsize1pt\vss% \vrule height17pt depth6pt width1pt\vskip8pt\vss\par}% \hskip2.12498pt}\hss}} \def\MATHtStueck{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{% \vrule height4.5pt depth-3.5pt width2.12497pt \vrule height4.5pt depth2pt width1pt \vrule height4.5pt depth-3.5pt width2.12498pt}\hss}} \def\MATHbackslash{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\backslash$\hss}\hss}} \def\MATHlbrace{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\{$\hss}\hss}} \def\MATHrbrace{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\}$\hss}\hss}} \def\MATHkleiner{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\langle$\hss}\hss}} \def\MATHgroesser{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$\rangle$\hss}\hss}} \def\MATHhoch{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss$^\land$\hss}\hss}} \def\MATHtief{\leavevmode \hbox to 0pt{\hbox to 5.24995pt{\hss\vrule height0pt depth.8pt width3pt\hss}\hss}} \def\Name#1\Description{\par\nopagebreak\vskip6pt\nopagebreak \hrule height1pt depth0pt width18cm \penalty-5000\bigskip\penalty-5000 \rightheadtext\nofrills{\smc Alphabetic List: \tenpoint\bf#1}% \leftheadtext\nofrills{\smc Alphabetic List: \tenpoint\bf#1}% \bf#1\Description} \def\Description{\nopagebreak\vskip6pt\nopagebreak \rm\hangafter1 \hangindent10pt \underbar{Description}: } \def\Usage{\vskip6pt \hangafter1 \hangindent10pt \rm \underbar{Usage}: \tt} \def\Example{\vskip6pt \rm\hangafter1 \hangindent10pt \underbar{Example(s)}: \nopagebreak\vskip6pt\nopagebreak} \def\Seealso{\nopagebreak\vskip6pt\nopagebreak \rm\hangafter1 \hangindent10pt \underbar{See also}: \tt} \topmatter \title \fourteenpoint\bf HYPQ \endtitle \author C.~Krattenthaler \endauthor \affil Institut f\"ur Mathematik der Universit\"at Wien,\\ Strudlhofgasse 4, A-1090 Wien, Austria.\\ e-mail: KRATT\@AP.UNIVIE.AC.AT\\ WWW: \tt http://www.mat.univie.ac.at/People/kratt \endaffil %\thanks{}\endthanks \endtopmatter \leftheadtext{The MATHEMATICA package \tenpoint\bf HYPQ} \rightheadtext{The MATHEMATICA package \tenpoint\bf HYPQ} \document This is a MATHEMATICA package for handling basic hypergeometric series. It provides quite a few tools for \roster \item "(A)" manipulating $q$-factorial expressions \item "(B)" transforming $q$-binomial sums into basic hypergeometric notation \item "(C)" summing basic hypergeometric series \item "(D)" transforming basic hypergeometric series \item "(E)" applying contiguous relations \item "(F)" doing formal limits of basic hypergeometric expressions \item "(G)" transforming basic hypergeometric MATHEMATICA expressions into \TeX-code. \endroster The tools for items (A), (B), (F), (G) are contained in the file \hbox{\tt hyp.q}, the basic package. This file must be loaded at the very beginning of your MATHEMATICA session. (Ignore error messages occuring when loading \hbox{\tt hyp.q}.) The file \hbox{\tt hyp.q} defines the basic objects, the rules and functions for items (A), (B), (F), and (G), and predefines all the remaining ones. The tools for (C) are the contents of the file \hbox{\tt summatio.q}, those for (D) are the contents of the files \hbox{\tt transfor.q} and \hbox{\tt transfor.qli}, and those for (E) are the contents of the file \hbox{\tt contig.q}. You also have access to summation and transformation formulas in form of equations. This is the contents of the files \hbox{\tt summatio.qgl} and \hbox{\tt transfor.qgl}, respectively. The file \hbox{\tt output} defines some nice MATHEMATICA output features for \hbox{\tt SUM}, \hbox{\tt Product}, \hbox{\tt Integrate}, \hbox{\tt Abs}, \hbox{\tt Floor}, \hbox{\tt Ceiling}, \hbox{\tt Pi}, and \hbox{\tt Infinity}. However, the philosophy of this package is: \bigskip \centerline{\twelvepoint\it Do it by yourself!} \bigskip The idea is that you should be able to control each step in a series of manipulations by yourself. So, for instance, this package does not make any attempt to sum or transform a series automatically. So, it is you who has to tell the package which command, summation, or transformation has to be applied next. Therefore a basic knowledge of basic hypergeometric series is required (cf\. \cite{\GaRaAA, pp.~1--6}). This handbook provides you with a list of the rules, functions, summations, transformations that are available. The main source for identities that are included in this package has been the book \cite{\GaRaAA}, which contains a fairly comprehensive collection of known summation and transformation formulas for basic hypergeometric series. In particular, the (almost) complete Appendix of \cite{\GaRaAA} is included in this package. Finally you should be warned that there is no guarantee that a formula that has been obtained using this package is actually valid. Many formulas or operations are only valid under certain restrictions for the parameters. This package only helps you to do calculations fast. It is up to you to check that the manipulations you are doing are actually being allowed. For a brief summary of the main features of this package the user is referred to \cite{\KratAT} which is the contents of the \AmSTeX\ file \hbox{\tt hyp$_{\text {--}}$hypq.tex}. \head Basic hypergeometric notation\endhead All the notation and terminology is adopted from \cite{\GaRaAA, pp.~1--6}. Given a (fixed) complex number $q$ with $|q|<1$, the {\it basic hypergeometric series\/} is defined by $${}_r\phi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q, z\right]=\sum _{n=0} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}} {\poq{q}{n}\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom n2}\right)^{s-r+1}z^n\ ,$$ where the rising $q$-factorial $(a;q)_n$ is given by $(a;q)_n:=(1-a)(1-aq)\cdots(1-aq^{n-1})$, $n\ge1$, $(a)_0:=1$. Also the infinite $q$-factorial $(a;q)_\infty:=\prod _{i=0} ^{\infty}(1-aq^{i})$ is used. A basic hypergeometric series $_{r+1}\phi_r$ is called {\it very well-poised\/} if $a_ib_i=qa_0$ for $i=1,2,\dots,r$, and among the parameters $a_i$ occur both $q\sqrt{a_0}$ and $-q\sqrt{a_0}$. We use the standard abbreviation for very well-poised basic hypergeometric series, $$_{r+1}W_r(a_0;a_3,a_4,\dots,a_r;z):={}_{r+1}\phi_r\!\[\matrix a_0,q\sqrt{a_0},-q\sqrt{a_0},a_3,a_4,\dots,a_r\\ \sqrt{a_0},-\sqrt{a_0},qa_0/a_2,qa_0/a_3,\dots,qa_0/a_r\endmatrix; z\].$$ A convenient notation for {\it basic hypergeometric series\/} with several bases is $$\multline {}_r\Phi_s\!\left[\matrix a^{(1)}_1,\dots,a^{(1)}_{r_1}; \dots ;a^{(p)}_1,\dots,a^{(p)}_{r_p}\\ b^{(1)}_1,\dots,b^{(1)}_{s_1};\dots; b^{(p)}_1,\dots,b^{(p)}_{s_p}\endmatrix; q_1,\dots,q_p; z\right]\\ =\sum _{n=0} ^{\infty}\frac {z^n} {(q_1;q_1)_n}(-1)^nq_1^{\binom n2} \prod _{i=1} ^{p} \frac {(a^{(i)}_1;q_i)_n\cdots(a^{(i)}_{r_i};q_i)_n} {(b^{(i)}_1;q_i)_n\cdots(b^{(i)}_{s_i};q_i)_n} \left((-1)^nq^{\binom n2}\right)^{s_i-r_i}\ . \endmultline$$ The {\it bilateral basic hypergeometric series\/} is defined by $${}_r\psi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q, z\right]=\sum _{n=-\infty} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}} {\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom n2}\right)^{s-r+1}z^n\ ,$$ We also use the compact Gasper-Rahman notation $$ {(a_1,a_2,\dots,a_r;q)_n} :=(a_1;q)_n\,(a_2;q)_n\,\cdots\,(a_r;q)_n.$$ \head The file \tt hyp.q\endhead The objects which are defined in the file hyp.q are \medskip {\leftskip20pt \rightskip20pt \noindent \hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller}, \hbox{\tt AbsUndetermined}, \hbox{\tt Add}, \hbox{\tt AmSLaTeX}, \hbox{\tt AmSTeX}, \hbox{\tt baszerl1}, \hbox{\tt baszerl2}, \hbox{\tt baszus1},\linebreak \hbox{\tt baszus2}, \hbox{\tt Binomialpq}, \hbox{\tt Binomialq}, \hbox{\tt Div}, \hbox{\tt Drucke}, \hbox{\tt Ers}, \hbox{\tt erw1}, \hbox{\tt erw2}, \hbox{\tt Expandq}, \hbox{\tt Factorialpq}, \hbox{\tt Factorialq}, \hbox{\tt Gleichung}, \hbox{\tt GlTausche}, \hbox{\tt Hoch}, \hbox{\tt hypqAttributes}, \hbox{\tt inv1}, \hbox{\tt inv2}, \hbox{\tt LaTeX}, \hbox{\tt Limes}, \hbox{\tt lina1}, \hbox{\tt lina2}, \hbox{\tt linz}, \hbox{\tt LS}, \hbox{\tt Mal}, \hbox{\tt ManipulationsListe}, \hbox{\tt MinusOne}, \hbox{\tt Multinomialpq}, \hbox{\tt Multinomialq}, \hbox{\tt neg1}, \hbox{\tt neg2}, \hbox{\tt ph}, \hbox{\tt Ph}, \hbox{\tt phCancel}, \hbox{\tt PhEinf}, \hbox{\tt phEinf}, \hbox{\tt phFormat}, \hbox{\tt Phinv}, \hbox{\tt phinv}, \hbox{\tt PhOrdne}, \hbox{\tt phOrdne}, \hbox{\tt PhPerm}, \hbox{\tt phPerm}, \hbox{\tt Phph}, \hbox{\tt phPh}, \hbox{\tt phps}, \hbox{\tt PhSUM}, \hbox{\tt phSUM}, \hbox{\tt phTausche}, \hbox{\tt PosListe}, \hbox{\tt pq}, \hbox{\tt PQ}, \hbox{\tt pqaufl}, \hbox{\tt pqinf}, \hbox{\tt pqinfzerl}, \hbox{\tt pqinfzus}, \hbox{\tt PQSort}, \hbox{\tt pqzerl}, \hbox{\tt pqzus}, \hbox{\tt ps}, \hbox{\tt psEinf}, \hbox{\tt psinv}, \hbox{\tt psOrdne}, \hbox{\tt psPerm}, \hbox{\tt psph}, \hbox{\tt psShift}, \hbox{\tt psSUM}, \hbox{\tt RS}, \hbox{\tt SchreibeZahl}, \hbox{\tt SimplifyPQ}, \hbox{\tt Sub}, \hbox{\tt Subst}, \hbox{\tt SUM}, \hbox{\tt SUMErw1}, \hbox{\tt SUMErw2}, \hbox{\tt SUMExpand}, \hbox{\tt SUMInfinity}, \hbox{\tt SUMph}, \hbox{\tt SUMPh}, \hbox{\tt SUMps}, \hbox{\tt SUMRegeln}, \hbox{\tt SUMSammle}, \hbox{\tt SUMShift}, \hbox{\tt SUMTausche}, \hbox{\tt SUMUmkehr}, \hbox{\tt SUMZerl}, \hbox{\tt TeX}, \hbox{\tt TeXMat}, \hbox{\tt TeXphW}, \hbox{\tt trans}, \hbox{\tt W}, \hbox{\tt zerl}, \hbox{\tt zus1}, \hbox{\tt zus2}, \hbox{\tt zus3}. \par} \medskip\noindent These objects can be divided into 9 groups: There are the basic objects, \medskip\noindent {\leftskip20pt\rightskip20pt\noindent \hbox{\tt Binomialpq}, \hbox{\tt Binomialq}, \hbox{\tt Factorialpq}, \hbox{\tt Factorialq}, \hbox{\tt Multinomialpq}, \hbox{\tt Multinomialq}, \hbox{\tt ph}, \hbox{\tt Ph}, \hbox{\tt pq}, \hbox{\tt pqinf}, \hbox{\tt ps}, \hbox{\tt SUM}, \hbox{\tt W},\par} \medskip\noindent the rules for manipulating $q$-factorial expressions \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt baszerl1}, \hbox{\tt baszerl2}, \hbox{\tt baszus1}, \hbox{\tt baszus2}, \hbox{\tt erw1}, \hbox{\tt erw2}, \hbox{\tt Expandq}, \hbox{\tt inv1}, \hbox{\tt inv2}, \hbox{\tt lina1}, \hbox{\tt lina2}, \hbox{\tt linz}, \hbox{\tt MinusOne}, \hbox{\tt neg1}, \hbox{\tt neg2}, \hbox{\tt pqaufl}, \hbox{\tt pqinfzerl}, \hbox{\tt pqinfzus}, \hbox{\tt pqzerl}, \hbox{\tt pqzus}, \hbox{\tt trans}, \hbox{\tt zerl}, \hbox{\tt zus1}, \hbox{\tt zus2}, \hbox{\tt zus3}, \par} \medskip\noindent the rules for manipulating sums and basic hypergeometric series, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt phCancel}, \hbox{\tt phEinf}, \hbox{\tt PhEinf}, \hbox{\tt phFormat}, \hbox{\tt Phinv}, \hbox{\tt phinv}, \hbox{\tt PhOrdne}, \hbox{\tt phOrdne}, \hbox{\tt PhPerm}, \hbox{\tt phPerm}, \hbox{\tt Phph}, \hbox{\tt phPh}, \hbox{\tt phps}, \hbox{\tt PhSUM}, \hbox{\tt phSUM}, \hbox{\tt phTausche}, \hbox{\tt ps}, \hbox{\tt psEinf}, \hbox{\tt psinv}, \hbox{\tt psOrdne}, \hbox{\tt psPerm}, \hbox{\tt psph}, \hbox{\tt psShift}, \hbox{\tt psSUM}, \hbox{\tt SUMErw1}, \hbox{\tt SUMErw2}, \hbox{\tt SUMExpand}, \hbox{\tt SUMInfinity}, \hbox{\tt SUMph}, \hbox{\tt SUMPh}, \hbox{\tt SUMps}, \hbox{\tt SUMRegeln}, \hbox{\tt SUMSammle}, \hbox{\tt SUMShift}, \hbox{\tt SUMTausche}, \hbox{\tt SUMUmkehr}, \hbox{\tt SUMZerl}, \par} \medskip\noindent two functions for controlled use of rules, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt Ers}, \hbox{\tt Posliste}, \par} \medskip\noindent one function for substitution of an expression instead of another expression, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt Subst}, \par} \medskip\noindent some objects for doing limits of basic hypergeometric expressions, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller}, \hbox{\tt AbsUndetermined}, \hbox{\tt Limes}, \par} \medskip\noindent one object for simplifying arguments in basic hypergeometric expressions, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt SimplifyPQ}, \par} \medskip\noindent some objects for converting expressions into \TeX-code, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt AmSLaTeX}, \hbox{\tt AmSTeX}, \hbox{\tt LaTeX}, \hbox{\tt TeX}, \hbox{\tt TeXMat}, \hbox{\tt TeXphW}, \par} \medskip\noindent two objects for on-line help, \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt hypqAttributes}, \hbox{\tt ManipulationsListe}, \par} \medskip\noindent and the function \medskip\noindent {\leftskip20pt\rightskip20pt \tt Drucke \par} \medskip\noindent which enables you to directly send an expression to the printer. Finally there are \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt Add}, \hbox{\tt Div}, \hbox{\tt Gleichung}, \hbox{\tt GlTausche}, \hbox{\tt Hoch}, \hbox{\tt LS}, \hbox{\tt Mal}, \hbox{\tt PQSort}, \hbox{\tt RS}, \hbox{\tt Sub}\par} \medskip\noindent for manipulating equations and writing expressions in a ``normalized" form (\hbox{\tt PQSort}) in order to be able to quickly check if two expressions agree. These objects are particularly important when using objects from \hbox{\tt summatio.qgl} and\linebreak \hbox{\tt transfor.qgl}. \bigskip Most of the tools for manipulating expressions that are provided by this package are rules. This has the advantage that very often you do not have to specify to which part of an expression you want to apply a rule, since there is just one subexpression to which the rule applies. However, if there are more subexpressions to which a rule applies, you will sometimes want to apply the rule only to some of the subexpressions. To handle this conveniently, there are the functions \hbox{\tt Ers} and \hbox{\tt PosListe}. \head The file \tt summatio.q\endhead This file contains most of the summation formulas of the book \cite{\GaRaAA}, including the (almost) complete Appendix~II of \cite{\GaRaAA}, in form of rules. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by \hbox{\tt summatio.q} are \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt S0110}, \hbox{\tt S1001}, \hbox{\tt S1101}, \hbox{\tt S1102}, \hbox{\tt S1110}, \hbox{\tt S2101}, \hbox{\tt S2102}, \hbox{\tt S2103}, \hbox{\tt S2104}, \hbox{\tt S2105}, \hbox{\tt S2106}, \hbox{\tt S2107}, \hbox{\tt S2161}, \hbox{\tt S2201}, \hbox{\tt S2202}, \hbox{\tt S2210}, \hbox{\tt S3201}, \hbox{\tt S3202}, \hbox{\tt S3203}, \hbox{\tt S3204}, \hbox{\tt S3261}, \hbox{\tt S3310}, \hbox{\tt S4301}, \hbox{\tt S4302}, \hbox{\tt S4303}, \hbox{\tt S4304}, \hbox{\tt S4305}, \hbox{\tt S4306}, \hbox{\tt S4307}, \hbox{\tt S4308}, \hbox{\tt S4361}, \hbox{\tt S4410}, \hbox{\tt S5401}, \hbox{\tt S5402}, \hbox{\tt S5501}, \hbox{\tt S6501}, \hbox{\tt S6502}, \hbox{\tt S6610}, \hbox{\tt S8701}, \hbox{\tt S8702}, \hbox{\tt S8703}, \hbox{\tt S8704}, \hbox{\tt S8761}, \hbox{\tt S10901}, \hbox{\tt SListe}, \hbox{\tt SumListe}. \par} \medskip\noindent The numbering of each rule is {\tt S$\langle$d1$\rangle$$\langle$d2$\rangle $$\langle$n1$\rangle$} following the following system: The number {\tt $\langle$d1$\rangle$} is the number of the upper parameters, the number {\tt $\langle$d2$\rangle$} is the number of the lower parameters of the basic hypergeometric series to which the rule applies. The number $\langle$n1$\rangle$ allows to distinguish the rules applying to basic hypergeometric series with equal numbers of upper and lower parameters. $\langle$n1$\rangle$ is within the range 01--30 if the summation is a one-term summation, it is within the range 61--90 if the summation is a two- or more-term summation. For terminating series there is a check if one of the parameters is of the form $q^{-n}$ where $n$ is a nonnegative integer. Depending on your input you might be asked if some expression is a nonnegative integer (see the examples for \hbox{\tt S3201}). Be sure to give an affirmative answer only for {\it one} of several expressions, otherwise the package will try to find the minimum of all of these, which might cause problems. This remark also applies to other rules which put this question, e.g\. \hbox{\tt SUMUmkehr}, \hbox{\tt phSUM}, or in case that automatic evaluating is active (cf\. \hbox{\tt PQ}). The rule \hbox{\tt SListe} enables you to quickly check if one of the summation rules \hbox{\tt S0110}--\hbox{\tt S8761} can be directly applied. \head The file \tt transfor.q\endhead This file contains most of the transformation formulas of the book \cite{\GaRaAA}, including the (almost) complete Appendix~III of \cite{\GaRaAA}, in form of rules. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by \hbox{\tt transfor.q} are \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt T2101}, \hbox{\tt T2102}, \hbox{\tt T2103}, \hbox{\tt T2104}, \hbox{\tt T2105}, \hbox{\tt T2106}, \hbox{\tt T2107}, \hbox{\tt T2108}, \hbox{\tt T2109}, \hbox{\tt T2110}, \hbox{\tt T2111}, \hbox{\tt T2112}, \hbox{\tt T2161}, \hbox{\tt T2162}, \hbox{\tt T2163}, \hbox{\tt T2201}, \hbox{\tt T2202}, \hbox{\tt T3101}, \hbox{\tt T3201}, \hbox{\tt T3202}, \hbox{\tt T3203}, \hbox{\tt T3204}, \hbox{\tt T3205}, \hbox{\tt T3206}, \hbox{\tt T3207}, \hbox{\tt T3208}, \hbox{\tt T3209}, \hbox{\tt T3210}, \hbox{\tt T3211}, \hbox{\tt T3212}, \hbox{\tt T3213}, \hbox{\tt T3214}, \hbox{\tt T3215}, \hbox{\tt T3216}, \hbox{\tt T3217}, \hbox{\tt T3261}, \hbox{\tt T3262}, \hbox{\tt T3263}, \hbox{\tt T3264}, \hbox{\tt T3265}, \hbox{\tt T3266}, \hbox{\tt T3267}, \hbox{\tt T3268}, \hbox{\tt T3269}, \hbox{\tt T4301}, \hbox{\tt T4302}, \hbox{\tt T4303}, \hbox{\tt T4304}, \hbox{\tt T4305}, \hbox{\tt T4306}, \hbox{\tt T4307}, \hbox{\tt T4308}, \hbox{\tt T4309}, \hbox{\tt T4310}, \hbox{\tt T4311}, \hbox{\tt T4312}, \hbox{\tt T4313}, \hbox{\tt T4361}, \hbox{\tt T4362}, \hbox{\tt T5401}, \hbox{\tt T5402}, \hbox{\tt T5403}, \hbox{\tt T5404}, \hbox{\tt T5405}, \hbox{\tt T5461}, \hbox{\tt T5462}, \hbox{\tt T5463}, \hbox{\tt T5464}, \hbox{\tt T5465}, \hbox{\tt T5466}, \hbox{\tt T5467}, \hbox{\tt T5468}, \hbox{\tt T5469}, \hbox{\tt T6501}, \hbox{\tt T7601}, \hbox{\tt T7701}, \hbox{\tt T8701}, \hbox{\tt T8702}, \hbox{\tt T8703}, \hbox{\tt T8704}, \hbox{\tt T8705}, \hbox{\tt T8706}, \hbox{\tt T8707}, \hbox{\tt T8708}, \hbox{\tt T8709}, \hbox{\tt T8710}, \hbox{\tt T8711}, \hbox{\tt T8761}, \hbox{\tt T8762}, \hbox{\tt T8763}, \hbox{\tt T8764}, \hbox{\tt T8810}, \hbox{\tt T10901}, \hbox{\tt T10902}, \hbox{\tt T10903}, \hbox{\tt T10904}, \hbox{\tt T10905}, \hbox{\tt T10906}, \hbox{\tt T10907}, \hbox{\tt T10961}, \hbox{\tt T10962}, \hbox{\tt T10963}, \hbox{\tt T101010}, \hbox{\tt T121101}, \hbox{\tt T121102}, \hbox{\tt T121103}, \hbox{\tt T121104}, \hbox{\tt T121105}, \hbox{\tt T121106}, \hbox{\tt T121107}, \hbox{\tt T121161}, \hbox{\tt TListe}, \hbox{\tt TransListe}. \par} \medskip\noindent The comments for the file \hbox{\tt summatio.q} regarding the numbering of the rules and optional questions for input also apply here. The rule \hbox{\tt TListe} enables you to quickly check if one of the transformation rules \hbox{\tt T2101}--\hbox{\tt T10961} can be directly applied. \head The file \tt transfor.qli\endhead Each of the objects of this file corresponds to a transformation rule of the file \hbox{\tt transfor.q}. Each object gives a list of all the outcomes under application of a particular transformation after before having permuted the upper and lower parameters of the involved basic hypergeometric series. All the objects in this file are rules. These rules help to prove conjectured transformation formulas quickly. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by \hbox{\tt transfor.qli} are \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt Tli2101}, \hbox{\tt Tli2102}, \hbox{\tt Tli2103}, \hbox{\tt Tli2104}, \hbox{\tt Tli2105}, \hbox{\tt Tli2106}, \hbox{\tt Tli2107}, \hbox{\tt Tli2108}, \hbox{\tt Tli2109}, \hbox{\tt Tli2110}, \hbox{\tt Tli2111}, \hbox{\tt Tli2112}, \hbox{\tt Tli2161}, \hbox{\tt Tli2162}, \hbox{\tt Tli2163}, \hbox{\tt Tli2201}, \hbox{\tt Tli2202}, \hbox{\tt Tli3101}, \hbox{\tt Tli3201}, \hbox{\tt Tli3202}, \hbox{\tt Tli3203}, \hbox{\tt Tli3204}, \hbox{\tt Tli3205}, \hbox{\tt Tli3206}, \hbox{\tt Tli3207}, \hbox{\tt Tli3208}, \hbox{\tt Tli3209}, \hbox{\tt Tli3210}, \hbox{\tt Tli3211}, \hbox{\tt Tli3212}, \hbox{\tt Tli3213}, \hbox{\tt Tli3214}, \hbox{\tt Tli3215}, \hbox{\tt Tli3216}, \hbox{\tt Tli3261}, \hbox{\tt Tli3262}, \hbox{\tt Tli3263}, \hbox{\tt Tli3264}, \hbox{\tt Tli3265}, \hbox{\tt Tli3266}, \hbox{\tt Tli3267}, \hbox{\tt Tli3268}, \hbox{\tt Tli3269}, \hbox{\tt Tli4301}, \hbox{\tt Tli4302}, \hbox{\tt Tli4303}, \hbox{\tt Tli4304}, \hbox{\tt Tli4305}, \hbox{\tt Tli4306}, \hbox{\tt Tli4307}, \hbox{\tt Tli4308}, \hbox{\tt Tli4309}, \hbox{\tt Tli4310}, \hbox{\tt Tli4311}, \hbox{\tt Tli4312}, \hbox{\tt Tli4313}, \hbox{\tt Tli4361}, \hbox{\tt Tli4362}, \hbox{\tt Tli5401}, \hbox{\tt Tli5402}, \hbox{\tt Tli5403}, \hbox{\tt Tli5404}, \hbox{\tt Tli5405}, \hbox{\tt Tli5461}, \hbox{\tt Tli5462}, \hbox{\tt Tli5463}, \hbox{\tt Tli5464}, \hbox{\tt Tli5465}, \hbox{\tt Tli5466}, \hbox{\tt Tli5467}, \hbox{\tt Tli5468}, \hbox{\tt Tli5469}, \hbox{\tt Tli6501}, \hbox{\tt Tli7601}, \hbox{\tt Tli8701}, \hbox{\tt Tli8702}, \hbox{\tt Tli8703}, \hbox{\tt Tli8704}, \hbox{\tt Tli8705}, \hbox{\tt Tli8706}, \hbox{\tt Tli8707}, \hbox{\tt Tli8708}, \hbox{\tt Tli8709}, \hbox{\tt Tli8710}, \hbox{\tt Tli8711}, \hbox{\tt Tli8761}, \hbox{\tt Tli8762}, \hbox{\tt Tli8763}, \hbox{\tt Tli8764}, \hbox{\tt Tli8810}, \hbox{\tt Tli10901}, \hbox{\tt Tli10902}, \hbox{\tt Tli10903}, \hbox{\tt Tli10904}, \hbox{\tt Tli10905}, \hbox{\tt Tli10906}, \hbox{\tt Tli10907}, \hbox{\tt Tli10961}, \hbox{\tt Tli10962}, \hbox{\tt Tli10963}, \hbox{\tt Tli101010}, \hbox{\tt Tli121101}, \hbox{\tt Tli121102}, \hbox{\tt Tli121103}, \hbox{\tt Tli121104}, \hbox{\tt Tli121105}, \hbox{\tt Tli121106}, \hbox{\tt Tli121107}, \hbox{\tt Tli121161}. \par} \head The files \tt summatio.qgl \rm and \tt transfor.qgl\endhead These files contain the same summations, respectively transformations, as \hbox{\tt summatio.q}, respectively \hbox{\tt transfor.q}, but in form of equations. You do not have to load these files by hand since they are loaded automatically once an object of this file is called. The respective objects are \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt Sgl0110}, \hbox{\tt Sgl1001}, \hbox{\tt Sgl1101}, \hbox{\tt Sgl1102}, \hbox{\tt Sgl1110}, \hbox{\tt Sgl2101}, \hbox{\tt Sgl2102}, \hbox{\tt Sgl2103}, \hbox{\tt Sgl2104}, \hbox{\tt Sgl2105}, \hbox{\tt Sgl2106}, \hbox{\tt Sgl2107}, \hbox{\tt Sgl2161}, \hbox{\tt Sgl2201}, \hbox{\tt Sgl2202}, \hbox{\tt Sgl2210}, \hbox{\tt Sgl3201}, \hbox{\tt Sgl3202}, \hbox{\tt Sgl3203}, \hbox{\tt Sgl3204}, \hbox{\tt Sgl3261}, \hbox{\tt Sgl3310}, \hbox{\tt Sgl4301}, \hbox{\tt Sgl4302}, \hbox{\tt Sgl4303}, \hbox{\tt Sgl4304}, \hbox{\tt Sgl4305}, \hbox{\tt Sgl4306}, \hbox{\tt Sgl4307}, \hbox{\tt Sgl4308}, \hbox{\tt Sgl4361}, \hbox{\tt Sgl4410}, \hbox{\tt Sgl5401}, \hbox{\tt Sgl5402}, \hbox{\tt Sgl5501}, \hbox{\tt Sgl6501}, \hbox{\tt Sgl6502}, \hbox{\tt Sgl6610}, \hbox{\tt Sgl8701}, \hbox{\tt Sgl8702}, \hbox{\tt Sgl8703}, \hbox{\tt Sgl8704}, \hbox{\tt Sgl8761}, \hbox{\tt Sgl10901},\linebreak \hbox{\tt SumListe\$gl}, \par} \medskip\noindent and \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt Tgl2101}, \hbox{\tt Tgl2102}, \hbox{\tt Tgl2103}, \hbox{\tt Tgl2104}, \hbox{\tt Tgl2105}, \hbox{\tt Tgl2106}, \hbox{\tt Tgl2107}, \hbox{\tt Tgl2108}, \hbox{\tt Tgl2109}, \hbox{\tt Tgl2110}, \hbox{\tt Tgl2111}, \hbox{\tt Tgl2112}, \hbox{\tt Tgl2161}, \hbox{\tt Tgl2162}, \hbox{\tt Tgl2163}, \hbox{\tt Tgl2201}, \hbox{\tt Tgl2202}, \hbox{\tt Tgl3101}, \hbox{\tt Tgl3201}, \hbox{\tt Tgl3202}, \hbox{\tt Tgl3203}, \hbox{\tt Tgl3204}, \hbox{\tt Tgl3205}, \hbox{\tt Tgl3206}, \hbox{\tt Tgl3207}, \hbox{\tt Tgl3208}, \hbox{\tt Tgl3209}, \hbox{\tt Tgl3210}, \hbox{\tt Tgl3211}, \hbox{\tt Tgl3212}, \hbox{\tt Tgl3213}, \hbox{\tt Tgl3214}, \hbox{\tt Tgl3215}, \hbox{\tt Tgl3216}, \hbox{\tt Tgl3261}, \hbox{\tt Tgl3262}, \hbox{\tt Tgl3263}, \hbox{\tt Tgl3264}, \hbox{\tt Tgl3265}, \hbox{\tt Tgl3266}, \hbox{\tt Tgl3267}, \hbox{\tt Tgl3268}, \hbox{\tt Tgl3269}, \hbox{\tt Tgl4301}, \hbox{\tt Tgl4302}, \hbox{\tt Tgl4303}, \hbox{\tt Tgl4304}, \hbox{\tt Tgl4305}, \hbox{\tt Tgl4306}, \hbox{\tt Tgl4307}, \hbox{\tt Tgl4308}, \hbox{\tt Tgl4309}, \hbox{\tt Tgl4310}, \hbox{\tt Tgl4311}, \hbox{\tt Tgl4312}, \hbox{\tt Tgl4313}, \hbox{\tt Tgl4361}, \hbox{\tt Tgl4362}, \hbox{\tt Tgl5401}, \hbox{\tt Tgl5402}, \hbox{\tt Tgl5403}, \hbox{\tt Tgl5404}, \hbox{\tt Tgl5405}, \hbox{\tt Tgl5461}, \hbox{\tt Tgl5462}, \hbox{\tt Tgl5463}, \hbox{\tt Tgl5464}, \hbox{\tt Tgl5465}, \hbox{\tt Tgl5466}, \hbox{\tt Tgl5467}, \hbox{\tt Tgl5468}, \hbox{\tt Tgl5469}, \hbox{\tt Tgl6501}, \hbox{\tt Tgl7601}, \hbox{\tt Tgl8701}, \hbox{\tt Tgl8702}, \hbox{\tt Tgl8703}, \hbox{\tt Tgl8704}, \hbox{\tt Tgl8705}, \hbox{\tt Tgl8706}, \hbox{\tt Tgl8707}, \hbox{\tt Tgl8708}, \hbox{\tt Tgl8709}, \hbox{\tt Tgl8710}, \hbox{\tt Tgl8711}, \hbox{\tt Tgl8761}, \hbox{\tt Tgl8762}, \hbox{\tt Tgl8763}, \hbox{\tt Tgl8764}, \hbox{\tt Tgl8810}, \hbox{\tt Tgl10901}, \hbox{\tt Tgl10902}, \hbox{\tt Tgl10903}, \hbox{\tt Tgl10904}, \hbox{\tt Tgl10905}, \hbox{\tt Tgl10906}, \hbox{\tt Tgl10907}, \hbox{\tt Tgl10961}, \hbox{\tt Tgl10962}, \hbox{\tt Tgl10963}, \hbox{\tt Tgl101010}, \hbox{\tt Tgl121101}, \hbox{\tt Tgl121102}, \hbox{\tt Tgl121103}, \hbox{\tt Tgl121104}, \hbox{\tt Tgl121105}, \hbox{\tt Tgl121106}, \hbox{\tt Tgl121107}, \hbox{\tt Tgl121161}, \hbox{\tt TransListe\$gl}. \par} \medskip\noindent When calling one of these objects you will be put two questions. If the variables of the called summation or transformation are undefined, the first question is \medskip \MATH Do you want to set values for the equation? [y|n]: \endMATH \medskip\noindent Enter \hbox{\tt y} if you want to set values, even only for some of them, if you do not need to set values enter \hbox{\tt n}. If some of the variables of the called summation or transformation are already defined, you will be asked as first question \medskip \MATH Some variables have a value. Should the variables $\{$[V,a,r,i,a,b,l,e,s]$\}$ be cleared? Do you want to set values for the equation (v)? [y|n|yv|nv]: \endMATH \medskip\noindent Now you have four options depending on if you want to set values or not and if you want to clear the already defined variables or not. For example, if you want to set values but do not want to clear the defined variables, enter \hbox{\tt nv}. The second question concerns the base \hbox{\tt q}. Either it is \medskip \MATH Do you want to set a value for q in the equation? [y|n]: \endMATH \medskip\noindent or \medskip\noindent \MATH q has a value. Should q be cleared? Do you want to set a value for q in the equation (v)? [y|n|yv|nv]: \endMATH \medskip\noindent The explanations for the first question also apply here. (Cf\. the examples in \hbox{\tt Sgl2101}). In addition there are the functions and variables \medskip\noindent {\leftskip20pt\rightskip20pt\tt Add, Div, Gleichung, GlTausche, Hoch, LS, Mal, RS, Sub,\par} \medskip\noindent for manipulating equations. In fact, once you have called one of the objects {\tt Sgl$*$} or {\tt Tgl$*$}, the right-hand side of the displayed equation will have been assigned to {\tt RS}, the left-hand side to {\tt LS}, and thus the equation itself to \hbox{\tt Gleichung} (cf\. the example in \hbox{\tt Gleichung}). The functions {\tt Add, Div, GlTausche, Hoch, Mal, Sub}, and also {\tt Ers}, allow you to manipulate the equation. \head The file \tt contig.q\endhead This file contains a vast number of contiguous relations in form of rules. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by {\tt contig.q} are \medskip\noindent {\leftskip20pt\rightskip20pt \hbox{\tt C01}, \hbox{\tt C02}, \hbox{\tt C11}, \hbox{\tt C12}, \hbox{\tt C13}, \hbox{\tt C14}, \hbox{\tt C15}, \hbox{\tt C16}, \hbox{\tt C17}, \hbox{\tt C18}, \hbox{\tt C19}, \hbox{\tt C20}, \hbox{\tt C21}, \hbox{\tt C22}, \hbox{\tt C23}, \hbox{\tt C24}, \hbox{\tt C25}, \hbox{\tt C26}, \hbox{\tt C27}, \hbox{\tt C28}, \hbox{\tt C29}, \hbox{\tt C30}, \hbox{\tt C31}, \hbox{\tt C32}, \hbox{\tt C33}, \hbox{\tt C34}, \hbox{\tt C35}, \hbox{\tt C36}, \hbox{\tt C37}, \hbox{\tt C38}, \hbox{\tt C39}, \hbox{\tt C40}, \hbox{\tt C41}, \hbox{\tt C42}, \hbox{\tt C43}, \hbox{\tt C44}, \hbox{\tt C45}, \hbox{\tt C46}, \hbox{\tt C47}, \hbox{\tt C48}, \hbox{\tt C49}, \hbox{\tt C50}, \hbox{\tt C51}, \hbox{\tt C52}, \hbox{\tt C53}, \hbox{\tt C54}, \hbox{\tt C55}, \hbox{\tt C56}, \hbox{\tt C57}, \hbox{\tt C58}, \hbox{\tt C59}, \hbox{\tt C60}, \hbox{\tt C61}, \hbox{\tt C62}, \hbox{\tt C63}, \hbox{\tt C64}, \hbox{\tt C65}, \hbox{\tt C66}, \hbox{\tt C67}, \hbox{\tt C68}, \hbox{\tt C69}, \hbox{\tt C70}, \hbox{\tt C71}, \hbox{\tt C72}, \hbox{\tt C73}, \hbox{\tt C74}, \hbox{\tt C75}, \hbox{\tt C76}, \hbox{\tt C77}, \hbox{\tt C78}, \hbox{\tt C79}, \hbox{\tt C80}, \hbox{\tt C81}, \hbox{\tt C82}, \hbox{\tt C83}, \hbox{\tt C84}, \hbox{\tt C85}, \hbox{\tt C86}, \hbox{\tt C87}, \hbox{\tt C88}, \hbox{\tt C89}, \hbox{\tt C90}, \hbox{\tt C91}, \hbox{\tt C92}, \hbox{\tt C93}, \hbox{\tt C94}, \hbox{\tt C95}, \hbox{\tt C96}, \hbox{\tt C97}, \hbox{\tt C98}, \hbox{\tt C99}, \hbox{\tt C100}, \hbox{\tt C101}, \hbox{\tt C102}, \hbox{\tt C103}, \hbox{\tt C104}, \hbox{\tt C105}, \hbox{\tt C106}, \hbox{\tt C107}, \hbox{\tt C108}, \hbox{\tt C109}, \hbox{\tt C110}, \hbox{\tt C111}, \hbox{\tt C112}, \hbox{\tt C113}, \hbox{\tt C114}, \hbox{\tt C115}, \hbox{\tt C116}, \hbox{\tt C117}, \hbox{\tt C118}, \hbox{\tt C119}, \hbox{\tt C120}, \hbox{\tt C121}, \hbox{\tt ContigListe}. \par} \medskip\noindent \head Simultaneous use of HYP and HYPQ \endhead It is possible to load both packages, HYP and HYPQ. In this case, the objects of the package that is loaded last will override the respective objects of the other package which have identical names. However, you can use the overrided objects by calling them by their {\it full} names. To determine the full name of an object the following rule applies: \medskip\noindent {\leftskip20pt\rightskip20pt If the object \hbox{\tt Object} is defined in the file \hbox{\tt File.ext}, then the full name of \hbox{\tt Object} is \hbox{\tt File`ext`Object}. \par} \medskip\noindent For instance, if you load \hbox{\tt hyp.q} first and then \hbox{\tt hyp.m} and want to use \hbox{\tt Limes} with a basic hypergeometric expression, then you have to type \hbox{\tt Hyp`q`Limes} instead of \hbox{\tt Limes}. (Calling \hbox{\tt Limes} would invoke the {\it ordinary} hypergeometric \hbox{\tt Limes}.) For information on contexts in MATHEMATICA confer \cite{\WolfAA}. \head On-line help \endhead For each object of this package on-line help is supported in the usual way. For instance, quick information about \hbox{\tt Limes} (not having the manual at hand) is available in the following way. \MATH In[1]:= ?Limes Description: Function for doing formal limits of basic hypergeometric expressions. If required for taking the limit, you will be asked whether or not the absolute value of some variable or expression is smaller than 1. Your decision is stored for the rest of your MATHEMATICA session. If you want to change your decision later, use "AbsGreater", "AbsSmaller", or "AbsUndetermined", respectively. By default the absolute value of q is defined to be smaller than 1. Also this can be changed by "AbsGreater", "AbsSmaller", or "AbsUndetermined", respectively. Warning: This function uses primitive algebraic techniques to do the limit. There is no check if taking the limit is actually allowed. So it is left to you to check the validity of a result of "Limes". Usage: Limes[Expr, x-\MATHgroesser x0]. See also: AbsGreater, AbsSmaller, AbsUndetermined, MinusOne. \endMATH \head The screen output \endhead The screen output of the examples in this manual imitates the output under usage of code tables 437, 860, 863, or 865 (cf\. the \hbox{\tt read.me}). The output under other code tables is a little bit less attractive, but similar. For instance, the examples for \hbox{\tt ph} then would read as follows. \MATH In[1]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z] \goodbreakpoint% [ ] | a, b | Out[1]= ph | ; q, z | 2 1| c | [ ] \goodbreakpoint% In[2]:= ph[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,e,0% \MATHrbrace ,q,z] \goodbreakpoint% [ ] | a, b, c | Out[2]= ph | ; q, z | 3 3| d, e, 0 | [ ] \endMATH \newpage \head A brief dictionary \endhead Most of the names of the objects are obviously German influenced. To help those who are not so familiar with German, brief German--English and English--German vocabularies are provided. \vbox{ \head \tenpoint \bf A German--English vocabulary \endhead $$\smatrix \format\sa\l\s\l\s\l\se\\ \hlinefor7\\ &\text {\eightpoint German}&&\text {\eightpoint English}&& \text {\eightpoint {\sl Mathematica} objects in HYPQ }&\\ &&&&&\text {\eightpoint containing the word}&\\ \hlinefor7\\ &\text {abspalten}&& \text {split}&& \text {\tt lina1, lina2}&\\ \hlinefor7\\ &\text {aufl\"osen}&& \text {dissolve}&& \text {\tt pqaufl}&\\ \hlinefor7\\ &\text {drucken}&& \text {print}&& \text {\tt Drucke}&\\ \hlinefor7\\ &\text {einf\"ugen}&& \text {insert}&& \text {\tt phEinf}&\\ \hlinefor7\\ &\text {ersetzen}&&\text {replace}&& \text {\tt Ers}&\\ \hlinefor7\\ &\text {erweitern}&& \text {extend}&& \text {\tt erw1, erw2, SUMErw1, SUMErw2}&\\ \hlinefor7\\ &\text {Gleichung}&& \text {equation}&&\text {\tt Gleichung}&\\ \hlinefor7\\ &\text {"2 hoch 3"}&& \text {"2 to the 3"}&& \text {\tt Hoch}&\\ \hlinefor7\\ &\text {"2 mal 3"}&& \text {"2 times 3"}&& \text {\tt Mal}&\\ \hlinefor7\\ &\text {ordnen}&& \text {order}&& \text {\tt phOrdne}&\\ \hlinefor7\\ &\text {Regel}&& \text {rule}&& \text {\tt SUMRegeln}&\\ \hlinefor7\\ &\text {sammeln}&& \text {collect}&& \text {\tt SUMSammle}&\\ \hlinefor7\\ &\text {schreiben}&& \text {write}&& \text {\tt SchreibeZahl}&\\ \hlinefor7\\ &\text {tauschen}&&\text {exchange, interchange}&& \text {\tt SUMTausche, GlTausche}&\\ \hlinefor7\\ &\text {umkehren}&&\text {reverse}&& \text {\tt SUMUmkehr}&\\ \hlinefor7\\ &\text {Zahl}&& \text {number}&& \text {\tt SchreibeZahl}&\\ \hlinefor7\\ &\text {zerlegen}&&\text {split}&& \text {\tt zerl1, pqzerl, pqinfzerl, SUMZerl}&\\ \hlinefor7\\ &\text {zusammenfassen}&&\text {put together}&&\text {\tt zus1, pqzus, pqinfzus}&\\ \hlinefor7\\ \endsmatrix $$ } \newpage \vbox{ \head \tenpoint \bf An English--German vocabulary \endhead $$\smatrix \format\sa\l\s\l\s\l\se\\ \hlinefor7\\ &\text {\eightpoint English}&&\text {\eightpoint German}&& \text {\eightpoint {\sl Mathematica} objects in HYPQ }&\\ &&&&&\text {\eightpoint containing the word}&\\ \hlinefor7\\ &\text {collect}&& \text {sammeln}&& \text {\tt SUMSammle}&\\ \hlinefor7\\ &\text {dissolve}&& \text {aufl\"osen}&& \text {\tt pqaufl}&\\ \hlinefor7\\ &\text {equation}&&\text {Gleichung}&& \text {\tt Gleichung}&\\ \hlinefor7\\ &\text {exchange}&&\text {tauschen}&&\text {\tt SUMTausche, GlTausche}&\\ \hlinefor7\\ &\text {extend}&& \text {erweitern}&& \text {\tt erw1, erw2, SUMErw1, SUMErw2}&\\ \hlinefor7\\ &\text {insert}&& \text {einf\"ugen}&& \text {\tt phEinf}&\\ \hlinefor7\\ &\text {interchange}&& \text {tauschen}&&\text {\tt SUMTausche, GlTausche}&\\ \hlinefor7\\ &\text {number}&& \text {Zahl}&& \text {\tt SchreibeZahl}&\\ \hlinefor7\\ &\text {order}&& \text {ordnen}&& \text {\tt phOrdne}&\\ \hlinefor7\\ &\text {print}&& \text {drucken}&& \text {\tt Drucke}&\\ \hlinefor7\\ &\text {replace}&& \text {ersetzen}&&\text {\tt Ers}&\\ \hlinefor7\\ &\text {reverse}&& \text {umkehren}&&\text {\tt SUMUmkehr }&\\ \hlinefor7\\ &\text {rule}&& \text {Regel}&& \text {\tt SUMRegeln}&\\ \hlinefor7\\ &\text {split}&& \text {abspalten}&& \text {\tt lina1, lina2}&\\ \hlinefor7\\ &\text {split}&& \text {zerlegen}&&\text {\tt zerl1, pqzerl, pqinfzerl, SUMZerl}&\\ \hlinefor7\\ &\text {"2 times 3"}&& \text {"2 mal 3"}&& \text {\tt Mal}&\\ \hlinefor7\\ &\text {"2 to the 3"}&& \text {"2 hoch 3"}&& \text {\tt Hoch}&\\ \hlinefor7\\ &\text {put together}&&\text {zusammenfassen}&&\text {\tt zus1, pqzus, pqinfzus}&\\ \hlinefor7\\ &\text {write}&& \text {schreiben}&& \text {\tt SchreibeZahl}&\\ \hlinefor7\\ \endsmatrix $$ } \newpage \head Alphabetic List of the objects with desriptions \endhead \vskip1cm \parindent0pt %\tracingmacros=2 \tracingcommands=2 \Name AbsGreater \Description Function for declaring the absolute value of a variable or expression to be greater than 1. This declaration is used by \hbox{\tt Limes}. By default the absolute value of {\tt q} is defined to be smaller than 1. \Usage AbsGreater[Expr]. \Example \MATH In[1]:= Limes[pq[a,n,p],n-\MATHgroesser Infinity] Is % \MATHvStrich p% \MATHvStrich smaller than 1? [y|n|u]: y \goodbreakpoint% Out[1]= (a;p) \MATHinfty \goodbreakpoint% In[2]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity] Is n even, odd, or neither of both? [e|o|n]: n Is % \MATHvStrich a% \MATHvStrich smaller than 1? [y|n|u]: u Is % \MATHvStrich a% \MATHvStrich smaller than 1? [y|n|u]: u \goodbreakpoint% The expression \goodbreakpoint% 2 n n -n/2 - n /2 p (-1) a p (-;p) a \MATHinfty \goodbreakpoint% was obtained. \goodbreakpoint% Therefore the limit n -\MATHgroesser \MATHinfty could not be determined. Here is your expression: \goodbreakpoint% a Out[2]= (--; p) n n p \goodbreakpoint% In[3]:= AbsGreater[p] \goodbreakpoint% In[4]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity] Is 2 n even, odd, or neither of both? [e|o|n]: e \goodbreakpoint% a 1 Out[4]= (-;-) p p \MATHinfty \endMATH \Seealso AbsSmaller, AbsUndetermined, Limes. \Name AbsSmaller \Description Function for declaring the absolute value of a variable or expression to be smaller than 1. This declaration is used by \hbox{\tt Limes}. By default the absolute value of {\tt q} is defined to be smaller than 1. \Usage AbsSmaller[Expr]. \Example \MATH In[1]:= Limes[pq[a,n,1/p],n-\MATHgroesser Infinity] Is % \MATHvStrich p% \MATHvStrich smaller than 1? [y|n|u]: n \goodbreakpoint% 1 Out[1]= (a;-) p \MATHinfty \goodbreakpoint% In[2]:= Limes[pq[a*p\MATHhoch n,n,1/p],n-\MATHgroesser Infinity] Is n even, odd, or neither of both? [e|o|n]: n Is % \MATHvStrich a% \MATHvStrich smaller than 1? [y|n|u]: u Is % \MATHvStrich a% \MATHvStrich smaller than 1? [y|n|u]: u \goodbreakpoint% The expression \goodbreakpoint% 2 n n n/2 + n /2 1 1 (-1) a p (---;-) a p p \MATHinfty \goodbreakpoint% was obtained. \goodbreakpoint% Therefore the limit n -\MATHgroesser \MATHinfty could not be determined. Here is your expression: \goodbreakpoint% n 1 Out[2]= (a p ; -) p n \goodbreakpoint% In[3]:= AbsSmaller[p] \goodbreakpoint% In[4]:= Limes[pq[a*p\MATHhoch n,n,1/p],n-\MATHgroesser Infinity] Is 2 n even, odd, or neither of both? [e|o|n]: e \goodbreakpoint% Out[4]= (a p;p) \MATHinfty \endMATH \Seealso AbsGreater, AbsUndetermined, Limes. \Name AbsUndetermined \Description Function for declaring the absolute value of a variable or expression to be neither smaller nor greater than 1. This declaration is used by \hbox{\tt Limes}. By default the absolute value of {\tt q} is defined to be smaller than 1. \Usage AbsUndetermined[Expr]. \Example \MATH In[1]:= AbsSmaller[p] \goodbreakpoint% In[2]:= Limes[pq[a,n,p],n-\MATHgroesser Infinity] \goodbreakpoint% Out[2]= (a;p) \MATHinfty \goodbreakpoint% In[3]:= AbsGreater[p] \goodbreakpoint% In[4]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity] Is 2 n even, odd, or neither of both? [e|o|n]: e \goodbreakpoint% a 1 Out[4]= (-;-) p p \MATHinfty \goodbreakpoint% In[5]:= AbsUndetermined[p] \goodbreakpoint% In[6]:= Limes[pq[a,n,p],n-\MATHgroesser Infinity] Is % \MATHvStrich p% \MATHvStrich smaller than 1? [y|n|u]: u \goodbreakpoint% The expression \goodbreakpoint% Indeterminate \goodbreakpoint% was obtained. \goodbreakpoint% Therefore the limit n -\MATHgroesser \MATHinfty could not be determined. Here is your expression: \goodbreakpoint% Out[6]= (a; p) n \goodbreakpoint% In[7]:= Limes[pq[a*p\MATHhoch -n,n,p],n-\MATHgroesser Infinity] Is % \MATHvStrich p% \MATHvStrich smaller than 1? [y|n|u]: u \goodbreakpoint% The expression \goodbreakpoint% Indeterminate \goodbreakpoint% was obtained. \goodbreakpoint% Therefore the limit n -\MATHgroesser \MATHinfty could not be determined. Here is your expression: \goodbreakpoint% a Out[7]= (--; p) n n p \endMATH \Seealso AbsGreater, AbsSmaller, Limes. \Name Add \Description Function that adds \hbox{\tt Expr} to \hbox{\tt Gleichung}. \Usage Add[Expr]. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= Add[pq[c/a,n]] \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich c c a n Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich + (-; q) == (-; q) + % ---------- 2 1\MATHvStrich \MATHvStrich a n a n (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[3]:= Gleichung \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich c c a n Out[3]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich + (-; q) == (-; q) + % ---------- 2 1\MATHvStrich \MATHvStrich a n a n (c; q) \MATHloEck c \MATHroEck n \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Div, Sub, Hoch, GlTausche, Ers. \Name AmSLaTeX \Description Switch that changes the output of TeXForm to be usable with \AmS-\LaTeX. By default the output of TeXForm is usable with \AmSTeX. \Usage AmSLaTeX. \Example \MATH In[1]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[2]:= TeXForm[ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z]] \goodbreakpoint% Out[2]//TeXForm= % \MATHlbrace % \MATHrbrace \MATHtief % \MATHlbrace 2% \MATHrbrace \MATHbackslash phi \MATHtief % \MATHlbrace 1% \MATHrbrace \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash \MATHbackslash let \MATHbackslash over / c\MATHbackslash endmatrix ;q, % \MATHlbrace \MATHbackslash displaystyle z% \MATHrbrace \MATHbackslash right ] \goodbreakpoint% In[3]:= AmSLaTeX \goodbreakpoint% In[4]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-LaTeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[5]:= TeXForm[ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z]] \goodbreakpoint% Out[5]//TeXForm= % \MATHlbrace % \MATHrbrace \MATHtief % \MATHlbrace 2% \MATHrbrace \MATHbackslash phi \MATHtief % \MATHlbrace 1% \MATHrbrace \MATHbackslash ! \MATHbackslash left [ \MATHbackslash begin% \MATHlbrace matrix% \MATHrbrace \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash \MATHbackslash let \MATHbackslash over / c\MATHbackslash end% \MATHlbrace matrix% \MATHrbrace ;q, % \MATHlbrace \MATHbackslash displaystyle z% \MATHrbrace \MATHbackslash right ] \endMATH \Seealso AmSTeX, LaTeX, TeX, TeXMat, TeXphW. \Name AmSTeX \Description Switch that changes the output of TeXForm to be usable with \AmSTeX. By default the output of TeXForm is usable with \AmSTeX. \Usage AmSTeX. \Example \MATH In[1]:= TeX \goodbreakpoint% In[2]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with Plain-TeX and LaTeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[3]:= TeXForm[ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z]] \goodbreakpoint% Out[3]//TeXForm= % \MATHlbrace % \MATHrbrace \MATHtief % \MATHlbrace 2% \MATHrbrace \MATHbackslash phi \MATHtief % \MATHlbrace 1% \MATHrbrace \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix % \MATHlbrace \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash cr \MATHbackslash let \MATHbackslash over / c% \MATHrbrace ;q, % \MATHlbrace \MATHbackslash displaystyle z% \MATHrbrace \MATHbackslash right ] \goodbreakpoint% In[4]:= AmSTeX \goodbreakpoint% In[5]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[6]:= TeXForm[ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z]] \goodbreakpoint% Out[6]//TeXForm= % \MATHlbrace % \MATHrbrace \MATHtief % \MATHlbrace 2% \MATHrbrace \MATHbackslash phi \MATHtief % \MATHlbrace 1% \MATHrbrace \MATHbackslash ! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash \MATHbackslash \MATHbackslash let \MATHbackslash over / c\MATHbackslash endmatrix ;q, % \MATHlbrace \MATHbackslash displaystyle z% \MATHrbrace \MATHbackslash right ] \endMATH \Seealso AmSLaTeX, LaTeX, TeX, TeXMat, TeXphW. \Name baszerl1 \Description \vtab $(a;q)_n \to \prod _{k=0} ^{m-1} (aq^k;q^m)_{n/m}$,\\ $(a;q)_\infty \to \prod _{k=0} ^{m-1} (aq^k;q^m)_\infty$. \endvtab \vskip6pt \hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.baszerl1. \Example \MATH In[1]:= pq[a,n] \goodbreakpoint% Out[1]= (a; q) n \goodbreakpoint% In[2]:= \%/.baszerl1 split into ? terms: 2 \goodbreakpoint% 2 2 Out[2]= (a; q ) (a q; q ) n/2 n/2 \goodbreakpoint% In[3]:= pq[b,4*m] \goodbreakpoint% Out[3]= (b; q) 4 m \goodbreakpoint% In[4]:= \%/.baszerl1 split into ? terms: 4 \goodbreakpoint% 4 4 2 4 3 4 Out[4]= (b; q ) (b q; q ) (b q ; q ) (b q ; q ) m m m m \goodbreakpoint% In[5]:= pqinf[c,q\MATHhoch (1/2)] \goodbreakpoint% Out[5]= (c;Sqrt[q]) \MATHinfty \goodbreakpoint% In[6]:= \%/.baszerl1 split into ? terms: 4 \goodbreakpoint% 2 2 2 3/2 2 Out[6]= (c;q ) (c Sqrt[q];q ) (c q;q ) (c q ;q ) \MATHinfty \MATHinfty \MATHinfty \MATHinfty \endMATH \Seealso baszerl2, baszus1, baszus2, Ers, PosListe, ManipulationsListe. \Name baszerl2 \Description \vtab $(a;q)_n \to \prod _{k=0} ^{m-1} (e^{2\pi i k/m}a^{1/m};q^{1/m})_n$,\\ $(a;q)_\infty \to \prod _{k=0}^{m-1}(e^{2\pi i k/m}a^{1/m};q^{1/m})_\infty$. \endvtab \vskip6pt \hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.baszerl2. \Example \MATH In[1]:= pq[a,n] \goodbreakpoint% Out[1]= (a; q) n \goodbreakpoint% In[2]:= \%/.baszerl2 split into ? terms: 2 \goodbreakpoint% Out[2]= (-Sqrt[a]; Sqrt[q]) (Sqrt[a]; Sqrt[q]) n n \goodbreakpoint% In[3]:= pq[a\MATHhoch 2,n,q\MATHhoch 4] \goodbreakpoint% 2 4 Out[3]= (a ; q ) n \goodbreakpoint% In[4]:= \%/.baszerl2 split into ? terms: 4 \goodbreakpoint% Out[4]= (-Sqrt[a]; q) (-I Sqrt[a]; q) (I Sqrt[a]; q) (Sqrt[a]; q) n n n n \goodbreakpoint% In[5]:= pqinf[c,q\MATHhoch 3] \goodbreakpoint% 3 Out[5]= (c;q ) \MATHinfty \goodbreakpoint% In[6]:= \%/.baszerl2 split into ? terms: 3 \goodbreakpoint% 1/3 (2 I)/3 \MATHpi 1/3 (4 I)/3 \MATHpi 1/3 Out[6]= (c ;q) (E c ;q) (E c ;q) \MATHinfty \MATHinfty \MATHinfty \endMATH \Seealso Ers, PosListe, ManipulationsListe. \Name baszus1 \Description \vtab $(a;q)_n \to (a;q^{1/m})_{mn}/ \prod _{k=1} ^{m-1} (aq^{k/m};q)_n$,\\ $(a;q)_\infty \to (a;q^{1/m})_\infty/ \prod _{k=1} ^{m-1} (aq^{k/m};q)_\infty$. \endvtab \vskip6pt \hskip10pt The parameter \hbox{\tt m} has to be entered on request. This operation is basically the inverse of \hbox{\tt baszerl1}. \Usage Expr/.baszus1. \Example \MATH In[1]:= pq[a,n,q\MATHhoch 2]*pq[a*q,n,q\MATHhoch 2] \goodbreakpoint% 2 2 Out[1]= (a; q ) (a q; q ) n n \goodbreakpoint% In[2]:= Ers[\%,baszus1,\MATHlbrace 1\MATHrbrace ] put together ? terms: 2 put together ? terms: 2 \goodbreakpoint% Out[2]= (a; q) 2 n \goodbreakpoint% In[3]:= pqinf[a/q,q\MATHhoch 2]*pqinf[a,q\MATHhoch 2]*pq[b,m]*pq[b*q\MATHhoch (1/3),m]*pq[b*q\MATHhoch (2/3),m] \goodbreakpoint% 2 a 2 1/3 2/3 Out[3]= (a;q ) (-;q ) (b; q) (b q ; q) (b q ; q) \MATHinfty q \MATHinfty m m m \goodbreakpoint% In[4]:= Ers[\%,baszus1,\MATHlbrace 3\MATHrbrace ] put together ? terms: 3 put together ? terms: 3 \goodbreakpoint% 2 a 2 1/3 Out[4]= (a;q ) (-;q ) (b; q ) \MATHinfty q \MATHinfty 3 m \goodbreakpoint% In[5]:= Ers[\%,baszus1,\MATHlbrace 2\MATHrbrace ] put together ? terms: 2 put together ? terms: 2 \goodbreakpoint% a 1/3 Out[5]= (-;q) (b; q ) q \MATHinfty 3 m \endMATH \Seealso Ers, PosListe, ManipulationsListe. \Name baszus2 \Description \vtab $(a;q)_n \to (a^m;q^m)_n/\prod _{k=1} ^{m-1}(e^{2\pi i k/m}a;q)_n$,\\ $(a;q)_\infty \to (a^m;q^m)_\infty/ \prod _{k=1} ^{m-1}(e^{2\pi i k/m}a;q)_\infty$. \endvtab \vskip6pt \hskip10pt The parameter \hbox{\tt m} has to be entered on request. This operation is basically the inverse of \hbox{\tt baszerl2}. \Usage Expr/.baszus2. \Example \MATH In[1]:= pq[a,m,q\MATHhoch 2]*pq[-a,m,q\MATHhoch 2] \goodbreakpoint% 2 2 Out[1]= (-a; q ) (a; q ) m m \goodbreakpoint% In[2]:= Ers[\%,baszus2,\MATHlbrace 1\MATHrbrace ] put together ? terms: 2 put together ? terms: 2 \goodbreakpoint% 2 4 Out[2]= (a ; q ) m \goodbreakpoint% In[3]:= pqinf[-A,q\MATHhoch (1/2)]*pqinf[A,q\MATHhoch (1/2)]*pq[-a,n]*pq[-I*a,n]*pq[I*a,n]* pq[a,n] \goodbreakpoint% Out[3]= (-A;Sqrt[q]) (A;Sqrt[q]) (-a; q) (-I a; q) (I a; q) (a; q) \MATHinfty \MATHinfty n n n n \goodbreakpoint% In[4]:= Ers[\%,baszus2,\MATHlbrace 3\MATHrbrace ] put together ? terms: 4 put together ? terms: 4 \goodbreakpoint% 4 4 Out[4]= (-A;Sqrt[q]) (A;Sqrt[q]) (a ; q ) \MATHinfty \MATHinfty n \goodbreakpoint% In[5]:= Ers[\%,baszus2,\MATHlbrace 1\MATHrbrace ] put together ? terms: 2 put together ? terms: 2 \goodbreakpoint% 2 4 4 Out[5]= (A ;q) (a ; q ) \MATHinfty n \endMATH Whenever you use a rule, that asks you for an input, within \hbox{\tt Ers}, you will get the reqest for input twice. The value which is entered last is used. \Seealso Ers, PosListe, ManipulationsListe. \Name Binomialpq \Description \hbox{\tt Binomialpq[n,k,q]} is the $q$-binomial coefficient, written in terms of $q$-factorial symbols \hbox{\tt pq}. The parameter \hbox{\tt q} is optional. It will be set equal \hbox{\tt q} if it is omitted. \Usage Binomialpq[n,k,q] \leavevmode\hphantom{Usa}\rm or: \tt Binomialpq[n,k]. \Example \MATH In[1]:= Binomialpq[n,k] \goodbreakpoint% 1 - k + n (q ; q) k Out[1]= ---------------- (q; q) k \goodbreakpoint% In[2]:= Binomialpq[6,3] \goodbreakpoint% 4 (q ; q) 3 Out[2]= -------- (q; q) 3 \endMATH \Seealso Binomialq, Multinomialpq, Multinomialq, Factorialq, Factorialpq. \Name Binomialq \Description \hbox{\tt Binomialq[n,k,q]} is the $q$-binomial coefficient, expanded into a $q$-series, if possible. The parameter \hbox{\tt q} is optional. It will be set equal \hbox{\tt q} if it is omitted. \Usage Binomialq[n,k,q] \leavevmode\hphantom{Usa}\rm or: \tt Binomialq[n,k]. \Example \MATH In[1]:= Binomialq[n+k,n-k] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich k + n \MATHvStrich Out[1]= \MATHvStrich \MATHvStrich \MATHvStrich -k + n \MATHvStrich \MATHloEck \MATHroEck q \goodbreakpoint% In[2]:= Binomialq[6,3] \goodbreakpoint% 2 3 4 5 6 7 8 9 Out[2]= 1 + q + 2 q + 3 q + 3 q + 3 q + 3 q + 2 q + q + q \endMATH \Seealso Binomialpq, Multinomialpq, Multinomialq, Factorialq, Factorialpq. \Name C01 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow 1 + {{{{\left( -1 \right) }^{1 - r + s}} z }\over {\left( 1 - q \right) }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / q, \qAi\\ \let \over / {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$ \Usage Expr/.C01. \Seealso C64, ContigListe, Ers, PosListe. \Name C02 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai, q\\ \let \over/ \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow -{{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {z }} {{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} + {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {z } } {{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over /\Aiq, q\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] $$ \Usage Expr/.C02. \Seealso C64, ContigListe, Ers, PosListe. \Name C11 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ] + {{\left( -1 \right) }^{1 - r + s}} z {{ \prodl_{i = 1}^{r} (1 - \ai ) }\over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$ \Usage Expr/.C11. \Seealso C64, ContigListe, Ers, PosListe. \Name C12 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - {{{{\left( -1 \right) }^{1 - r + s}} z }\over q} {{ \prodl_{i = 1}^{r} (1 - \ai ) }\over { \prodl_{i = 1}^{s} (1 - \bi ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ]$$ \Usage Expr/.C12. \Seealso C64, ContigListe, Ers, PosListe. \Name C13 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z} {{ \prodl_{i = 1}^{s} (1 - {{\bi}\over q} ) }\over { \prodl_{i = 1}^{r} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z} {{ \prodl_{i = 1}^{s} (1 - {{\bi}\over q} ) }\over { \prodl_{i = 1}^{r} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ] \endmultline$$ \Usage Expr/.C13. \Seealso C64, ContigListe, Ers, PosListe. \Name C14 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} a z }\over q} {{ \prodl_{i = 1}^{r-1} (1 - \ai ) }\over { \prodl_{i = 1}^{s} (1 - \bi ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]+ {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] $$ \Usage Expr/.C14[m1].\newline \rm {\tt m1} is the position of the special upper parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C15 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] - {{\left( -1 \right) }^{1 - r + s}} a z {{ \prodl_{i = 1}^{r - 1} (1 - \ai ) } \over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$ \Usage Expr/.C15[m1].\newline \rm {\tt m1} is the position of the special upper parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C16 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{2 - p + q}} }\over {az}} {{ \prodl_{i = 1}^{s} (1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} {q^{2 - p + q}}}\over{az}} {{ \prodl_{i = 1}^{s} (1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \endmultline$$ \Usage Expr/.C16[m1].\newline \rm {\tt m1} is the position of the special upper parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C17 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -1 \right) }^{1 - r + s}} z {{ \prodl_{i = 1}^{r - 1} (1 - \ai ) }\over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]+ {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ] $$ \Usage Expr/.C17[m1].\newline \rm {\tt m1} is the position of the special upper parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C18 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - {{{{\left( -1 \right) }^{1 - r + s}} z }\over q} {{ \prodl_{i = 1}^{r - 1} (1 - \ai ) }\over { \prodl_{i = 1}^{s} (1 - \bi ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ]$$ \Usage Expr/.C18[m1].\newline \rm {\tt m1} is the position of the special upper parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C19 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z} {{ \prodl_{i = 1}^{s} (1 - {{\bi}\over q} ) }\over { \prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over z} {{ \prodl_{i = 1}^{s} (1 - {{\bi}\over q} ) }\over { \prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} ) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ] \endmultline$$ \Usage Expr/.C19[m1].\newline \rm {\tt m1} is the position of the special upper parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C20 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{{{\left( -1 \right) }^{1 - r + s}} b z } \over {\left( 1 - b \right) \left( 1 - b q \right) }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over{\prodl_{i = 1}^{s - 1} (1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] $$ \Usage Expr/.C20[n1].\newline \rm {\tt n1} is the position of the special lower parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C21 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] - {{{{\left( -1 \right) }^{1 - r + s}} b z }\over {\left( 1 - b \right) \left( 1 - {b\over q} \right) q }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s - 1} (1 - \bi )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$ \Usage Expr/.C21[n1].\newline \rm {\tt n1} is the position of the special lower parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C22 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{3 - p + q}} }\over {b z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{3 - p + q}} }\over {b z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \endmultline$$ \Usage Expr/.C22[n1].\newline \rm {\tt n1} is the position of the special lower parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C23 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle q z} \right ] + {{{{\left( -1 \right) }^{1 - r + s}} z } \over {\left( 1 - b \right) \left( 1 - b q \right) }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s - 1} (1 - \bi )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] $$ \Usage Expr/.C23[n1].\newline \rm {\tt n1} is the position of the special lower parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C24 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - {{{{\left( -1 \right) }^{1 - r + s}} z }\over {\left( 1 - b \right) \left( 1 - {b\over q} \right) q }} {{ \prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s - 1} (1 - \bi )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / b q, \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ]$$ \Usage Expr/.C24[n1].\newline \rm {\tt n1} is the position of the special lower parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C25 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} \ } \over {z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ] \endmultline$$ \Usage Expr/.C25[n1].\newline \rm {\tt n1} is the position of the special lower parameter. \Seealso C64, ContigListe, Ers, PosListe. \Name C26 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - b \right) }\over {a - b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]+ {{\left( 1 - a \right) }\over {-a + b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]$$ \Usage Expr/.C26[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C27 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( b - {a\over q} \right) }\over {1 - {a\over q}}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ] + {{\left( 1 - b \right) }\over {1 - {a\over q}}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C27[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C28 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{a \left( 1 - b \right) }\over {a - b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{\left( 1 - a \right) b }\over {-a + b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C28[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C29 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( b - {a\over q} \right) }\over {b \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{a \left( 1 - b \right) }\over {b \left( 1 - {a\over q} \right) q}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C29[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C30 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -1 \right) }^{1 - r + s}} \left( -b + {a\over q} \right) z {{ \prodl_{i = 1}^{r - 2} (1 - \ai ) } \over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \qAi\\ \let \over / \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] + {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] $$ \Usage Expr/.C30[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C31 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {\left( {a\over q} - {b\over q} \right) z }} {{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {b\over q}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {\left( {a\over q} - {b\over q} \right) z }} {{\prodl_{i = 1}^{s} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \endmultline$$ \Usage Expr/.C31[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C32 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {b\over q}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ] \\+ {{\left( -1 \right) }^{1 - r + s}} \left( 1 - {{a b}\over q} \right) z {{ \prodl_{i = 1}^{r - 2} (1 - \ai ) }\over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a, b, a b, \qAi\\ \let \over / {{a b}\over q}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline$$ \Usage Expr/.C32[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C33 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b q, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b q \right) z }\over {q }} {{\prodl_{i = 1}^{r - 2} (1 - \ai )}\over {\prodl_{i = 1}^{s} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a q, b q, a b {q^2}, \qAi\\ \let \over / a b q, \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ] \endmultline$$ \Usage Expr/.C33[m1,m2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C34 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {b\over q} \right) }\over {a - {b\over q}}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] + {{\left( 1 - a \right) }\over {-a + {b\over q}}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]$$ \Usage Expr/.C34[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C35 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -a + b \right) }\over {\left( 1 - {a\over q} \right) q}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle q z} \right ]+ {{\left( 1 - {b\over q} \right) } \over {1 - {a\over q}}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C35[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C36 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( a - b \right) }\over {1 - b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle q z} \right ] + {{\left( 1 - a \right) }\over {1 - b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C36[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C37 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{a \left( 1 - {b\over q} \right) }\over {a - {b\over q}}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{\left( 1 - a \right) b }\over {\left( -a + {b\over q} \right) q}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C37[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C38 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -a + b \right) }\over {b \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{a \left( 1 - {b\over q} \right) } \over {b \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C38[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C39 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( a - b \right) } \over {a \left( 1 - b \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{\left( 1 - a \right) b }\over {a \left( 1 - b \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C39[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C40 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( {a\over q} - {b\over q} \right) z } \over {\left( 1 - b \right) \left( 1 - {b\over q} \right) }} {{\prodl_{i = 1}^{s - 1} (1 - \bi )}\over {\prodl_{i = 1}^{r - 1} (1 - \ai )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \qAi\\ \let \over / b q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] + {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] $$ \Usage Expr/.C40[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C41 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle z} \right ] - {{{{\left( -1 \right) }^{1 - r + s}} \left( a - b \right) z }\over {\left( 1 - b \right) \left( 1 - b q \right) }} {{\prodl_{i = 1}^{r - 1} (1 - \ai )}\over {\prodl_{i = 1}^{s - 1} (1 - \bi )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \qAi\\ \let \over / b {q^2}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ]$$ \Usage Expr/.C41[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C42 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( -{b\over {{q^2}}} + {a\over q} \right) z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over / {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( -{b\over {{q^2}}} + {a\over q} \right) z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\ \let \over / {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \endmultline$$ \Usage Expr/.C42[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C43 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / b q, \Bi\endmatrix ;q, {\displaystyle q z} \right ] + {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b \right) z }\over {\left( 1 - b \right) \left( 1 - b q \right) }} {{\prodl_{i = 1}^{r - 1} (1 - \ai )}\over {\prodl_{i = 1}^{s - 1} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a, a b q, \qAi\\ \let \over / b {q^2}, a b, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] $$ \Usage Expr/.C43[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C44 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] - {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b \right) z }\over {\left( 1 - b \right) \left( 1 - {b\over q} \right) q }} {{\prodl_{i = 1}^{r - 1} (1 - \ai )}\over {\prodl_{i = 1}^{s - 1} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a q, a b q, \qAi\\ \let \over / b q, a b, \qBi\endmatrix ;q, {\displaystyle {q^{-p + q}} z} \right ] $$ \Usage Expr/.C44[m1,n1].\newline \rm {\tt m1} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C45 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {b\over q} \right) q }\over {a - b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] + {{\left( 1 - {a\over q} \right) q } \over {-a + b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, b, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]$$ \Usage Expr/.C45[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C46 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -a + {b\over q} \right) }\over {1 - a}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, b, \Bi\endmatrix ;q, {\displaystyle q z} \right ]+ {{\left( 1 - {b\over q} \right) }\over {1 - a}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C46[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C47 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{a \left( 1 - {b\over q} \right) }\over {a - b} } {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{b \left( 1 - {a\over q} \right) }\over {-a + b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, b, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C47[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C48 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -a + {b\over q} \right) q }\over {\left( 1 - a \right) b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, b, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{a \left( 1 - {b\over q} \right) q }\over {\left( 1 - a \right) b}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C48[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C49 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, {b\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\+ {{{{\left( -1 \right) }^{1 - r + s}} \left( a - {b\over q} \right) z }\over {\left( 1 - a \right) \left( 1 - b \right) \left( 1 - {b\over q} \right) \left( 1 - a q \right) }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s - 2} (1 - \bi )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \qAi\\ \let \over / a {q^2}, b q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline$$ \Usage Expr/.C49[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C50 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\\longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( {a\over {{q^2}}} - {b\over {{q^2}}} \right) z }} {{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {a\over {{q^2}}}, {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( {a\over {{q^2}}} - {b\over {{q^2}}} \right) z }} {{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r} (1 - {{\ai}\over q} )}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {a\over q}, {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \endmultline$$ \Usage Expr/.C50[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C51 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, b q, \Bi\endmatrix ;q, {\displaystyle q z} \right ] \\+ {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - a b q \right) z }\over {\left( 1 - a \right) \left( 1 - b \right) \left( 1 - a q \right) \left( 1 - b q \right) }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s - 2} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a b {q^2}, \qAi\\ \let \over / a {q^2}, b {q^2}, a b q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline$$ \Usage Expr/.C51[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C52 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, {b\over q}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {{a b}\over q} \right) z }\over {\left( 1 - a \right) \left( 1 - b \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) q }} {{\prodl_{i = 1}^{r} (1 - \ai )}\over {\prodl_{i = 1}^{s - 2} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a b, \qAi\\ \let \over / a q, b q, {{a b}\over q}, \qBi\endmatrix ;q, {\displaystyle {q^{-r + s}} z} \right ] \endmultline$$ \Usage Expr/.C52[n1,n2].\newline \rm {\tt m1}, {\tt m2} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C53 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, c, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - b \right) \left( a - {c\over q} \right) }\over {\left( a - b \right) \left( 1 - {c\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, {c\over q}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{\left( 1 - a \right) \left( b - {c\over q} \right) }\over {\left( -a + b \right) \left( 1 - {c\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, {c\over q}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C53[m1,m2,m3].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C54 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / c, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - b \right) \left( a - c \right) }\over {\left( a - b \right) \left( 1 - c \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let \over / c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{\left( 1 - a \right) \left( b - c \right) }\over {\left( -a + b \right) \left( 1 - c \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let \over / c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C54[m1,m2,n1].\newline \rm {\tt m1}, {\tt m2} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C55 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / c, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -b + {a\over q} \right) \left( 1 - {c\over q} \right) } \over {\left( 1 - {a\over q} \right) \left( -b + {c\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\ \let \over / {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{\left( 1 - b \right) \left( {a\over q} - {c\over q} \right) }\over {\left( 1 - {a\over q} \right) \left( b - {c\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\ \let \over / c, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C55[m1,m2,n1].\newline \rm {\tt m1}, {\tt m2} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C56 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, c, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( a - c \right) \left( 1 - {b\over q} \right) }\over {\left( 1 - c \right) \left( a - {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {b\over q}, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]+ {{\left( 1 - a \right) \left( -c + {b\over q} \right) }\over {\left( 1 - c \right) \left( -a + {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / b, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C56[m1,n1,n2].\newline \rm {\tt m1} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C57 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / b, c, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( {a\over q} - {b\over q} \right) \left( 1 - {c\over q} \right) }\over {\left( 1 - {a\over q} \right) \left( -{b\over q} + {c\over q} \right) } } {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / b, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{\left( 1 - {b\over q} \right) \left( {a\over q} - {c\over q} \right) }\over {\left( 1 - {a\over q} \right) \left( {b\over q} - {c\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {b\over q}, c, \Bi\endmatrix ;q, {\displaystyle z} \right ]$$ \Usage Expr/.C57[m1,n1,n2].\newline \rm {\tt m1} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C58 \Description Contiguous relation in form of a rule. $$ {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, b, c, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -c + {a\over q} \right) \left( 1 - {b\over q} \right) }\over {\left( 1 - c \right) \left( {a\over q} - {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {b\over q}, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ] + {{\left( 1 - {a\over q} \right) \left( -c + {b\over q} \right) }\over {\left( 1 - c \right) \left( -{a\over q} + {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, b, c q, \Bi\endmatrix ;q, {\displaystyle z} \right ] $$ \Usage Expr/.C58[n1,n2,n3].\newline \rm {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C59 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c {q^2}}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( b - {a\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {a\over q} \right) \left( 1 - {{c q}\over a} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c q}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle q z} \right ] \\+ {{\left( 1 - b \right) \left( 1 - {c\over b} \right) }\over {\left( 1 - {a\over q} \right) \left( 1 - {{c q}\over a} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c q}\over a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C59[m1,m2,m3,m4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C60 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) }\over {\left( b - a \right) \left( 1 - {c\over {a b}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\ -{{\left( 1 - b \right) \left( 1 - {c\over b} \right) }\over {\left( b - a \right) \left( 1 - {c\over {a b}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \endmultline $$ \Usage Expr/.C60[m1,m2,m3,m4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C61 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - b \right) c }\over {a b \left( 1 - {c\over {a q}} \right) \left( 1 - {c\over {b q}} \right) q}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {c\over {a q}}, b q, {c\over {b q}}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over q} \right) \left( 1 - {c\over {a b q}} \right) }\over {\left( 1 - {c\over {a q}} \right) \left( 1 - {c\over {b q}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}} , a, {c\over {a q}}, b, {c\over {b q}}, \Ai\\ \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C61[m1,m2,m3,m4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C62 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {{c q^2}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b/q, {c*q\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( {b\over q} - a \right) \left( 1 - {c*q\over {a b}} \right) \left( 1 - c q \right) z }} {{\prodl_{i = 1}^{r-4}(1 - \ai) }\over {\prodl_{i = 1}^{s}(1 - \bi)}} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b, {{c q^2}\over b}, c {q^2}, \qAi\\ \let \over / c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C62[m1,m2,m3,m4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C63 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c {q^2}}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over {d q}} \right) \left( 1 - {{c q}\over {a d}} \right) }\over {\left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) \left( 1 - {a\over q} \right) \left( 1 - {{c q}\over a} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c q}\over a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( -b + {a\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) d \left( 1 - {a\over q} \right) \left( 1 - {{c q}\over a} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q}, {{c q}\over a}, b, {c\over b}, d q, {{c q}\over d}, \Ai\\ \let \over / d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C63[m1,m2,m3,m4,$d$].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} are the positions of the special upper parameters, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C64 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over {a b}} \right) \left( b - {a\over q} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ]\\ + {{\left( 1 - b \right) \left( 1 - {c\over {b q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over a}, b q, \Ai\\ \let \over / {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C64[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Example \MATH In[1]:= ph[% \MATHlbrace A,B,q*C,D% \MATHrbrace ,% \MATHlbrace E,A*C/D,F% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck A, B, C q, D % \MATHruEck % \MATHvStrich % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich A C ; q, z % \MATHvStrich 4 3% \MATHvStrich E, ---, F % \MATHvStrich % \MATHloEck D % \MATHroEck \goodbreakpoint% In[2]:= \%/.C64[3,1,4,2] \goodbreakpoint% A % \MATHluEck C, -, D, B % \MATHruEck % \MATHvStrich q % \MATHvStrich C A D \MATHphi % \MATHvStrich ; q, q z % \MATHvStrich (-; q) (---; q) 4 3% \MATHvStrich A C % \MATHvStrich D 1 D q 1 % \MATHloEck ---, E, F % \MATHroEck D Out[2]= --------------------------------------------- + A (C; q) (-; q) 1 q 1 A % \MATHluEck C, -, D q, B % \MATHruEck % \MATHvStrich q % \MATHvStrich A C \MATHphi % \MATHvStrich ; q, z % \MATHvStrich (D; q) (---; q) 4 3% \MATHvStrich A C % \MATHvStrich 1 D q 1 % \MATHloEck ---, E, F % \MATHroEck D q \MATHgroesser ------------------------------------------- A (C; q) (-; q) 1 q 1 \goodbreakpoint% In[3]:= \%/.pqaufl \goodbreakpoint% A % \MATHluEck C, -, D, B % \MATHruEck C A % \MATHvStrich q % \MATHvStrich (1 - -) D (1 - ---) \MATHphi % \MATHvStrich ; q, q z % \MATHvStrich D D q 4 3% \MATHvStrich A C % \MATHvStrich % \MATHloEck ---, E, F % \MATHroEck D Out[3]= --------------------------------------------- + A (1 - C) (1 - -) q A % \MATHluEck C, -, D q, B % \MATHruEck A C % \MATHvStrich q % \MATHvStrich (1 - D) (1 - ---) \MATHphi % \MATHvStrich ; q, z % \MATHvStrich D q 4 3% \MATHvStrich A C % \MATHvStrich % \MATHloEck ---, E, F % \MATHroEck D q \MATHgroesser ------------------------------------------- A (1 - C) (1 - -) q \endMATH \rm The third, first, and fourth upper parameters in {\tt Out[1]} are {\tt q C}, {\tt A}, and {\tt D}, respectively, the second lower parameter in {\tt Out[1]} is $\frac {\text{\tt A C}}{\text {{\tt D}}}$. Hence {\tt C64} can be applied with the replacements $a\to \text {{\tt q C}}$, $b\to \text {{\tt D}}$ and $c\to \text {{\tt A C}}$. \Seealso ContigListe, Ers, PosListe. \Name C65 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) }\over {\left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( {b\over q}-a \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ] \endmultline $$ \Usage Expr/.C65[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C66 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) }\over {\left( b - a \right) \left( 1 - {c\over {a b}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\ -{{\left( 1 - b \right) \left( 1 - {c\over b} \right) }\over {\left( b - a \right) \left( 1 - {c\over {a b}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b q, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \endmultline $$ \Usage Expr/.C66[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C67 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - b \right) c }\over {a b \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {c\over a}, b q, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, {c\over a}, b, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C67[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C68 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{a b \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }\over {c \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) q}} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {{c q}\over a}, {b\over q}, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{a b \left( 1 - {c\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {c \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) q}} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}} , {a\over q}, {c\over a}, {b\over q}, \Ai\\ \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C68[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C69 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( {b\over q}-a \right) \left( 1 - c q \right) \left( 1 - {{c q}\over {a b}} \right) z }\over {\left( 1 - {{c q}\over b} \right) \left( 1 - {{c {q^2}}\over b} \right) }} {{\prodl_{i = 1}^{r-3}(1 - \ai) }\over {\prodl_{i = 1}^{s-1}(1 - \bi) }} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b, c {q^2}, \qAi\\ \let \over / {{c {q^3}}\over b}, c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C69[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C70 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over a}, b q, \Ai\\ \let \over / {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - c \right) \left( 1 - {c\over {a b}} \right) \left( b - {a\over q} \right) z }\over {\left( 1 - {c\over b} \right) \left( 1 - {c\over {b q}} \right) }} {{\prodl_{i = 1}^{r-3}(1 - \ai) }\over {\prodl_{i = 1}^{s-1}(1 - \bi) }} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b q, c q, \qAi\\ \let \over / {{c q}\over b}, c, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C70[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C71 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - b \right) \left( 1 - {c\over {a d}} \right) \left( 1 - {c\over {b q}} \right) \left( 1 - {a\over {d q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {a\over q} \right) \left( 1 - {c\over {b d q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over a}, b q, \Ai\\ \let \over / {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over {a b}} \right) \left( 1 - d \right) \left( -b + {a\over q} \right) \left( 1 - {c\over {d q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) d \left( 1 - {a\over q} \right) \left( 1 - {c\over {b d q}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q}, {c\over a}, b, d q, {c\over d}, \Ai\\ \let \over / {c\over b}, d, {c\over {d q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C71[m1,m2,m3,n1,$d$].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C72 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over {d q}} \right) \left( 1 - {{c q}\over {b d}} \right) }\over {\left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) \left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, {b\over q}, \Ai\\ \let \over / {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( a - {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d \left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / a, {c\over a}, {b\over q}, d q, {{c q}\over d}, \Ai\\ \let \over / {{c {q^2}}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C72[m1,m2,m3,n1,$d$].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C73 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, a b, \Ai\\ \let \over / {{a b}\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {\left( 1 - {{a b}\over q} \right) z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 3} (1 - {{\ai}\over q} )}} {} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / a, b, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {\left( 1 - {{a b}\over q} \right) z }} {{\prodl_{i = 1}^{s - 1} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 3} (1 - {{\ai}\over q} )}} {} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / {a\over q}, {b\over q}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ] \endmultline$$ \Usage Expr/.C73[m1,m2,m3,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C74 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {{a b}\over c} \right) c \left( -b + {a\over q} \right) }\over {a b \left( 1 - {c\over a} \right) \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b, \Ai\\ \let \over / {{c q}\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ] \\+ {{\left( 1 - b \right) \left( 1 - {c\over {b q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\ \let \over / {{c q}\over a}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline$$ \Usage Expr/.C74[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C75 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) a b \left( 1 - {c\over {a q}} \right) q }\over {\left( a - b \right) c \left( 1 - {{a b q}\over c} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let \over / {c\over {a q}}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\ -{{a \left( 1 - b \right) b \left( 1 - {c\over {b q}} \right) q }\over {\left( a - b \right) c \left( 1 - {{a b q}\over c} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let \over / {c\over a}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \endmultline$$ \Usage Expr/.C75[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C76 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - b \right) c }\over {a b \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, b q, \Ai\\ \let \over / {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, b, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C76[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C77 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{a b \left( 1 - {c\over {a q}} \right) \left( 1 - {c\over {b q}} \right) }\over {c \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {b\over q}, \Ai\\ \let \over / {c\over {a q}}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{a b \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over {{q^2}}} \right) }\over {c \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}}, -{\sqrt{c}}, {a\over q}, {b\over q}, \Ai\\ \let \over / {{{\sqrt{c}}}\over q}, -{{{\sqrt{c}}}\over q}, {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C77[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C78 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {b\over q}, \Ai\\ \let \over / {c\over {a q}}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\+ {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - c \right) \left( 1 - {c\over {a b}} \right) \left( -a + {b\over q} \right) z }\over {\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {c\over {a q}} \right) \left( 1 - {{c q}\over b} \right) }} {{\prodl_{i = 1}^{r - 2} (1 - \ai )}\over {\prodl_{i = 1}^{s - 2} (1 - \bi )}} {} _{r+1} \phi _{s+1} \! \left [ \matrix \let \over / a q, b, c q, \qAi\\ \let \over / {{c q}\over a}, {{c {q^2}}\over b}, c, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline$$ \Usage Expr/.C78[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C79 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - b \right) \left( 1 - {c\over {a d}} \right) \left( 1 - {c\over {b q}} \right) \left( 1 - {a\over {d q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {a\over q} \right) \left( 1 - {c\over {b d q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, b q, \Ai\\ \let \over / {{c q}\over a}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over {a b}} \right) \left( 1 - d \right) \left( -b + {a\over q} \right) \left( 1 - {c\over {d q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) d \left( 1 - {a\over q} \right) \left( 1 - {c\over {b d q}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q}, b, d q, {c\over d}, \Ai\\ \let \over / {{c q}\over a}, {c\over b}, d, {c\over {d q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C79[m1,m2,n1,n2,$d$].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Example \MATH In[1]:= ph[% \MATHlbrace A,B,B*F/E,D% \MATHrbrace ,% \MATHlbrace E,F% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck B F % \MATHruEck % \MATHvStrich A, B, ---, D % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich E ; q, z % \MATHvStrich 4 2% \MATHvStrich % \MATHvStrich % \MATHloEck E, F % \MATHroEck \goodbreakpoint% In[2]:= \%/.C79[2,3,2,1,x] \goodbreakpoint% B B F q % \MATHluEck -, -----, A, D % \MATHruEck B F E F B % \MATHvStrich q E % \MATHvStrich (1 - ---) (1 - -) (1 - -) (1 - ---) \MATHphi % \MATHvStrich ; q, z % \MATHvStrich E q x q x 4 2% \MATHvStrich E % \MATHvStrich % \MATHloEck F q, - % \MATHroEck q Out[2]= --------------------------------------------------------------- + B B F E (1 - F) (1 - -) (1 - ---) (1 - ---) q E x q x \goodbreakpoint% \MATHgroesser B B F B F % \MATHluEck -, ---, q x, ---, A, D % \MATHruEck E B F B B F % \MATHvStrich q E x % \MATHvStrich (1 - -) (-(---) + -) (1 - ---) (1 - x) \MATHphi % \MATHvStrich ; q, z % \MATHvStrich B E q q x 6 4% \MATHvStrich B F % \MATHvStrich % \MATHloEck F q, E, x, --- % \MATHroEck q x \MATHgroesser -------------------------------------------------------------------------- B B F E (1 - F) (1 - -) (1 - ---) (1 - ---) x q E x q x \endMATH \rm The second and third upper parameters in {\tt Out[1]} are {\tt B}, and $\frac {\text{\tt B F}}{\text {\tt E}}$, respectively, the second and first lower parameters in {\tt Out[1]} are {\tt F} and {\tt E}, respectively. Hence {\tt C79} can be applied with the replacements $a\to \text {{\tt B}}$, $b\to \frac {\text{\tt B F}}{\text {\tt E}}$ and $c\to \text {{\tt B F}}$. The {\tt x} replaces $d$ in the right-hand side expression. \Seealso ContigListe, Ers, PosListe. \Name C80 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) }\over {\left( 1 - {a\over q} \right) \left( 1 - {c\over {a q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ]\\ + {{\left( 1 - {b\over q} \right) \left( 1 - {c\over {b q}} \right) }\over {\left( 1 - {a\over q} \right) \left( 1 - {c\over {a q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, \Ai\\ \let \over / {b\over q}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C80[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C81 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, \Ai\\ \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( b - a \right) \left( 1 - {c\over {a b}} \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ] \endmultline $$ \Usage Expr/.C81[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C82 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) }\over {\left( {b\over q}-a \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, \Ai\\ \let \over / b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ]\\ -{{\left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( {b\over q}-a \right) \left( 1 - {{c q}\over {a b}} \right) } } {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \endmultline $$ \Usage Expr/.C82[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C83 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) c \left( 1 - {b\over q} \right) }\over {a b \left( 1 - {c\over b} \right) \left( 1 - {c\over {a q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {c\over {a q}}, \Ai\\ \let \over / {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over {a b}} \right) \left( 1 - {c\over q} \right) }\over {\left( 1 - {c\over b} \right) \left( 1 - {c\over {a q}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}} , a, {c\over {a q}}, \Ai\\ \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C83[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C84 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, \Ai\\ \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( b - a \right) \left( 1 - {c\over {a b}} \right) \left( 1 - c q \right) z } \over {\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - b q \right) \left( 1 - {{c q}\over b} \right) }} {{\prodl_{i = 1}^{r-2}(1 - \ai) }\over {\prodl_{i = 1}^{s-2}(1 - \bi) }} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, c {q^2}, \qAi\\ \let \over / b {q^2}, {{c {q^2}}\over b}, c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C84[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C85 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, \Ai\\ \let \over / {b\over q}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {b\over q} \right) \left( 1 - {c\over {b q}} \right) }} {{\prodl_{i = 1}^{r-2}(1 - \ai) }\over {\prodl_{i = 1}^{s-2}(1 - \bi) }} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a, {c\over a}, c, \qAi\\ \let \over / b q, {{c q}\over b}, {c\over q}, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C85[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C86 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {b\over q} \right) \left( 1 - {c\over {b q}} \right) \left( 1 - {a\over {d q}} \right) \left( 1 - {c\over {a d q}} \right) }\over {\left( 1 - {a\over q} \right) \left( 1 - {c\over {a q}} \right) \left( 1 - {b\over {d q}} \right) \left( 1 - {c\over {b d q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, \Ai\\ \let \over / {b\over q}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over {a b}} \right) \left( 1 - d \right) \left( 1 - {c\over {d {q^2}}} \right) \left( {a\over q} - {b\over q} \right) }\over {d \left( 1 - {a\over q} \right) \left( 1 - {c\over {a q}} \right) \left( 1 - {b\over {d q}} \right) \left( 1 - {c\over {b d q}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, d q, {c\over {d q}}, \Ai\\ \let \over / b, {c\over b}, d, {c\over {d {q^2}}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C86[m1,m2,n1,n2,$d$].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C87 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, {{c q}\over a}, \Ai\\ \let \over / b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d}} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / a, {c\over a}, d q, {{c q}\over d}, \Ai\\ \let \over / b q, {{c q}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C87[m1,m2,n1,n2,$d$].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C88 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {{a b}\over q}, \Ai\\ \let \over / b, {{a b}\over {{q^2}}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( 1 - {{a b}\over {{q^2}}} \right) z }} {{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}} {} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over / {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]\\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( 1 - {{a b}\over {{q^2}}} \right) z }} {{\prodl_{i = 1}^{s - 2} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 2} (1 - {{\ai}\over q} )}} {} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\ \let \over / {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ] \endmultline$$ \Usage Expr/.C88[m1,m2,n1,n2].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C89 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( -{a\over q} + {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {{c q}\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ]\\ + {{\left( 1 - {c\over b} \right) \left( 1 - {b\over q} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {{c q}\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C89[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C90 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {c\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( b - a \right) \left( 1 - {c\over {a b}} \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c q}\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ] \endmultline $$ \Usage Expr/.C90[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C91 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over {a q}} \right) }\over {\left( 1 - {c\over {a b}} \right) \left( {b\over q}-a \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {c\over {a q}}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\ -{{\left( 1 - {c\over b} \right) \left( 1 - {b\over q} \right) }\over {\left( 1 - {c\over {a b}} \right) \left( {b\over q}-a \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \endmultline $$ \Usage Expr/.C91[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C92 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) c \left( 1 - {b\over q} \right) q }\over {a b \left( 1 - {c\over a} \right) \left( 1 - {{c q}\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {{c q}\over a}, {b\over q}, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - c \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {{c q}\over b} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C92[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C93 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{a b \left( 1 - {c\over a} \right) \left( 1 - {c\over {b q}} \right) }\over {\left( 1 - b \right) c \left( 1 - {a\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {c\over a}, b q, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{a b \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over q} \right) }\over {\left( 1 - b \right) c \left( 1 - {a\over q} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}} , {a\over q}, \Ai\\ \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, {{c q}\over a}, b q, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C93[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C94 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {c\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( b - a \right) \left( 1 - {c\over {a b}} \right) \left( 1 - c q \right) z }\over {\left( 1 - b \right) \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - b q \right) \left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over b} \right) }} {{\prodl_{i = 1}^{r-1}(1 - \ai) }\over {\prodl_{i = 1}^{s-3}(1 - \bi) }} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a q, c {q^2}, \qAi\\ \let \over / {{c {q^2}}\over a}, b {q^2}, {{c {q^2}}\over b}, c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C94[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C95 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {{c q}\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - c \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) z }\over {\left( 1 - b \right) \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {b\over q} \right) \left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over b} \right) }} {{\prodl_{i = 1}^{r-1}(1 - \ai) }\over {\prodl_{i = 1}^{s-3}(1 - \bi) }} {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / a, c q, \qAi\\ \let \over / {{c {q^2}}\over a}, b q, {{c {q^2}}\over b}, c, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C95[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C96 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over b} \right) \left( 1 - {c\over {a d}} \right) \left( 1 - {b\over q} \right) \left( 1 - {a\over {d q}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {c\over {b d}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over {d q}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / {a\over q}, \Ai\\ \let \over / {{c q}\over a}, {b\over q}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - d \right) \left( {a\over q} - {b\over q} \right) \left( 1 - {c\over {d q}} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {c\over {b d}} \right) d \left( 1 - {a\over q} \right) \left( 1 - {b\over {d q}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {a\over q}, d q, {c\over d}, \Ai\\ \let \over / {{c q}\over a}, b, {{c q}\over b}, d, {c\over {d q}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C96[m1,n1,n2,n3,$d$].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C97 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c q}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {c\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }\over {\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d}} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / a, d q, {{c q}\over d}, \Ai\\ \let \over / {{c q}\over a}, b q, {{c q}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C97[m1,n1,n2,n3,$d$].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C98 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {{a b}\over {{q^2}}}, \Ai\\ \let \over / a, b, {{a b}\over {{q^3}}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\\longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( 1 - {{a b}\over {{q^3}}} \right) z }} {{\prodl_{i = 1}^{s - 3} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}} {} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {a\over {{q^2}}}, {b\over {{q^2}}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \\- {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {a\over {{q^2}}} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} }\over {\left( 1 - {{a b}\over {{q^3}}} \right) z }} {{\prodl_{i = 1}^{s - 3} (1 - {{\bi}\over q} )}\over {\prodl_{i = 1}^{r - 1} (1 - {{\ai}\over q} )}} {} _{r - 1} \phi _{s - 1} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {a\over q}, {b\over q}, \Biq\endmatrix ;q, {\displaystyle {q^{r - s}} z} \right ] \endmultline$$ \Usage Expr/.C98[m1,n1,n2,n3].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C99 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {c\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( {b\over q}-a \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - a \right) \left( 1 - {c\over a} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, {{c q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle q z} \right ]\\ + {{\left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - a \right) \left( 1 - {c\over a} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C99[n1,n2,n3,n4].\newline \rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C100 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {c\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {a\over q} \right) \left( 1 - {c\over {a q}} \right) }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, {c\over {a q}}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \\ -{{\left( 1 - {b\over q} \right) \left( 1 - {c\over {b q}} \right) }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {c\over a}, {b\over q}, {c\over {b q}}, \Bi\endmatrix ;q, {\displaystyle {z\over q}} \right ] \endmultline $$ \Usage Expr/.C100[n1,n2,n3,n4].\newline \rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C101 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {c\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{c \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) q }\over {a b \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / {\sqrt{c}} {\sqrt{q}}, - {\sqrt{c}} {\sqrt{q}} , \Ai\\ \let \over / {{{\sqrt{c}}}\over {{\sqrt{q}}}}, -{{{\sqrt{c}}}\over {{\sqrt{q}}}}, a, {{c q}\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C101[n1,n2,n3,n4].\newline \rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C102 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {{c {q^2}}\over a}, b, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, {{c q}\over a}, b q, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{{{\left( -1 \right) }^{1 - r + s}} \left( b - {a\over q} \right) \left( 1 - c q \right) \left( 1 - {{c q}\over {a b}} \right) z }\over {\left( 1 - a \right) \left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over q} \right) \left( 1 - b q \right) \left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over b} \right) \left( 1 - {{c {q^2}}\over a} \right) }} {{\prodl_{i = 1}^{r}(1 - \ai) }\over {\prodl_{i = 1}^{s-4}(1 - \bi) }}\\ {} _{1 + r} \phi _{1 + s} \! \left [ \matrix \let \over / c {q^2}, \qAi\\ \let \over / a q, {{c {q^3}}\over a}, b {q^2}, {{c {q^2}}\over b}, c q, \qBi\endmatrix ;q, {\displaystyle {q^{1 - r + s}} z} \right ] \endmultline $$ \Usage Expr/.C102[n1,n2,n3,n4].\newline \rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower parameters. \Seealso C64, ContigListe, Ers, PosListe. \Name C103 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {c\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) \left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over {d q}} \right) \left( 1 - {{c q}\over {b d}} \right) }} {} _{r} \phi _{s} \! \left [ \matrix \let \over / \Ai\\ \let \over / a q, {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ + {{\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( a - {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }\over {\left( 1 - a \right) \left( 1 - {c\over a} \right) d \left( 1 - {b\over {d q}} \right) \left( 1 - {{c q}\over {b d}} \right) }} {} _{2 + r} \phi _{2 + s} \! \left [ \matrix \let \over / d q, {{c q}\over d}, \Ai\\ \let \over / a q, {{c q}\over a}, b, {{c {q^2}}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C103[n1,n2,n3,n4,$d$].\newline \rm {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special lower parameters, $d$ is the additional parameter at the right hand side. \Seealso C79, ContigListe, Ers, PosListe. \Name C104 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over b}, c, \Ai\\ \let \over / {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{\prodl_{i = 1}^{s-1}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-5}(1 - {{\ai}\over q} )}} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, b, {c\over b}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\ -{{{{\left( -1 \right) }^{1 - r + s}} {q^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{\prodl_{i = 1}^{s-1}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-5}(1 - {{\ai}\over q} )}} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {c\over a}, {b\over q}, {c\over {b q}}, \Aiq\\ \let \over / \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ] \endmultline $$ \Usage Expr/.C104[m1,m2,m3,m4,m5,n1].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5} and {\tt n1} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C105 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, c, \Ai\\ \let \over / {{c q}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over b} \right) \left( 1 - {c\over {b q}} \right) {q^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{\prodl_{i = 1}^{s-2}(1 - {{\bi}\over q}) }\over {\prodl_{i = 1}^{r-4}(1 - {{\ai}\over q})}} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, b, \Aiq\\ \let \over / {c\over {b q}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ] \\ -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over b} \right) \left( 1 - {c\over {b q}} \right) {q^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{\prodl_{i = 1}^{s-2}(1 - {{\bi}\over q}) }\over {\prodl_{i = 1}^{r-4}(1 - {{\ai}\over q})}} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {c\over a}, {b\over q}, \Aiq\\ \let \over / {c\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ] \endmultline $$ \Usage Expr/.C105[m1,m2,m3,m4,n1,n2].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C106 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, c, \Ai\\ \let \over / {{c q}\over a}, {{c q}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow \\ {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {c\over {a q}} \right) \left( 1 - {c\over {b q}} \right) {{ q }^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{ \prodl_{i = 1}^{s-3}1 - {{\bi}\over q} }\over { \prodl_{i = 1}^{r-3}1 - {{\ai}\over q} }} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q}, b, \Aiq\\ \let \over / {c\over a}, {c\over {b q}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ]\\ -{{{{ (-1) }^{1 - r + s}} \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {c\over {a q}} \right) \left( 1 - {c\over {b q}} \right) {{ q }^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{ \prodl_{i = 1}^{s-3}1 - {{\bi}\over q} }\over { \prodl_{i = 1}^{r-3}1 - {{\ai}\over q} }} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {b\over q}, \Aiq\\ \let \over / {c\over {a q}}, {c\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1 + r - s}} z} \right ] \endmultline $$ \Usage Expr/.C106[m1,m2,m3,n1,n2,n3]. \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C107 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, c, \Ai\\ \let \over / b q, {{c q}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {b\over q} \right) \left( 1 - {c\over {b q}} \right) {q^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{\prodl_{i = 1}^{s-3}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-3}(1 - {{\ai}\over q} )}} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q}, {c\over {a q}}, \Aiq\\ \let \over / {b\over q}, {c\over {b q}}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\ -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {b\over q} \right) \left( 1 - {c\over {b q}} \right) {q^{1 - r + s}} }\over {\left( 1 - {c\over {a b}} \right) \left( -{a\over q} + {b\over q} \right) \left( 1 - {c\over q} \right) z }} {{\prodl_{i = 1}^{s-3}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-3}(1 - {{\ai}\over q} )}} {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, {c\over a}, \Aiq\\ \let \over / b, {c\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ] \endmultline $$ \Usage Expr/.C107[m1,m2,m3,n1,n2,n3].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C108 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, c, \Ai\\ \let \over / {{c q}\over a}, b, {{c {q^2}}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) \left( 1 - {c\over {a q}} \right) {q^{1 - r + s}} \left( 1 - {{c q}\over b} \right) }\over {\left( {b\over {{q^2}}} - {a\over q} \right) \left( 1 - {c\over q} \right) \left( 1 - {{c q}\over {a b}} \right) z }} {{\prodl_{i = 1}^{s-4}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-2}(1 - {{\ai}\over q} )}}\\ {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / {a\over q}, \Aiq\\ \let \over / {c\over a}, {b\over {{q^2}}}, {c\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\ -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {b\over q} \right) \left( 1 - {c\over {a q}} \right) {q^{1 - r + s}} \left( 1 - {{c q}\over b} \right) }\over {\left( {b\over {{q^2}}} - {a\over q} \right) \left( 1 - {c\over q} \right) \left( 1 - {{c q}\over {a b}} \right) z }} {{\prodl_{i = 1}^{s-4}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-2}(1 - {{\ai}\over q} )}}\\ {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / a, \Aiq\\ \let \over / {c\over {a q}}, {b\over q}, {{c q}\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ] \endmultline $$ \Usage Expr/.C108[m1,m2,n1,n2,n3,n4].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C109 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / c, \Ai\\ \let \over / a, {{c {q^2}}\over a}, b, {{c {q^2}}\over b}, {c\over q}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ \longrightarrow {{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over {{q^2}}} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} \left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( -{a\over {{q^2}}} + {b\over {{q^2}}} \right) \left( 1 - {c\over q} \right) \left( 1 - {{c {q^2}}\over {a b}} \right) z }} {{\prodl_{i = 1}^{s-5}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-1}(1 - {{\ai}\over q} )}}\\ {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {a\over q}, {{c q}\over a}, {b\over {{q^2}}}, {c\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ]\\ -{{{{\left( -1 \right) }^{1 - r + s}} \left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over {{q^2}}} \right) \left( 1 - {b\over {{q^2}}} \right) \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^{1 - r + s}} \left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( -{a\over {{q^2}}} + {b\over {{q^2}}} \right) \left( 1 - {c\over q} \right) \left( 1 - {{c {q^2}}\over {a b}} \right) z }} {{\prodl_{i = 1}^{s-5}(1 - {{\bi}\over q} ) }\over {\prodl_{i = 1}^{r-1}(1 - {{\ai}\over q} )}}\\ {} _{r-1} \phi _{s-1} \! \left [ \matrix \let \over / \Aiq\\ \let \over / {a\over {{q^2}}}, {c\over a}, {b\over q}, {{c q}\over b}, \Biq\endmatrix ;q, {\displaystyle {q^{-1+r-s}} z} \right ] \endmultline $$ \Usage Expr/.C109[m1,n1,n2,n3,n4,n5].\newline \rm {\tt m1} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C110 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, {c\over a}, b, {c\over b}, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }\over {\left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( 1 - a \right) \left( 1 - b \right) c }\over {a b \left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {c\over a}, b q, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C110[m1,m2,m3,m4,m5,m6,n1,n2].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5}, {\tt m6} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C111 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, {c\over b}, d q, {{c q}\over d}, \Ai\\ \let \over / d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) d }\over {\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b, {c\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d }\over {\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {c\over a}, b q, {{c q}\over b}, \Ai\\ \let \over / \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C111[m1,m2,m3,m4,m5,m6,n1,n2].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5}, {\tt m6} and {\tt n1}, {\tt n2} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C112 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, {c\over a}, b, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }\over {\left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {{c q}\over a}, b, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( 1 - a \right) \left( 1 - b \right) c }\over {a b \left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {c\over a}, b q, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C112[m1,m2,m3,m4,m5,n1,n2,n3].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C113 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, b, d q, {{c q}\over d}, \Ai\\ \let \over / {{c q}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) d }\over {\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {{c q}\over a}, b, \Ai\\ \let \over / {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d }\over {\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {c\over a}, b q, \Ai\\ \let \over / {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C113[m1,m2,m3,m4,m5,n1,n2,n3].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4}, {\tt m5} and {\tt n1}, {\tt n2}, {\tt n3} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C114 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, b, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over a} \right) \left( 1 - {c\over b} \right) }\over {\left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, b, \Ai\\ \let \over / {c\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( 1 - a \right) \left( 1 - b \right) c }\over {a b \left( 1 - c \right) \left( 1 - {c\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, b q, \Ai\\ \let \over / {{c q}\over a}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C114[m1,m2,m3,m4,n1,n2,n3,n4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C115 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, {c\over a}, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over a} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - c \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {{c q}\over a}, \Ai\\ \let \over / b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( 1 - a \right) c \left( 1 - {b\over q} \right) q }\over {a b \left( 1 - c \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {c\over a}, \Ai\\ \let \over / {b\over q}, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C115[m1,m2,m3,m4,n1,n2,n3,n4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C116 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, b, d q, {{c q}\over d}, \Ai\\ \let \over / {{c q}\over a}, {{c q}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) \left( 1 - {b\over d} \right) \left( 1 - {c\over {b d}} \right) d }\over {\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, b, \Ai\\ \let \over / {c\over a}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( 1 - b \right) \left( 1 - {c\over b} \right) \left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d }\over {\left( a - b \right) \left( 1 - {c\over {a b}} \right) \left( 1 - {c\over d} \right) \left( 1 - d \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, b q, \Ai\\ \let \over / {{c q}\over a}, {c\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C116[m1,m2,m3,m4,n1,n2,n3,n4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C117 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, {c\over a}, d q, {{c q}\over d}, \Ai\\ \let \over / b, {{c {q^2}}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) d \left( 1 - {b\over {d q}} \right) \left( 1 - {{c q}\over {b d}} \right) }\over {\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( a - {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, {{c q}\over a}, \Ai\\ \let \over / b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d \left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( a - {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) } } {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, {c\over a}, \Ai\\ \let \over / {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C117[m1,m2,m3,m4,n1,n2,n3,n4].\newline \rm {\tt m1}, {\tt m2}, {\tt m3}, {\tt m4} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C118 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , a, \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, {{c q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {c\over a} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - c \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {c\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{\left( 1 - a \right) c \left( 1 - {b\over q} \right) q }\over {a b \left( 1 - c \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {{c q}\over a}, {b\over q}, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C118[m1,m2,m3,n1,n2,n3,n4,n5].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C119 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / a, d q, {{c q}\over d}, \Ai\\ \let \over / {{c q}\over a}, b, {{c {q^2}}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - a \right) \left( 1 - {c\over a} \right) d \left( 1 - {b\over {d q}} \right) \left( 1 - {{c q}\over {b d}} \right) }\over {\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( a - {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a q, \Ai\\ \let \over / {c\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{\left( 1 - {a\over d} \right) \left( 1 - {c\over {a d}} \right) d \left( 1 - {b\over q} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( a - {b\over q} \right) \left( 1 - {{c q}\over {a b}} \right) } } {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / a, \Ai\\ \let \over / {{c q}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C119[m1,m2,m3,n1,n2,n3,n4,n5].\newline \rm {\tt m1}, {\tt m2}, {\tt m3} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C120 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / {\sqrt{c}} q, - {\sqrt{c}} q , \Ai\\ \let \over / {\sqrt{c}}, -{\sqrt{c}}, a, {{c {q^2}}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over b} \right) }\over {\left( 1 - c \right) \left( 1 - {{c {q^2}}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {{c q}\over a}, b, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ]\\ - {{c \left( 1 - {a\over q} \right) \left( 1 - {b\over q} \right) {q^2} }\over {a b \left( 1 - c \right) \left( 1 - {{c {q^2}}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, {{c {q^2}}\over a}, {b\over q}, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C120[m1,m2,n1,n2,n3,n4,n5,n6].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5}, {\tt n6} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name C121 \Description Contiguous relation in form of a rule. $$\multline {} _{r} \phi _{s} \! \left [ \matrix \let \over / d q, {{c q}\over d}, \Ai\\ \let \over / a, {{c {q^2}}\over a}, b, {{c {q^2}}\over b}, d, {c\over d}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ \longrightarrow {{d \left( 1 - {a\over q} \right) \left( 1 - {b\over {d q}} \right) \left( 1 - {{c q}\over a} \right) \left( 1 - {{c q}\over {b d}} \right) }\over {\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( {a\over q} - {b\over q} \right) \left( 1 - {{c {q^2}}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let \over / {a\over q}, {{c q}\over a}, b, {{c {q^2}}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \\ -{{d \left( 1 - {b\over q} \right) \left( 1 - {a\over {d q}} \right) \left( 1 - {{c q}\over b} \right) \left( 1 - {{c q}\over {a d}} \right) }\over {\left( 1 - {c\over d} \right) \left( 1 - d \right) \left( {a\over q} - {b\over q} \right) \left( 1 - {{c {q^2}}\over {a b}} \right) }} {} _{r-2} \phi _{s-2} \! \left [ \matrix \let \over / \Ai\\ \let \over / a, {{c {q^2}}\over a}, {b\over q}, {{c q}\over b}, \Bi\endmatrix ;q, {\displaystyle z} \right ] \endmultline $$ \Usage Expr/.C121[m1,m2,n1,n2,n3,n4,n5,n6].\newline \rm {\tt m1}, {\tt m2} and {\tt n1}, {\tt n2}, {\tt n3}, {\tt n4}, {\tt n5}, {\tt n6} are the positions of the special upper and lower parameters, respectively. \Seealso C64, ContigListe, Ers, PosListe. \Name ContigListe \Description List of all contiguous relations. \Usage ContigListe. \Name Div \Description Function that divides \hbox{\tt Gleichung} by \hbox{\tt Expr}. \Usage Div[Expr]. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= Div[a\MATHhoch n] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich c 2 1\MATHvStrich \MATHvStrich (-; q) \MATHloEck c \MATHroEck a n Out[2]= ------------------- == ------- n (c; q) a n \goodbreakpoint% In[3]:= Gleichung \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich c 2 1\MATHvStrich \MATHvStrich (-; q) \MATHloEck c \MATHroEck a n Out[3]= ------------------- == ------- n (c; q) a n \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Sub, Hoch, GlTausche, Ers. \Name Drucke \Description Function that directly sends an expression \hbox{\tt Expr} in the Form \hbox{\tt PrintedForm} to the printer. This function only works for DOS-machines with a printer directly connected. \hbox{\tt PrintedForm} is an optional parameter which can be any of the format types (\hbox{\tt InputForm}, \hbox{\tt OutputForm}, \hbox{\tt TeXForm}, \dots). The default is \hbox{\tt OutputForm}. \Usage Drucke[Expr,PrintedForm]. \Seealso TeXMat, TeX, TeXphW. \Name Ers \Description Function for controlled application of rules and functions. \Usage Ers[Expr,Rules,PosList].\newline \rm \hbox{\tt Rules} can be a rule, a list of rules, or a function. \hbox{\tt PosList} must be a list of positions in \hbox{\tt Expr} to which \hbox{\tt Rules} should be applied. For instance, if \hbox{\tt PosList=$\{\{1,2\},\{4\}\}$}, then \hbox{\tt Rules} is applied to \hbox{\tt Expr[[1,2]]} and \hbox{\tt Expr[[4]]} in \hbox{\tt Expr}. If \hbox{\tt PosList=$\{2,3\}$}, then \hbox{\tt Rules} is applied to \hbox{\tt Expr[[2]]} and \hbox{\tt Expr[[3]]} in \hbox{\tt Expr}. The positions of subexpressions can be determined by the function \hbox{\tt PosListe}. If \hbox{\tt Ers} is applied to an equation then the new left-hand and right-hand sides are automatically stored in the variables {\tt LS} and {\tt RS}. \vskip6pt \hangafter0 \hangindent10pt\rm There is an exceptional usage of \hbox{\tt Ers}, namely\newline \hbox{\hskip25pt\tt Ers[Rules]}.\newline \rm In this case the \hbox{\tt Rules} are applied to both sides of the equation that is currently stored in \hbox{\tt Gleichung}. Again, the new left-hand and right-hand sides are automatically stored in the variables {\tt LS} and {\tt RS}. \Example \MATH In[1]:= (-1)\MATHhoch n*pq[a*q,n]*pq[c,k]/pq[1/b,m]/pq[q/d,l] \goodbreakpoint% n (-1) (c; q) (a q; q) k n Out[1]= ----------------------- 1 q (-; q) (-; q) b m d l \goodbreakpoint% In[2]:= PosListe[\%] \goodbreakpoint% n 1 Out[2]= \MATHlbrace \MATHlbrace (-1) , \MATHlbrace \MATHlbrace 1\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace -------, \MATHlbrace \MATHlbrace 2\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace (c; q) , \MATHlbrace \MATHlbrace 3\MATHrbrace % \MATHrbrace \MATHrbrace , 1 k (-; q) b m \goodbreakpoint% 1 \MATHgroesser \MATHlbrace (a q; q) , \MATHlbrace \MATHlbrace 4\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace -------, \MATHlbrace \MATHlbrace 5\MATHrbrace % \MATHrbrace \MATHrbrace \MATHrbrace n q (-; q) d l \goodbreakpoint% In[3]:= Ers[\%\%,neg1,\MATHlbrace 5\MATHrbrace ] \goodbreakpoint% 1 + l n q (-1) (c; q) (a q; q) (------; q) k n d -l Out[3]= ------------------------------------- 1 (-; q) b m \goodbreakpoint% In[4]:= Ers[\%\%\%,neg1,\MATHlbrace 2,4\MATHrbrace ] \goodbreakpoint% m n q (-1) (c; q) (--; q) k b -m Out[4]= ----------------------- q 1 + n (-; q) (a q ; q) d l -n \goodbreakpoint% In[5]:= SUM[\%,\MATHlbrace k,0,Infinity\MATHrbrace ] \goodbreakpoint% 1 + l \MATHinfty n q \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1) (c; q) % (a q; q) (------; q) \MATHbackslash k n d -l Out[5]= \MATHgroesser ------------------------------------- / 1 \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck (-; q) k=0 b m \goodbreakpoint% In[6]:= PosListe[\%,2] \goodbreakpoint% n Out[6]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 2, 2\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace (-1) , \MATHlbrace \MATHlbrace 1, % 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace k, \MATHlbrace \MATHlbrace % 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace \MATHinfty , \MATHlbrace % \MATHlbrace 2, 3\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% 1 \MATHgroesser \MATHlbrace -------, \MATHlbrace \MATHlbrace 1, 2\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace (c; q) , \MATHlbrace \MATHlbrace 1, % 3\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (a q; q) , \MATHlbrace % \MATHlbrace 1, 4\MATHrbrace \MATHrbrace \MATHrbrace , 1 k n (-; q) b m \goodbreakpoint% 1 + l q \MATHgroesser \MATHlbrace (------; q) , \MATHlbrace \MATHlbrace 1, % 5\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace d -l \goodbreakpoint% In[7]:= Ers[\%\%,neg1,\MATHlbrace \MATHlbrace 1,3\MATHrbrace \MATHrbrace ] \goodbreakpoint% 1 + l \MATHinfty n q \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1) % (a q; q) (------; q) \MATHbackslash n d -l Out[7]= \MATHgroesser ----------------------------- / 1 k \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % (-; q) (c q ; q) k=0 b m -k \goodbreakpoint% In[8]:= Ers[\%\%\%,neg1,\MATHlbrace \MATHlbrace 1,2\MATHrbrace ,\MATHlbrace % 1,5\MATHrbrace \MATHrbrace ] \goodbreakpoint% m \MATHinfty n q \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1) % (c; q) (a q; q) (--; q) \MATHbackslash k n b -m Out[8]= \MATHgroesser --------------------------------- / q \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck (-; q) k=0 d l \goodbreakpoint% In[9]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[9]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[10]:= Ers[a-\MATHgroesser q/a] \goodbreakpoint% n a c \MATHluEck q -n \MATHruEck q (---; q) \MATHvStrich -, q \MATHvStrich q n Out[10]= \MATHphi \MATHvStrich a ; q, q \MATHvStrich == ------------ 2 1\MATHvStrich \MATHvStrich n \MATHloEck c \MATHroEck a (c; q) n \goodbreakpoint% In[11]:= Gleichung \goodbreakpoint% n a c \MATHluEck q -n \MATHruEck q (---; q) \MATHvStrich -, q \MATHvStrich q n Out[11]= \MATHphi \MATHvStrich a ; q, q \MATHvStrich == ------------ 2 1\MATHvStrich \MATHvStrich n \MATHloEck c \MATHroEck a (c; q) n \goodbreakpoint% In[12]:= PQ \goodbreakpoint% In[13]:= pq[a,4]+1/pq[b,3] \goodbreakpoint% 1 2 3 Out[13]= ---------------------------- + (1 - a) (1 - a q) (1 - a q ) (1 - a q ) 2 (1 - b) (1 - b q) (1 - b q ) \goodbreakpoint% In[14]:= Ers[\%,Expand,\MATHlbrace 2\MATHrbrace ] \goodbreakpoint% 2 2 2 2 3 2 3 3 3 2 4 Out[14]= 1 - a - a q + a q - a q + a q - a q + 2 a q - a q + a q - 3 4 2 5 3 5 3 6 4 6 1 \MATHgroesser a q + a q - a q - a q + a q + ---------------------------- 2 (1 - b) (1 - b q) (1 - b q ) \endMATH \Seealso PosListe, ManipulationsListe, Subst. \Name erw1 \Description \vtab $(a;q)_n \to (a;q)_{m+n}/(aq^n;q)_m$,\\ $(a;q)_n \to (a;q)_\infty/(aq^n;q)_\infty$. \endvtab \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. To apply the second rule, \hbox{\tt m} has to be \hbox{\tt Infinity}. \Usage Expr/.erw1. \Example \MATH In[1]:= pq[a,n,q\MATHhoch 2] \goodbreakpoint% 2 Out[1]= (a; q ) n \goodbreakpoint% In[2]:= \%/.erw1 top-extend by: m \goodbreakpoint% 2 (a; q ) m + n Out[2]= ------------- 2 n 2 (a q ; q ) m \goodbreakpoint% In[3]:= pq[a,m,1/q] \goodbreakpoint% 1 Out[3]= (a; -) q m \goodbreakpoint% In[4]:= \%/.erw1 top-extend by: Infinity \goodbreakpoint% 1 (a;-) q \MATHinfty Out[4]= ------- a 1 (--;-) m q \MATHinfty q \endMATH \Seealso erw2, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe. \Name erw2 \Description \vtab $(a;q)_n \to (a/q^m;q)_{m+n}/(a/q^m;q)_m$,\\ $(a;q)_\infty \to (a/q^m;q)_\infty/(a/q^m;q)_m$. \endvtab \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.erw2. \Example \MATH In[1]:= pq[a,n,q\MATHhoch 2] \goodbreakpoint% 2 Out[1]= (a; q ) n \goodbreakpoint% In[2]:= \%/.erw2 bottom-extend by: m \goodbreakpoint% a 2 (----; q ) 2 m m + n q Out[2]= --------------- a 2 (----; q ) 2 m m q \goodbreakpoint% In[3]:= pqinf[b,1/q] \goodbreakpoint% 1 Out[3]= (b;-) q \MATHinfty \goodbreakpoint% In[4]:= \%/.erw2 bottom-extend by: n \goodbreakpoint% n 1 (b q ;-) q \MATHinfty Out[4]= ---------- n 1 (b q ; -) q n \endMATH \Seealso erw1, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe. \Name Expandq \Description Rule that expands all the exponents in powers. \Usage Expr/.Expandq. \Example \MATH In[1]:= pq[a*q\MATHhoch 3,n]/.trans \goodbreakpoint% 2 -2 - n n n 3 n - (n - n )/2 q Out[1]= (-1) a q (-------; q) a n \goodbreakpoint% In[2]:= \%/.Expandq \goodbreakpoint% 2 -2 - n n n (5 n)/2 + n /2 q Out[2]= (-1) a q (-------; q) a n \endMATH \Seealso SimplifyPQ, MinusOne, SUMExpand, Ers, PosListe. \Name Factorialpq \Description \hbox{\tt Factorialpq[n,k,q]} is the usual $q$-factorial, written in terms of $q$-factorial symbols \hbox{\tt pq}. The parameter \hbox{\tt q} is optional. It will be set equal \hbox{\tt q} if it is omitted. \Usage Factorialpq[n,q] \leavevmode\hphantom{Usa}\rm or: \tt Factorialpq[n]. \Example \MATH In[1]:= Factorialpq[n] \goodbreakpoint% (q; q) n Out[1]= -------- n (1 - q) \goodbreakpoint% In[2]:= Factorialpq[5] \goodbreakpoint% (q; q) 5 Out[2]= -------- 5 (1 - q) \endMATH \Seealso Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialq. \Name Factorialq \Description \hbox{\tt Factorialq[n,k,q]} is the usual $q$-factorial, expanded into a $q$-series, if possible. The parameter \hbox{\tt q} is optional. It will be set equal \hbox{\tt q} if it is omitted. \Usage Factorialq[n,q] \leavevmode\hphantom{Usa}\rm or: \tt Factorialq[n]. \Example \MATH In[1]:= Factorialq[n] \goodbreakpoint% \MATHluEck \MATHruEck Out[1]= \MATHvStrich n \MATHvStrich ! \MATHloEck \MATHroEck q \goodbreakpoint% In[2]:= Factorialq[5] \goodbreakpoint% 2 3 4 5 6 7 8 Out[2]= 1 + 4 q + 9 q + 15 q + 20 q + 22 q + 20 q + 15 q + 9 q + 9 10 \MATHgroesser 4 q + q \endMATH \Seealso Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialpq. \Name Gleichung \Description Is a variable which stores equations. The equation \hbox{\tt Gleichung} can be manipulated using the functions \hbox{\tt Add}, \hbox{\tt Sub}, \hbox{\tt Mal}, \hbox{\tt Div}, \hbox{\tt Hoch}, \hbox{\tt GlTausche}, \hbox{\tt Ers}, and \hbox{\tt SUM[k,m,n]}, where \hbox{\tt m} and \hbox{\tt n} are integers or variables. The last command causes the equation to be summed over \hbox{\tt k} from \hbox{\tt m} to \hbox{\tt n}. The parameter \hbox{\tt k} is optional. It will be set \hbox{\tt kk}, \hbox{\tt ii}, \hbox{\tt jj}, \hbox{\tt ll}, \hbox{\tt mm}, or \hbox{\tt nn}, if it is omitted. \Usage Gleichung. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= Gleichung \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[3]:= Add[1] \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[3]= 1 + \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == 1 + ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[4]:= Gleichung \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[4]= 1 + \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == 1 + ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[5]:= LS=5 \goodbreakpoint% Out[5]= 5 \goodbreakpoint% In[6]:= Gleichung \goodbreakpoint% n c a (-; q) a n Out[6]= 5 == 1 + ---------- (c; q) n \goodbreakpoint% In[7]:= SUM[n,0,m] \goodbreakpoint% m m m n c \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck % \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck % \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck a (-; q) \MATHbackslash \MATHbackslash \MATHbackslash a n Out[7]= \MATHgroesser 5 == \MATHgroesser 1 + \MATHgroesser ---------- / / / (c; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck % \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck n n=0 n=0 n=0 \goodbreakpoint% In[8]:= SUM[0,M] \goodbreakpoint% M m M m M m n c \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck % \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck % \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich % \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich \MATHhStrich % \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck a (-; q) \MATHbackslash \MATHbackslash \MATHbackslash % \MATHbackslash \MATHbackslash \MATHbackslash a n Out[8]= \MATHgroesser \MATHgroesser 5 == \MATHgroesser % \MATHgroesser 1 + \MATHgroesser \MATHgroesser ---------- / / / / / / (c; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck % \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich % \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich \MATHhStrich % \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHroEck n kk=0 n=0 kk=0 n=0 kk=0 n=0 \goodbreakpoint% In[9]:= Gleichung \goodbreakpoint% M m M m M m n c \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck % \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck % \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich % \MATHhStrich \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich \MATHhStrich % \MATHhStrich \MATHruEck \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck a (-; q) \MATHbackslash \MATHbackslash \MATHbackslash % \MATHbackslash \MATHbackslash \MATHbackslash a n Out[9]= \MATHgroesser \MATHgroesser 5 == \MATHgroesser % \MATHgroesser 1 + \MATHgroesser \MATHgroesser ---------- / / / / / / (c; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck % \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich % \MATHhStrich \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich \MATHhStrich % \MATHhStrich \MATHroEck \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHroEck n kk=0 n=0 kk=0 n=0 kk=0 n=0 \endMATH \Seealso SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers, Subst,\linebreak PQSort. \Name GlTausche \Description \hbox{\tt GlTausche} interchanges right-hand and left-hand sides in \hbox{\tt Gleichung}. \Usage GlTausche. \Example \MATH In[1]:= Sgl3201 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% -n c c % \MATHluEck a, b, q % \MATHruEck (-; q) (-; q) % \MATHvStrich % \MATHvStrich a n b n Out[1]= \MATHphi % \MATHvStrich 1 - n; q, q % \MATHvStrich == ----------------- 3 2% \MATHvStrich a b q % \MATHvStrich c % \MATHloEck c, ---------- % \MATHroEck (c; q) (---; q) c n a b n \goodbreakpoint% In[2]:= Gleichung \goodbreakpoint% -n c c % \MATHluEck a, b, q % \MATHruEck (-; q) (-; q) % \MATHvStrich % \MATHvStrich a n b n Out[2]= \MATHphi % \MATHvStrich 1 - n; q, q % \MATHvStrich == ----------------- 3 2% \MATHvStrich a b q % \MATHvStrich c % \MATHloEck c, ---------- % \MATHroEck (c; q) (---; q) c n a b n \goodbreakpoint% In[3]:= GlTausche \goodbreakpoint% c c -n (-; q) (-; q) % \MATHluEck a, b, q % \MATHruEck a n b n % \MATHvStrich % \MATHvStrich Out[3]= ----------------- == \MATHphi % \MATHvStrich 1 - n; q, q % \MATHvStrich c 3 2% \MATHvStrich a b q % \MATHvStrich (c; q) (---; q) % \MATHloEck c, ---------- % \MATHroEck n a b n c \goodbreakpoint% In[4]:= Gleichung \goodbreakpoint% c c -n (-; q) (-; q) % \MATHluEck a, b, q % \MATHruEck a n b n % \MATHvStrich % \MATHvStrich Out[4]= ----------------- == \MATHphi % \MATHvStrich 1 - n; q, q % \MATHvStrich c 3 2% \MATHvStrich a b q % \MATHvStrich (c; q) (---; q) % \MATHloEck c, ---------- % \MATHroEck n a b n c \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, Sub, Hoch, Ers, Subst. \Name Hoch \Description Function that takes \hbox{\tt Gleichung} to the \hbox{\tt Expr}-th power. \Usage Hoch[Expr]. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= Hoch[3] \goodbreakpoint% 3 n c 3 \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich 3 a n Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ------------- 2 1\MATHvStrich \MATHvStrich 3 \MATHloEck c \MATHroEck (c; q) n In[3]:= Gleichung \goodbreakpoint% 3 n c 3 \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich 3 a n Out[3]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ------------- 2 1\MATHvStrich \MATHvStrich 3 \MATHloEck c \MATHroEck (c; q) n \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, Sub, GlTausche, Ers. \Name hypqAttributes \Description Shows the current setup of the session. The setup can be changed by the switches \hbox{\tt PQ}, \hbox{\tt phCancel}, \hbox{\tt TeX}, and \hbox{\tt TeXphW}. The default-setup is shown in the following Example. \Usage hypqAttributes. \Example \MATH In[1]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \endMATH \Seealso PQ, phCancel, TeX, TeXphW. \Name inv1 \Description $(a;q)_n \to (1/a;1/q)_n(-a)^nq^{\binom n2}$. \Usage Expr/.inv1. \Example \MATH In[1]:= pq[a\MATHhoch 2,n] \goodbreakpoint% 2 Out[1]= (a ; q) n \goodbreakpoint% In[2]:= \%/.inv1 \goodbreakpoint% n 2 n -2 1 (-1) a (a ; -) q n Out[2]= -------------------- 2 (n - n )/2 q \endMATH \Seealso inv2, Ers, PosListe, ManipulationsListe. \Name inv2 \Description $(a;q)_n \to (aq^{n-1};1/q)_n$. \Usage Expr/.inv2. \Example \MATH In[1]:= pq[a\MATHhoch 2,n] \goodbreakpoint% 2 Out[1]= (a ; q) n \goodbreakpoint% In[2]:= \%/.inv2 \goodbreakpoint% 2 -1 + n 1 Out[2]= (a q ; -) q n \endMATH \Seealso inv1, Ers, PosListe, ManipulationsListe. \Name LaTeX \Description Switch that changes the output of TeXForm to be usable with Plain-{\TeX} and \LaTeX. By default the output of TeXForm is usable with \AmSTeX. \Usage LaTeX. \Example \MATH In[1]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AMSTeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[2]:= TeXForm[ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]] \goodbreakpoint% Out[2]//TeXForm= \MATHlbrace \MATHrbrace \MATHtief \MATHlbrace 2\MATHrbrace % \MATHbackslash phi \MATHtief \MATHlbrace 1\MATHrbrace \MATHbackslash % ! \MATHbackslash left [ \MATHbackslash matrix \MATHbackslash let % \MATHbackslash over / a, b\MATHbackslash \MATHbackslash \MATHbackslash let % \MATHbackslash over / c\MATHbackslash endmatrix ;q, \MATHlbrace \MATHbackslash displaystyle % z\MATHrbrace \MATHbackslash right ] \goodbreakpoint% In[3]:= LaTeX \goodbreakpoint% In[4]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with Plain-TeX and LaTeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[5]:= TeXForm[ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z]] \goodbreakpoint% Out[5]//TeXForm= \MATHlbrace \MATHrbrace \MATHtief \MATHlbrace 2\MATHrbrace % \MATHbackslash phi \MATHtief \MATHlbrace 1\MATHrbrace \MATHbackslash % ! \MATHbackslash left [ \MATHbackslash matrix \MATHlbrace % \MATHbackslash let \MATHbackslash over / a, b\MATHbackslash cr % \MATHbackslash let \MATHbackslash over / c\MATHrbrace ;q, \MATHlbrace \MATHbackslash displaystyle z\MATHrbrace % \MATHbackslash right ] \endMATH \Seealso AmSTeX, AmSLaTeX, TeX, TeXMat, TeXphW. \Name Limes \Description Function for doing formal limits of basic hypergeometric expressions. If required for taking the limit, you will be asked whether or not the absolute value of some variable or expression is smaller than 1. You will be offered three options, \hbox{\tt [y|n|u]}. If the absolute value of the variable is smaller than 1 then enter {\tt y}, if it is greater than 1 then enter {\tt n}, if you do not want to make an explicit declaration then enter {\tt u} (for ``undetermined"). Your decision, if explicit, is stored for the rest of your MATHEMATICA session. If you want to change your decision later, use \hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller}, or \hbox{\tt AbsUndetermined}, respectively. By default the absolute value of {\tt q} is defined to be smaller than 1. Also this can be changed by \hbox{\tt AbsGreater}, \hbox{\tt AbsSmaller}, or \hbox{\tt AbsUndetermined}, respectively. \vskip6pt \hangafter0 \hangindent10pt\rm If you want to let a base $q$ tend to 1, then you should also have the basic file \hbox{\tt hyp.m} of the MATHEMATICA package HYP in your MAHEMATICA input directory, since in these situations the file \hbox{\tt hyp.m} will be loaded automatically. The \hbox{\tt \$ContextPath} (cf\. \cite{\WolfAA}) is set that objects of the package HYP do not override the respective objects of the package HYPQ. The situation is just as if you would have loaded \hbox{\tt hyp.m} first and then \hbox{\tt hyp.q}. Concerning a simultaneous use of the packages HYPQ and HYP confer the section {\it Simultaneous use of\/ {\rm HYP} and\/ {\rm HYPQ}} at the beginning of this handbook. \vskip6pt \hangafter1 \hangindent10pt\rm Hint: When using \hbox{\tt Limes} you sometimes have to do a little bit of ``preparation". For instance, when applying \hbox{\tt Limes[\dots,Regel]} to expressions of the form $(a;q)_n$, where $a$ and $n$ tend to $\infty$ under \hbox{\tt Regel}, then the package will be unable to do it, unless at least $1/aq^n$ tends to a definite limit. Hence, you will sometimes have to use \hbox{\tt zerl} before being able to apply \hbox{\tt Limes} (see the last example). \vskip6pt \hangafter1 \hangindent10pt\rm \underbar{Warning}: This function uses primitive algebraic techniques to do the limit. There is no check if taking the limit is actually allowed. So it is left to you to check the validity of a result of \hbox{\tt Limes}. \Usage Limes[Expr, x->x0]. \Example The derivation of the identity \cite{\GaRaAA, (2.7.6)} that leads to the Rogers--Ramanujan identities, starting from Watson's transformation Tgl8702. \vskip10pt \MATH In[1]:= Tgl8702 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% \MATHluEck -n 2 2 + n \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q), b, c, d, e, q a q Out[1]= \MATHphi \MATHvStrich ; q, % --------- 8 7\MATHvStrich a q a q a q a q 1 + n b c d e \MATHloEck Sqrt[a], -Sqrt[a], ---, ---, ---, ---, a q b c d e \goodbreakpoint% a q -n a q \MATHruEck \MATHluEck ---, d, e, q \MATHruEck (a q, ---; q) \MATHvStrich \MATHvStrich b c \MATHvStrich % d e n \MATHgroesser \MATHvStrich == \MATHphi \MATHvStrich % ; q, q \MATHvStrich -------------- \MATHvStrich 4 3\MATHvStrich a q a q d e \MATHvStrich % a q a q \MATHroEck \MATHloEck ---, ---, ---- \MATHroEck (---, % ---; q) b c n d e n a q \goodbreakpoint% In[2]:= Limes[\%,b-\MATHgroesser Infinity] \goodbreakpoint% \MATHluEck -n % 2 2 + n \MATHruEck \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q), c, d, e, q % a q \MATHvStrich Out[2]= \MATHphi \MATHvStrich % ; q, --------- \MATHvStrich == 7 7\MATHvStrich a q a q a q 1 + n % c d e \MATHvStrich \MATHloEck Sqrt[a], -Sqrt[a], 0, ---, ---, ---, a q % \MATHroEck c d e \goodbreakpoint% -n \MATHluEck d, e, q \MATHruEck \MATHvStrich \MATHvStrich a q \MATHphi \MATHvStrich a q d e ; q, q \MATHvStrich (a q; q) (---; q) 3 2\MATHvStrich ---, ---- \MATHvStrich n d e n \MATHloEck c n \MATHroEck a q \MATHgroesser ------------------------------------------ a q a q (---; q) (---; q) d n e n \goodbreakpoint% In[3]:= Limes[\%,c-\MATHgroesser Infinity] \goodbreakpoint% \MATHluEck -n 2 2 + n \MATHruEck \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q), d, e, q a q \MATHvStrich Out[3]= \MATHphi \MATHvStrich ; q, --------- \MATHvStrich == 6 7\MATHvStrich a q a q 1 + n d e \MATHvStrich \MATHloEck Sqrt[a], -Sqrt[a], 0, 0, ---, ---, a q \MATHroEck d e \goodbreakpoint% -n \MATHluEck d, e, q \MATHruEck \MATHvStrich \MATHvStrich a q \MATHphi \MATHvStrich d e ; q, q \MATHvStrich (a q; q) (---; q) 3 2\MATHvStrich 0, ---- \MATHvStrich n d e n \MATHloEck n \MATHroEck a q \MATHgroesser ------------------------------------------ a q a q (---; q) (---; q) d n e n \goodbreakpoint% In[4]:= Limes[\%,d-\MATHgroesser Infinity] \goodbreakpoint% \MATHluEck -n % 2 2 + n \MATHruEck \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q), e, q % a q \MATHvStrich Out[4]= \MATHphi \MATHvStrich ; % q, --------- \MATHvStrich == 5 7\MATHvStrich a q 1 + n % e \MATHvStrich \MATHloEck Sqrt[a], -Sqrt[a], 0, 0, 0, ---, a q % \MATHroEck e \goodbreakpoint% \MATHluEck 1 + n \MATHruEck \MATHvStrich -n a q \MATHvStrich \MATHphi \MATHvStrich e, q ; q, -------- \MATHvStrich (a q; q) 2 1\MATHvStrich e \MATHvStrich n \MATHloEck 0 \MATHroEck \MATHgroesser ------------------------------------ a q (---; q) e n \goodbreakpoint% In[5]:= Limes[\%,e-\MATHgroesser Infinity] \goodbreakpoint% \MATHluEck -n % \MATHruEck \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q), q % 2 2 + n \MATHvStrich Out[5]= \MATHphi \MATHvStrich % ; q, a q \MATHvStrich == 4 7\MATHvStrich 1 + n % \MATHvStrich \MATHloEck Sqrt[a], -Sqrt[a], 0, 0, 0, 0, a q \MATHroEck \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n 1 + n \MATHvStrich \MATHgroesser \MATHphi \MATHvStrich q ; q, a q \MATHvStrich (a q; q) 1 1\MATHvStrich \MATHvStrich n \MATHloEck 0 \MATHroEck \goodbreakpoint% In[6]:= Limes[\%,n-\MATHgroesser Infinity] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q) 2 2 \MATHvStrich Out[6]= \MATHphi \MATHvStrich % ; q, a q \MATHvStrich == 3 7\MATHvStrich Sqrt[a], -Sqrt[a], 0, 0, 0, 0, 0 \MATHvStrich \MATHloEck \MATHroEck \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich - \MATHvStrich \MATHgroesser (a q;q) \MATHphi \MATHvStrich ; q, a q \MATHvStrich \MATHinfty 0 1\MATHvStrich 0 \MATHvStrich \MATHloEck \MATHroEck \goodbreakpoint% In[7]:= \%/.phSUM/.SUMRegeln \goodbreakpoint% \MATHinfty \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck 2 \MATHbackslash kk 2 kk -kk/2 + (5 kk )/2 Out[7]= \MATHgroesser ((-1) a q (a; q) % (-(Sqrt[a] q); q) / kk kk \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk=0 \goodbreakpoint% \MATHgroesser (Sqrt[a] q; q) ) / ((-Sqrt[a]; q) (Sqrt[a]; % q) (q; q) ) == kk kk kk kk \goodbreakpoint% \MATHinfty 2 \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck kk kk \MATHbackslash a q \MATHgroesser (a q;q) ( \MATHgroesser --------) \MATHinfty / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk kk=0 \goodbreakpoint% In[8]:= PosListe[\%,3] \goodbreakpoint% kk 2 kk Out[8]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 1, 2, % 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (-1) , % \MATHlbrace \MATHlbrace 1, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , % \MATHlbrace a , \MATHlbrace \MATHlbrace 1, 1, 2\MATHrbrace \MATHrbrace % \MATHrbrace , \goodbreakpoint% \MATHgroesser \MATHlbrace kk, \MATHlbrace \MATHlbrace 1, 2, % 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q, \MATHlbrace \MATHlbrace % 2, 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace a q, \MATHlbrace % \MATHlbrace 2, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% 2 -kk/2 + (5 kk )/2 \MATHgroesser \MATHlbrace q , \MATHlbrace \MATHlbrace 1, % 1, 3\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace \MATHinfty , % \MATHlbrace \MATHlbrace 1, 2, 3\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% 1 \MATHgroesser \MATHlbrace \MATHlbrace kk, 0, \MATHinfty \MATHrbrace , % \MATHlbrace \MATHlbrace 2, 2, 2\MATHrbrace \MATHrbrace \MATHrbrace , % \MATHlbrace ---------------, \MATHlbrace \MATHlbrace 1, 1, 4\MATHrbrace \MATHrbrace % \MATHrbrace , (-Sqrt[a]; q) kk \goodbreakpoint% 1 \MATHgroesser \MATHlbrace --------------, \MATHlbrace \MATHlbrace 1, % 1, 5\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (a; q) , \MATHlbrace % \MATHlbrace 1, 1, 6\MATHrbrace \MATHrbrace \MATHrbrace , (Sqrt[a]; q) kk kk \goodbreakpoint% 2 kk kk 1 a q \MATHgroesser \MATHlbrace --------, \MATHlbrace \MATHlbrace 1, 1, % 7\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace --------, \MATHlbrace % \MATHlbrace 2, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , (q; q) (q; q) kk kk \goodbreakpoint% \MATHgroesser \MATHlbrace (-(Sqrt[a] q); q) , \MATHlbrace \MATHlbrace 1, % 1, 8\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (Sqrt[a] q; q) , % \MATHlbrace \MATHlbrace 1, 1, 9\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace kk kk \goodbreakpoint% In[9]:= Ers[\%\%,lina1,\MATHlbrace \MATHlbrace 1,1,4\MATHrbrace ,\MATHlbrace 1,1,5\MATHrbrace % \MATHrbrace ] \goodbreakpoint% \MATHinfty \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck 2 \MATHbackslash kk 2 kk -kk/2 + (5 kk )/2 Out[9]= \MATHgroesser ((-1) a q (a; q) % (-(Sqrt[a] q); q) / kk kk \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk=0 \goodbreakpoint% \MATHgroesser (Sqrt[a] q; q) ) / kk \goodbreakpoint% \MATHgroesser ((1 - Sqrt[a]) (1 + Sqrt[a]) (q; q) (-(Sqrt[a] q); q) kk -1 + kk \goodbreakpoint% \MATHinfty 2 \MATHluEck \MATHhStrich % \MATHhStrich \MATHhStrich \MATHruEck kk kk \MATHbackslash a q \MATHgroesser (Sqrt[a] q; q) ) == (a q;q) ( \MATHgroesser % --------) -1 + kk \MATHinfty / (q; q) \MATHloEck \MATHhStrich % \MATHhStrich \MATHhStrich \MATHroEck kk kk=0 \goodbreakpoint% In[10]:= PosListe[\%,3] \goodbreakpoint% kk Out[10]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 1, 2, % 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (-1) , \MATHlbrace % \MATHlbrace 1, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% 1 1 \MATHgroesser \MATHlbrace -----------, \MATHlbrace \MATHlbrace 1, % 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace -----------, % \MATHlbrace \MATHlbrace 1, 1, 3\MATHrbrace \MATHrbrace \MATHrbrace , 1 - Sqrt[a] 1 + Sqrt[a] \goodbreakpoint% 2 kk \MATHgroesser \MATHlbrace a , \MATHlbrace \MATHlbrace 1, 1, % 4\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace kk, \MATHlbrace \MATHlbrace % 1, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q, \MATHlbrace % \MATHlbrace 2, 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% 2 -kk/2 + (5 kk )/2 \MATHgroesser \MATHlbrace a q, \MATHlbrace \MATHlbrace 2, 1, % 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q % , \MATHlbrace \MATHlbrace 1, 1, 5\MATHrbrace \MATHrbrace \MATHrbrace , % \MATHlbrace \MATHinfty , \MATHlbrace \MATHlbrace 1, 2, 3\MATHrbrace % \MATHrbrace \MATHrbrace , \goodbreakpoint% \MATHgroesser \MATHlbrace \MATHlbrace kk, 0, \MATHinfty \MATHrbrace , % \MATHlbrace \MATHlbrace 2, 2, 2\MATHrbrace \MATHrbrace \MATHrbrace , % \MATHlbrace (a; q) , \MATHlbrace \MATHlbrace 1, 1, 6\MATHrbrace % \MATHrbrace \MATHrbrace , kk \goodbreakpoint% 2 kk kk 1 a q \MATHgroesser \MATHlbrace --------, \MATHlbrace \MATHlbrace 1, 1, % 7\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace --------, \MATHlbrace % \MATHlbrace 2, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , (q; q) (q; q) kk kk \goodbreakpoint% 1 \MATHgroesser \MATHlbrace ------------------------, \MATHlbrace \MATHlbrace % 1, 1, 8\MATHrbrace \MATHrbrace \MATHrbrace , (-(Sqrt[a] q); q) -1 + kk \goodbreakpoint% \MATHgroesser \MATHlbrace (-(Sqrt[a] q); q) , \MATHlbrace % \MATHlbrace 1, 1, 9\MATHrbrace \MATHrbrace \MATHrbrace , kk \goodbreakpoint% 1 \MATHgroesser \MATHlbrace ---------------------, \MATHlbrace \MATHlbrace % 1, 1, 10\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (Sqrt[a] q; q) % , \MATHlbrace \MATHlbrace 1, 1, 11\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace (Sqrt[a] q; q) kk -1 + kk \goodbreakpoint% In[11]:= Ers[\%\%,lina2,\MATHlbrace \MATHlbrace 1,1,9\MATHrbrace ,\MATHlbrace % 1,1,11\MATHrbrace \MATHrbrace ] \goodbreakpoint% \MATHinfty \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck 2 \MATHbackslash kk 2 kk -kk/2 + (5 kk )/2 kk Out[11]= \MATHgroesser ((-1) a q (1 - Sqrt[a] q ) / \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk=0 \goodbreakpoint% kk \MATHgroesser (1 + Sqrt[a] q ) (a; q) ) / ((1 - Sqrt[a]) % (1 + Sqrt[a]) (q; q) ) kk kk \goodbreakpoint% \MATHinfty 2 \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck kk kk \MATHbackslash a q \MATHgroesser == (a q;q) ( \MATHgroesser --------) \MATHinfty / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHroEck kk kk=0 \goodbreakpoint% In[12]:= Ers[\%,ExpandAll,\MATHlbrace \MATHlbrace 1,1\MATHrbrace \MATHrbrace ] \goodbreakpoint% 2 \MATHinfty kk 2 kk (5 kk )/2 \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1) % a q (a; q) \MATHbackslash kk Out[12]= \MATHgroesser --------------------------------- - / kk/2 kk/2 \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck q % (q; q) - a q (q; q) kk=0 kk kk \goodbreakpoint% 2 kk 1 + 2 kk 2 kk + (5 kk )/2 (-1) a q (a; q) kk \MATHgroesser ------------------------------------------- == kk/2 kk/2 q (q; q) - a q (q; q) kk kk \goodbreakpoint% \MATHinfty 2 \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich % \MATHruEck kk kk \MATHbackslash a q \MATHgroesser (a q;q) ( \MATHgroesser --------) \MATHinfty / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk kk=0 \goodbreakpoint% In[13]:= Ers[\%,Factor,\MATHlbrace \MATHlbrace 1,1\MATHrbrace \MATHrbrace ] \goodbreakpoint% 2 \MATHinfty kk 2 kk -kk/2 + (5 kk )/2 2 kk \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1) a q (-1 + a q ) (a; q) \MATHbackslash kk Out[13]= \MATHgroesser ------------------------------------------------------- == / (-1 + a) (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk kk=0 \goodbreakpoint% \MATHinfty 2 \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck kk kk \MATHbackslash a q \MATHgroesser (a q;q) ( \MATHgroesser --------) \MATHinfty / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck kk kk=0 \endMATH \vskip10pt\noindent Examples for letting the base $q$ tend to 1. \vskip10pt \MATH In[1]:= Sgl3201 Do you want to set values for the equation? [y|n]: y a=q\MATHhoch a b=q\MATHhoch b c=q\MATHhoch c n=n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% -a + c -b + c % \MATHluEck a b -n % \MATHruEck (q ; q) (q ; q) % \MATHvStrich q , q , q % \MATHvStrich n n Out[1]= \MATHphi % \MATHvStrich ; q, q % \MATHvStrich == --------------------------- 3 2% \MATHvStrich c 1 + a + b - c - n % \MATHvStrich c -a - b + c % \MATHloEck q , q % \MATHroEck (q ; q) (q ; q) n n \goodbreakpoint% In[2]:= Limes[\%,q-\MATHgroesser 1] \goodbreakpoint% % \MATHluEck % \MATHruEck (-a + c) (-b + c) % \MATHvStrich a, b, -n % \MATHvStrich n n Out[2]= F % \MATHvStrich ; 1 % \MATHvStrich == ------------------- 3 2% \MATHvStrich c, 1 + a + b - c - n % \MATHvStrich (c) (-a - b + c) % \MATHloEck % \MATHroEck n n \goodbreakpoint% In[3]:= Sgl2103 Do you want to set values for the equation? [y|n]: y a=q\MATHhoch a b=q\MATHhoch b c=q\MATHhoch c Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% -a + c -b + c % \MATHluEck a b % \MATHruEck (q , q ; q) % \MATHvStrich q , q -a - b + c % \MATHvStrich \MATHinfty Out[3]= \MATHphi % \MATHvStrich ; q, q % \MATHvStrich == ---------------------- 2 1% \MATHvStrich c % \MATHvStrich c -a - b + c % \MATHloEck q % \MATHroEck (q , q ; q) \MATHinfty \goodbreakpoint% In[4]:= Limes[\%,q-\MATHgroesser 1] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich \MATHGamma (c) \MATHGamma (-a - b + c) Out[4]= F % \MATHvStrich ; 1 % \MATHvStrich == ------------------- 2 1% \MATHvStrich c % \MATHvStrich \MATHGamma (-a + c) \MATHGamma (-b + c) % \MATHloEck % \MATHroEck \endMATH \vskip10pt\noindent An example that should illustrate the hint in the Description of \hbox{\tt Limes}. \vskip10pt \MATH In[1]:= pq[a/q\MATHhoch n,2*n]/a\MATHhoch n*(-1)\MATHhoch n*q\MATHhoch Binomial[n+1,2] \goodbreakpoint% n (n (1 + n))/2 a (-1) q (--; q) n 2 n q Out[1]= ------------------------------- n a \goodbreakpoint% In[2]:= Limes[\%,n-\MATHgroesser Infinity] \goodbreakpoint% The expression \goodbreakpoint% Indeterminate \goodbreakpoint% was obtained. \goodbreakpoint% Therefore the limit n -\MATHgroesser \MATHinfty could not be determined. Here is your expression: \goodbreakpoint% n (n (1 + n))/2 a (-1) q (--; q) n 2 n q Out[2]= ------------------------------- n a \goodbreakpoint% In[3]:= \%/.zerl bottom-split by: n \goodbreakpoint% n (n (1 + n))/2 a (-1) q (a; q) (--; q) n n n q Out[3]= ------------------------------------- n a \goodbreakpoint% In[4]:= Limes[\%,n-\MATHgroesser Infinity] Is 2 n even, odd, or neither of both? [e|o|n]: e \goodbreakpoint% q Out[4]= (a;q) (-;q) \MATHinfty a \MATHinfty \endMATH \Seealso AbsGreater, AbsSmaller, AbsUndetermined, MinusOne. \Name lina1 \Description \vtab $(a;q)_n \to (1-a)(aq;q)_{n-1}$,\\ $(a;q)_\infty \to (1-a)(aq;q)_\infty$. \endvtab \vskip6pt \leavevmode\hskip10pt\Usage Expr/.lina1. \Example \MATH In[1]:= pq[a,m,q\MATHhoch 2] \goodbreakpoint% 2 Out[1]= (a; q ) m \goodbreakpoint% In[2]:= \%/.lina1 \goodbreakpoint% 2 2 Out[2]= (1 - a) (a q ; q ) -1 + m \goodbreakpoint% In[3]:= 1/pqinf[a,q\MATHhoch 2] \goodbreakpoint% 1 Out[3]= ------- 2 (a;q ) \MATHinfty \goodbreakpoint% In[4]:= \%/.lina1 \goodbreakpoint% 1 Out[4]= ------------------ 2 2 (1 - a) (a q ;q ) \MATHinfty \endMATH \Seealso lina2, linz, Ers, PosListe, ManipulationsListe. \Name lina2 \Description $(a;q)_n \to (1-aq^{n-1})(a;q)_{n-1}$. \Usage Expr/.lina2. \Example \MATH In[1]:= pq[a,m,q\MATHhoch 2] \goodbreakpoint% 2 Out[1]= (a; q ) m \goodbreakpoint% In[2]:= \%/.lina2 \goodbreakpoint% -2 + 2 m 2 Out[2]= (1 - a q ) (a; q ) -1 + m \endMATH \Seealso lina1, linz, Ers, PosListe, ManipulationsListe. \Name linz \Description Rule that absorbs linear terms. \Usage Expr/.linz. \Example \MATH In[1]:= (1-a)*pq[a*q\MATHhoch 2,m,q\MATHhoch 2]/(1-a*q\MATHhoch (2*m-2))/pq[a,m-1,q\MATHhoch 2] \goodbreakpoint% 2 2 (1 - a) (a q ; q ) m Out[1]= ------------------------------- -2 + 2 m 2 (1 - a q ) (a; q ) -1 + m \goodbreakpoint% In[2]:= \%/.linz \goodbreakpoint% 2 (a; q ) 1 + m Out[2]= ------------------------------- -2 + 2 m 2 (1 - a q ) (a; q ) -1 + m \goodbreakpoint% In[3]:= \%/.linz \goodbreakpoint% 2 (a; q ) 1 + m Out[3]= ------------ 2 (a; q ) m \goodbreakpoint% In[4]:= 1/(1-b/q)/pqinf[b] \goodbreakpoint% 1 Out[4]= -------------- b (1 - -) (b;q) q \MATHinfty \goodbreakpoint% In[5]:= \%/.linz \goodbreakpoint% 1 Out[5]= ------ b (-;q) q \MATHinfty \endMATH \Seealso lina1, lina2, Ers, PosListe, ManipulationsListe. \Name LS \Description \hbox{\tt LS} is the left-hand side in \hbox{\tt Gleichung}. \Usage LS. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= LS \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich 2 1\MATHvStrich \MATHvStrich \MATHloEck c \MATHroEck \goodbreakpoint% In[3]:= Add[1] \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[3]= 1 + \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == 1 + ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[4]:= LS \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich Out[4]= 1 + \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich 2 1\MATHvStrich \MATHvStrich \MATHloEck c \MATHroEck \goodbreakpoint% In[5]:= LS=pq[a,m] \goodbreakpoint% Out[5]= (a; q) m \goodbreakpoint% In[6]:= Gleichung \goodbreakpoint% n c a (-; q) a n Out[6]= (a; q) == 1 + ---------- m (c; q) n \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, RS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers,\linebreak Subst. \Name Mal \Description Function that multiplies \hbox{\tt Gleichung} by \hbox{\tt Expr}. \Usage Mal[Expr]. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= Mal[pq[c,n,q]] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich n c Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich (c; q) == a (-; q) 2 1\MATHvStrich \MATHvStrich n a n \MATHloEck c \MATHroEck \goodbreakpoint% In[3]:= Gleichung \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich n c Out[3]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich (c; q) == a (-; q) 2 1\MATHvStrich \MATHvStrich n a n \MATHloEck c \MATHroEck \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Add, Div, Sub, Hoch, GlTausche, Ers. \Name ManipulationsListe \Description Gives a list of all available rules for manipulating finite and infinite $q$-factorial symbols. \Usage ManipulationsListe. \Name MinusOne \Description: Rule for getting rid of expressions of the form $(-1)^N$ where $N$ is an even or odd integer. \Usage Expr/.MinusOne. \MATH In[1]:= pq[a,2*n] \goodbreakpoint% Out[1]= (a; q) 2 n \goodbreakpoint% In[2]:= \%/.trans \goodbreakpoint% 1 - 2 n 2 n 2 n q (-1) a (--------; q) a 2 n Out[2]= ----------------------------- 2 (2 n - 4 n )/2 q \goodbreakpoint% In[3]:= \%/.MinusOne Is 2 n even, odd, or neither of both? [e|o|n]: e \goodbreakpoint% 1 - 2 n 2 n q a (--------; q) a 2 n Out[3]= --------------------- 2 (2 n - 4 n )/2 q \endMATH \Seealso SimplifyPQ, Expandq, SUMExpand. \Name Multinomialpq \Description: \hbox{\tt Multinomialq[n1,n2,\dots,q]} is the $q$-multinomial coefficient $\left[\smallmatrix {\sum_i n_i} \\{n_1,n_2,\dots}\endsmallmatrix\right]_q$, written in terms of $q$-factorial symbols {\tt pq}. \Usage Multinomialpq[n1,n2,\dots,q]. \MATH In[1]:= Multinomialpq[a,b,c,s] \goodbreakpoint% (s; s) a + b + c Out[1]= ----------------------- (s; s) (s; s) (s; s) a b c \goodbreakpoint% In[2]:= Multinomialpq[2,3,1,q] \goodbreakpoint% (q; q) 6 Out[2]= ----------------------- (q; q) (q; q) (q; q) 1 2 3 \endMATH \Seealso Multinomialq, Binomialq, Binomialpq, Factorialq, Factorialpq. \Name Multinomialq \Description: \hbox{\tt Multinomialq[n1,n2,\dots,q]} is the $q$-multinomial coefficient $\left[\smallmatrix {\sum_i n_i} \\{n_1,n_2,\dots}\endsmallmatrix\right]_q$, expanded into a $q$-series, if possible. \Usage Multinomialq[n1,n2,\dots,q]. \MATH In[1]:= Multinomialq[2,3,1,s] \goodbreakpoint% 2 3 4 5 6 7 8 9 Out[1]= 1 + 2 s + 4 s + 6 s + 8 s + 9 s + 9 s + 8 s + 6 s + 4 s + 10 11 \MATHgroesser 2 s + s \goodbreakpoint% In[2]:= Multinomialq[a,b,c,q] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a + b + c % \MATHvStrich Out[2]= % \MATHvStrich % \MATHvStrich % \MATHvStrich a, b, c % \MATHvStrich % \MATHloEck % \MATHroEck q \endMATH \Seealso Multinomialpq, Binomialq, Binomialpq, Factorialq, Factorialpq. \Name neg1 \Description $(a;q)_n \to 1/(aq^n;q)_{-n}$. \Usage Expr/.neg1. \Example \MATH In[1]:= pq[a\MATHhoch 2,-n] \goodbreakpoint% 2 Out[1]= (a ; q) -n \goodbreakpoint% In[2]:= \%/.neg1 \goodbreakpoint% 1 Out[2]= ------------ 2 -n (a q ; q) n \endMATH \Seealso neg2, Ers, PosListe, ManipulationsListe. \Name neg2 \Description $(a;q)_n \to q^{\binom {n+1}2}/((-q/a)^n(q/a;q)_{-n})$. \Usage Expr/.neg2. \Example \MATH In[1]:= pq[a\MATHhoch 2,-n] \goodbreakpoint% 2 Out[1]= (a ; q) -n \goodbreakpoint% In[2]:= \%/.neg2 \goodbreakpoint% 2 n/2 + n /2 n q Out[2]= (-1) -------------- 2 n q a (--; q) 2 n a \endMATH \Seealso neg1, Ers, PosListe, ManipulationsListe. \Name Ph \Description \hbox{\tt Ph[List1A,List1B,q1,List2A,List2B,q2,\dots,ListkA,ListkB,qk,z]} is the multibasic basic hypergeometric series with upper parameters \hbox{\tt List}1A and lower parameters \hbox{\tt List}1B for base \hbox{\tt q}1,\dots , with upper parameters \hbox{\tt List}kA and lower parameters \hbox{\tt List}kB for base \hbox{\tt q}k, and argument \hbox{\tt z}. \Usage Ph[List1A,List1B,q1,List2A,List2B,q2,\dots,ListkA,ListkB,qk,z]. \Example \MATH In[1]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,% \MATHlbrace e,f,g% \MATHrbrace ,% \MATHlbrace E,F,G% \MATHrbrace ,p,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b: e, f, g % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso SListe, TListe, SUMRegeln, SUMph, PhSUM, ph, ps, pq, pqinf, phCancel, phOrdne, phPerm,\linebreak phTausche, PQ, phFormat. \Name ph \Description \hbox{\tt ph[List1,List2,q,z]} is the basic hypergeometric series with upper parameters \hbox{\tt List}1, lower parameters \hbox{\tt List}2, base \hbox{\tt q}, and argument \hbox{\tt z}. \Usage ph[List1,List2,q,z]. \Example \MATH In[1]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich a, b \MATHvStrich Out[1]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 2 1\MATHvStrich c \MATHvStrich \MATHloEck \MATHroEck \goodbreakpoint% In[2]:= ph[\MATHlbrace a,b,c\MATHrbrace ,\MATHlbrace d,e,0\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich a, b, c \MATHvStrich Out[2]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 3 3\MATHvStrich d, e, 0 \MATHvStrich \MATHloEck \MATHroEck \endMATH \Seealso SListe, TListe, SUMRegeln, SUMph, phSUM, Ph, ph, pq, pqinf, phCancel, phOrdne, phPerm, phTausche, PQ, phFormat.\NoBlackBoxes\par\BlackBoxes \Name phCancel \Description Switch that activates automatic cancelling of the upper and lower parameters in \hbox{\tt ph[]}, \hbox{\tt Ph[]} and \hbox{\tt ps[]}, or makes it inactive, respectively. By default automatic cancelling is active. \Usage phCancel. \Example \MATH In[1]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[2]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich b \MATHvStrich Out[2]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 1 1\MATHvStrich c \MATHvStrich \MATHloEck \MATHroEck \goodbreakpoint% In[3]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,n,q\MATHhoch 2] \goodbreakpoint% 2 (b; q ) n Out[3]= -------- 2 (c; q ) n \goodbreakpoint% In[4]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ] \goodbreakpoint% (b; q) \MATHinfty Out[4]= ------- (c; q) \MATHinfty \goodbreakpoint% In[5]:= phCancel \goodbreakpoint% In[6]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph inactive. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[7]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich a, b \MATHvStrich Out[7]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 2 2\MATHvStrich a, c \MATHvStrich \MATHloEck \MATHroEck \goodbreakpoint% In[8]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ,n,q\MATHhoch 2] \goodbreakpoint% 2 (b; q ) n Out[8]= -------- 2 (c; q ) n \goodbreakpoint% In[9]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace a,c\MATHrbrace ] \goodbreakpoint% (b; q) \MATHinfty Out[9]= ------- (c; q) \MATHinfty \goodbreakpoint% In[10]:= phCancel \goodbreakpoint% In[11]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \endMATH \Seealso ph, W, hypqAttributes. \Name PhEinf \Description Rule that inactivates automatic cancelling in \hbox{\tt Ph[]} and then adds a parameter which has to be entered on request, together with the information to which base it belongs, to the upper and lower parameters of \hbox{\tt Ph[]}. \Usage Expr/.PhEinf. \Example \MATH In[1]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,% \MATHlbrace e,f,g% \MATHrbrace ,% \MATHlbrace E,F,G% \MATHrbrace ,p,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b: e, f, g % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.PhEinf Add the parameter: A to the parameters belonging to the i-th base, where i=1 \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich A, a, b: e, f, g % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich A, c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[3]:= \%/.PhEinf Add the parameter: B to the parameters belonging to the i-th base, where i=2 \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich A, a, b: B, e, f, g % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich A, c: B, E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso phCancel, phOrdne, PhPerm, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name phEinf \Description Rule that inactivates automatic cancelling in \hbox{\tt ph[]} and then adds a parameter which has to be entered on request to the upper and lower parameters of \hbox{\tt ph[]}. \Usage Expr/.phEinf. \Example \MATH In[1]:= ph[% \MATHlbrace b,c,q% \MATHrbrace ,% \MATHlbrace q*a/c,q*a/b% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich a q a q; q, z % \MATHvStrich 3 2% \MATHvStrich ---, --- % \MATHvStrich % \MATHloEck c b % \MATHroEck \goodbreakpoint% In[2]:= \%/.phEinf Add the parameter: a \goodbreakpoint% % \MATHluEck a, b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich a q a q; q, z % \MATHvStrich 4 3% \MATHvStrich a, ---, --- % \MATHvStrich % \MATHloEck c b % \MATHroEck \endMATH \Seealso phOrdne, phPerm, phTausche, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name phFormat \Description Switch that activates basic hypergeometric output, or makes it inactive, respectively. By default basic hypergeometric output is active. \Usage phFormat. \Example \MATH In[1]:= pq[a,n]/pqinf[b,q\MATHhoch 2]*ph[\MATHlbrace c,d\MATHrbrace ,\MATHlbrace c*d\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich c, d \MATHvStrich \MATHphi \MATHvStrich ; q, z \MATHvStrich (a; q) 2 1\MATHvStrich c d \MATHvStrich n \MATHloEck \MATHroEck Out[1]= ------------------------- 2 (b;q ) \MATHinfty \goodbreakpoint% In[2]:= Tgl5402 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% Format::toobig: Expression too big for output. Enter "phFormat" and retry. \goodbreakpoint% In[3]:= phFormat \goodbreakpoint% In[4]:= \%\% \goodbreakpoint% 1 - n 1 - n 1 - n -n q q q 2 2 2 -2 + 2 n Out[4]= ph[\MATHlbrace q , b, c, d, e\MATHrbrace , \MATHlbrace ------, ------, ------, b c d e q \MATHrbrace , b c d \goodbreakpoint% 1 - 2 n 3/2 - n q q \MATHgroesser q, q] == ph[\MATHlbrace --------, -----------------------, b c d Sqrt[b] Sqrt[c] Sqrt[d] \goodbreakpoint% 3/2 - n 1 - n 1 - n 1 - n q q q q -n/2 -n/2 \MATHgroesser -(-----------------------), ------, ------, ------, q , -q , Sqrt[b] Sqrt[c] Sqrt[d] c d b d b c \goodbreakpoint% 3 - 3 n 1/2 - n/2 1/2 - n/2 q \MATHgroesser q , -q , e, ----------\MATHrbrace , 2 2 2 b c d e \goodbreakpoint% 1/2 - n 1/2 - n 1 - n 1 - n q q q q \MATHgroesser \MATHlbrace -----------------------, -(-----------------------), ------, ------, Sqrt[b] Sqrt[c] Sqrt[d] Sqrt[b] Sqrt[c] Sqrt[d] b c \goodbreakpoint% 1 - n 2 - (3 n)/2 2 - (3 n)/2 3/2 - (3 n)/2 q q q q \MATHgroesser ------, ------------, -(------------), --------------, d b c d b c d b c d \goodbreakpoint% 3/2 - (3 n)/2 2 - 2 n q q -1 + n \MATHgroesser -(--------------), --------, b c d e q \MATHrbrace , q, q] b c d b c d e \goodbreakpoint% 2 - 2 n 3 - 3 n 2 - 2 n 3 - 3 n q q q q \MATHgroesser pq[\MATHlbrace --------, --------\MATHrbrace , \MATHlbrace --------, ----------\MATHrbrace , n, q] b c d e 2 2 2 b c d 2 2 2 b c d b c d e \goodbreakpoint% In[5]:= \%1 \goodbreakpoint% ph[\MATHlbrace c, d\MATHrbrace , \MATHlbrace c d\MATHrbrace , q, z] pq[a, n, q] Out[5]= ----------------------------------- 2 pqinf[b, q ] \goodbreakpoint% In[6]:= phFormat \goodbreakpoint% In[7]:= \%1 \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich c, d \MATHvStrich \MATHphi \MATHvStrich ; q, z \MATHvStrich (a; q) 2 1\MATHvStrich c d \MATHvStrich n \MATHloEck \MATHroEck Out[7]= ------------------------- 2 (b;q ) \MATHinfty \endMATH \Seealso ph. \Name Phinv \Description Rule that transforms a multibasic hypergeometric series\linebreak \hbox{\tt Ph[List1A,List1B,q1,List2A,List2B,q2,\dots,ListkA,ListkB,qk,z]} with bases \hbox{\tt q}1,\dots, into a multibasic hypergeometric series \hbox{\tt Ph[\dots,1/q1,,\dots,1/q2,\dots,\dots,1/qk,z]} with bases 1/\hbox{\tt q}1,\dots \Usage Expr/.Phinv. \Example \MATH In[1]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,% \MATHlbrace e,f,g% \MATHrbrace ,% \MATHlbrace E,F,G% \MATHrbrace ,p,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b: e, f, g % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.Phinv \goodbreakpoint% 1 1 1 1 1 % \MATHluEck -, -: -, -, - % \MATHruEck % \MATHvStrich a b e f g 1 1 a b e f g z % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; -, -; ----------- % \MATHvStrich % \MATHvStrich 1 1 1 1 q p c E F G q % \MATHvStrich % \MATHloEck -: -, -, - % \MATHroEck c E F G \goodbreakpoint% In[3]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c,0% \MATHrbrace ,q,% \MATHlbrace e,f,g% \MATHrbrace ,% \MATHlbrace E,F% \MATHrbrace ,p,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b: e, f, g % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich c, 0: E, F % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= \%/.Phinv \goodbreakpoint% 1 1 1 1 1 % \MATHluEck -, -, 0: -, -, - % \MATHruEck % \MATHvStrich a b e f g 1 1 a b e f g z % \MATHvStrich Out[4]= \MATHphi % \MATHvStrich ; -, -; ----------- % \MATHvStrich % \MATHvStrich 1 1 1 q p c E F q % \MATHvStrich % \MATHloEck -: -, -, 0 % \MATHroEck c E F \endMATH \Seealso SUMRegeln, Ers, PosListe, phinv, psinv. \Name phinv \Description Rule that transforms a basic hypergeometric series \hbox{\tt ph[List1,List2,q,z]} with base {\tt q} into a basic hypergeometric series \hbox{\tt ph[\dots,1/q,..]} with base \hbox{\tt 1/q}. \Usage Expr/.phinv. \Example \MATH In[1]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 2 1% \MATHvStrich c % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.phinv \goodbreakpoint% 1 1 % \MATHluEck -, - % \MATHruEck % \MATHvStrich a b 1 a b z % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; -, ----- % \MATHvStrich 2 1% \MATHvStrich 1 q c q % \MATHvStrich % \MATHloEck - % \MATHroEck c \goodbreakpoint% In[3]:= ph[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,0% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 3 2% \MATHvStrich d, 0 % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= \%/.phinv \goodbreakpoint% 1 1 1 % \MATHluEck -, -, - % \MATHruEck % \MATHvStrich a b c 1 a b c z % \MATHvStrich Out[4]= \MATHphi % \MATHvStrich ; -, ------- % \MATHvStrich 3 1% \MATHvStrich 1 q d q % \MATHvStrich % \MATHloEck - % \MATHroEck d \endMATH \Seealso SUMRegeln, Ers, PosListe. \Name phOrdne \Description Rule that tries to order the parameters of a basic hypergeometric series in ``well-poised" order. If there is an upper parameter of the form $q^{-n}$, where $n$ might be a nonnegative integer, then it is put at the very last place in the upper list. If the parameters could be paired such that the product of each pair equals \hbox{\tt A}$q$, however \hbox{\tt A} is missing in the upper parameters, then you have to add \hbox{\tt A} to the upper and lower parameters by \hbox{\tt phEinf} before applying \hbox{\tt phOrdne}. \Usage Expr/.phOrdne. \Example \MATH In[1]:= ph[\MATHlbrace q\MATHhoch -n,b,q*Sqrt[a],a,-q*Sqrt[a]\MATHrbrace % ,\MATHlbrace a*q/b,-Sqrt[a],Sqrt[a], a*q\MATHhoch (1+n)\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck -n \MATHruEck \MATHvStrich q , b, Sqrt[a] q, a, -(Sqrt[a] q) \MATHvStrich Out[1]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 5 4\MATHvStrich a q 1 + n \MATHvStrich \MATHloEck ---, -Sqrt[a], Sqrt[a], a q \MATHroEck b \goodbreakpoint% In[2]:= \%/.phOrdne \goodbreakpoint% \MATHluEck -n \MATHruEck \MATHvStrich a, Sqrt[a] q, -(Sqrt[a] q), b, q \MATHvStrich Out[2]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 5 4\MATHvStrich a q 1 + n \MATHvStrich \MATHloEck Sqrt[a], -Sqrt[a], ---, a q \MATHroEck b \goodbreakpoint% In[3]:= ph[\MATHlbrace q\MATHhoch -n,b,c\MATHrbrace ,\MATHlbrace d,e\MATHrbrace % ,q\MATHhoch 2,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n 2 \MATHvStrich Out[3]= \MATHphi \MATHvStrich q , b, c; q , z \MATHvStrich 3 2\MATHvStrich \MATHvStrich \MATHloEck d, e \MATHroEck \goodbreakpoint% In[4]:= \%/.phOrdne \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n 2 \MATHvStrich Out[4]= \MATHphi \MATHvStrich b, c, q ; q , z \MATHvStrich 3 2\MATHvStrich \MATHvStrich \MATHloEck e, d \MATHroEck In[5]:= ph[% \MATHlbrace b,c,q% \MATHrbrace ,% \MATHlbrace a*q/c,a*q/b% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[5]= \MATHphi % \MATHvStrich a q a q; q, z % \MATHvStrich 3 2% \MATHvStrich ---, --- % \MATHvStrich % \MATHloEck c b % \MATHroEck \goodbreakpoint% In[6]:= \%/.phOrdne \goodbreakpoint% % \MATHluEck b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[6]= \MATHphi % \MATHvStrich a q a q; q, z % \MATHvStrich 3 2% \MATHvStrich ---, --- % \MATHvStrich % \MATHloEck c b % \MATHroEck \goodbreakpoint% In[7]:= \%/.phEinf Add the parameter: a \goodbreakpoint% % \MATHluEck a, b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[7]= \MATHphi % \MATHvStrich a q a q; q, z % \MATHvStrich 4 3% \MATHvStrich a, ---, --- % \MATHvStrich % \MATHloEck c b % \MATHroEck \goodbreakpoint% In[8]:= \%/.phOrdne \goodbreakpoint% % \MATHluEck a, b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[8]= \MATHphi % \MATHvStrich a q a q ; q, z % \MATHvStrich 4 3% \MATHvStrich ---, ---, a % \MATHvStrich % \MATHloEck b c % \MATHroEck \goodbreakpoint% In[9]:= phCancel \goodbreakpoint% In[10]:= \%\%/.phOrdne \goodbreakpoint% % \MATHluEck b, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[10]= \MATHphi % \MATHvStrich a q a q; q, z % \MATHvStrich 3 2% \MATHvStrich ---, --- % \MATHvStrich % \MATHloEck b c % \MATHroEck \endMATH \Seealso phEinf, phPerm, phTausche, ph, W, PQSort, Ers, PosListe. \Name phPerm \Description Rule for permuting parameters in basic hypergeometric series. \Usage Expr/.phPerm[,x].\newline \rm {\tt x} can be {\tt u}, {\tt l}, {\tt b}. u causes a permutation of upper parameters, {\tt l} causes a permutation of lower parameters, {\tt b} causes a simultaneous permutation of respective upper and lower parameters. \hbox{\tt Permutation} must be a sequence of positive numbers forming a permutation. Under the options {\tt u} and {\tt l} the effect is that the new parameter at position {\tt i} is the old parameter from position \hbox{\tt Permutation[i]}. However, the behaviour of \hbox{\tt FPerm} under the option {\tt b} is special. The option {\tt b} is especially designed for the permutation of parameters of {\it well-poised} series. Hence, the first upper parameter is not moved, whereas the new {\it upper} parameter at position {\tt i+1} is the old upper parameter from position \hbox{\tt Permutation[i]+1}, and the new {\it lower} parameter at position {\tt i} is the old lower parameter from position \hbox{\tt Permutation[i]}. \Example \MATH In[1]:= ph[% \MATHlbrace a,b,c,d% \MATHrbrace ,% \MATHlbrace e,f,g% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c, d % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 4 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%1/.phPerm[3,2,1,u] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich c, b, a, d % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 4 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[3]:= \%1/.phPerm[3,2,1,l] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c, d % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 4 3% \MATHvStrich g, f, e % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= \%1/.phPerm[2,1,b] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, c, b, d % \MATHvStrich Out[4]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 4 3% \MATHvStrich f, e, g % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso phTausche, phOrdne, ph, W, PQSort, Ers, PosListe. \Name Phph \Description Rule that transforms a \hbox{\tt Ph[]} into ``ordinary" hypergeometric notation, if possible. \Usage Expr/.Phph. \Example \MATH In[1]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,% \MATHlbrace e,f,g% \MATHrbrace ,% \MATHlbrace E,F,G% \MATHrbrace ,q\MATHhoch 2,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b: e, f, g 2 % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, q ; z % \MATHvStrich % \MATHvStrich c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.Phph \goodbreakpoint% % \MATHluEck % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich 8 7% \MATHvStrich % \MATHloEck % \MATHruEck a, b, Sqrt[e], -Sqrt[e], Sqrt[f], -Sqrt[f], Sqrt[g], -Sqrt[g] % \MATHvStrich \MATHgroesser ; q, z % \MATHvStrich c, Sqrt[E], -Sqrt[E], Sqrt[F], -Sqrt[F], Sqrt[G], -Sqrt[G] % \MATHvStrich % \MATHroEck \endMATH \Seealso Ph, ph, phPh, Ers, PosListe. \Name phPh \Description Rule that transforms a \hbox{\tt ph[]} into multibasic notation. \Usage Expr/.phPh. \Example \MATH In[1]:= ph[% \MATHlbrace a,d,e\MATHhoch (1/2),-e\MATHhoch (1/2),f\MATHhoch (1/2),-f\MATHhoch (1/2),g\MATHhoch (1/2),-g\MATHhoch (1/2)% \MATHrbrace , % \MATHlbrace c,E\MATHhoch (1/2),-E\MATHhoch (1/2),F\MATHhoch (1/2),-F\MATHhoch (1/2),G\MATHhoch (1/2),-G\MATHhoch (1/2)% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich 8 7% \MATHvStrich % \MATHloEck % \MATHruEck a, d, Sqrt[e], -Sqrt[e], Sqrt[f], -Sqrt[f], Sqrt[g], -Sqrt[g] % \MATHvStrich \MATHgroesser ; q, z % \MATHvStrich c, Sqrt[E], -Sqrt[E], Sqrt[F], -Sqrt[F], Sqrt[G], -Sqrt[G] % \MATHvStrich % \MATHroEck \goodbreakpoint% In[2]:= \%/.phPh \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, d: e, f, g 2 % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; q, q ; z % \MATHvStrich % \MATHvStrich c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso ph, Ph, Phph, Ers, PosListe. \Name phps \Description Rule that transforms a ph[] into a difference of a \hbox{\tt ps[]} and a \hbox{\tt ph[]}. \Usage Expr/.phps. \Example \MATH In[1]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 2 1% \MATHvStrich c % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.phps \goodbreakpoint% 2 q c % \MATHluEck --, q, q % \MATHruEck (-, 1; q) % \MATHvStrich c c q % \MATHvStrich q 1 \MATHphi % \MATHvStrich ; q, ----- % \MATHvStrich ---------- 3 2% \MATHvStrich 2 2 a b z % \MATHvStrich a b % \MATHloEck q q % \MATHroEck (-, -; q) --, -- q q 1 % \MATHluEck % \MATHruEck a b % \MATHvStrich a, b % \MATHvStrich Out[2]= -(------------------------------------) + ps % \MATHvStrich ; q, z % \MATHvStrich z 2 2% \MATHvStrich c, q % \MATHvStrich % \MATHloEck % \MATHroEck In[3]:= \%/.pqaufl \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich Out[3]= ps % \MATHvStrich ; q, z % \MATHvStrich 2 2% \MATHvStrich c, q % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= ph[% \MATHlbrace a,b,q% \MATHrbrace ,% \MATHlbrace c,d% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, q % \MATHvStrich Out[4]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 3 2% \MATHvStrich c, d % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[5]:= \%/.phps \goodbreakpoint% 2 2 q q c d % \MATHluEck --, --, q % \MATHruEck (-, -; q) % \MATHvStrich c d c d % \MATHvStrich q q 1 \MATHphi % \MATHvStrich ; q, ----- % \MATHvStrich ---------- 3 2% \MATHvStrich 2 2 a b z % \MATHvStrich a b % \MATHloEck q q % \MATHroEck (-, -; q) --, -- q q 1 % \MATHluEck % \MATHruEck a b % \MATHvStrich a, b % \MATHvStrich Out[5]= -(-------------------------------------) + ps % \MATHvStrich ; q, z % \MATHvStrich z 2 2% \MATHvStrich c, d % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso ph, ps, psph, Ers, PosListe. \Name PhSUM \Description Rule that transforms a \hbox{\tt Ph[]} into a \hbox{\tt SUM[]}. \Usage Expr/.PhSUM. \Example \MATH In[1]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,% \MATHlbrace e,f,g% \MATHrbrace ,% \MATHlbrace E,F,G% \MATHrbrace ,p,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b: e, f, g % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q, p; z % \MATHvStrich % \MATHvStrich c: E, F, G % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.PhSUM A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% \MATHinfty k % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q) (b; q) (e; p) (f; p) (g; p) \MATHbackslash k k k k k Out[2]= \MATHgroesser ------------------------------------------ / (c; q) (E; p) (F; p) (G; p) (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k k k k k=0 \goodbreakpoint% In[3]:= Ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,% \MATHlbrace e,f,p\MATHhoch -n% \MATHrbrace ,% \MATHlbrace E,F,G% \MATHrbrace ,p,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -n % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich a, b: e, f, p ; q, p; z % \MATHvStrich % \MATHvStrich % \MATHvStrich % \MATHloEck c: E, F, G % \MATHroEck \goodbreakpoint% In[4]:= \%/.PhSUM Is n a nonnegative integer? [y|n]: y A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% n k -n % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q) (b; q) (e; p) (f; p) (p ; p) \MATHbackslash k k k k k Out[4]= \MATHgroesser -------------------------------------------- / (c; q) (E; p) (F; p) (G; p) (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k k k k k=0 \endMATH \Seealso Ph, SUM, SUMPh, Ers, PosListe. \Name phSUM \Description Rule that transforms a \hbox{\tt ph[]} into a \hbox{\tt SUM[]}. \Usage Expr/.phSUM. \Example \MATH In[1]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q\MATHhoch 2,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b 2 % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q , z % \MATHvStrich 2 1% \MATHvStrich c % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.phSUM A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% \MATHinfty k 2 2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q ) (b; q ) \MATHbackslash k k Out[2]= \MATHgroesser -------------------- / 2 2 2 % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck (c; q ) (q ; q ) k=0 k k \goodbreakpoint% In[3]:= ph[% \MATHlbrace a,q\MATHhoch (-2*n)% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q\MATHhoch 2,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -2 n 2 % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich a, q ; q , z % \MATHvStrich 2 1% \MATHvStrich % \MATHvStrich % \MATHloEck c % \MATHroEck \goodbreakpoint% In[4]:= \%/.phSUM Is n a nonnegative integer? [y|n]: y A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: j \goodbreakpoint% n j 2 -2 n 2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q ) (q ; q ) \MATHbackslash j j Out[4]= \MATHgroesser ------------------------ / 2 2 2 % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck (c; q ) (q ; q ) j=0 j j \goodbreakpoint% In[5]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c,d% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich Out[5]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 2 2% \MATHvStrich c, d % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[6]:= \%/.phSUM A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% 2 \MATHinfty k -k/2 + k /2 k % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck (-1) q z (a; q) (b; q) \MATHbackslash k k Out[6]= \MATHgroesser ------------------------------------- / (c; q) (d; q) (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k k k=0 \endMATH \Seealso ph, SUM, SUMph, Ers, PosListe. \Name phTausche \Description Rule for reordering parameters in basic hypergeometric series. \Usage Expr/.phTausche[n1,n2,x].\newline \rm \hbox{\tt x} can be \hbox{\tt u}, \hbox{\tt l}, \hbox{\tt b}. \hbox{\tt u} causes a reordering of upper parameters, \hbox{\tt l} causes a reordering of lower parameters, \hbox{\tt b} causes a simultaneous reordering of respective upper and lower parameters. \hbox{\tt n1} is the position of the parameter to be reordered, \hbox{\tt n2} is the new position. \Example \MATH In[1]:= ph[\MATHlbrace q\MATHhoch -n,b,q*Sqrt[a],a,-q*Sqrt[a]\MATHrbrace % ,\MATHlbrace a*q/b,-Sqrt[a],Sqrt[a], a*q\MATHhoch (1+n)\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck -n \MATHruEck \MATHvStrich q , b, Sqrt[a] q, a, -(Sqrt[a] q) \MATHvStrich Out[1]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 5 4\MATHvStrich a q 1 + n \MATHvStrich \MATHloEck ---, -Sqrt[a], Sqrt[a], a q \MATHroEck b \goodbreakpoint% In[2]:= \%/.phTausche[1,3,u] \goodbreakpoint% \MATHluEck -n \MATHruEck \MATHvStrich b, Sqrt[a] q, q , a, -(Sqrt[a] q) \MATHvStrich Out[2]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 5 4\MATHvStrich a q 1 + n \MATHvStrich \MATHloEck ---, -Sqrt[a], Sqrt[a], a q \MATHroEck b \goodbreakpoint% In[3]:= \%/.phTausche[4,2,l] \goodbreakpoint% \MATHluEck -n \MATHruEck \MATHvStrich b, Sqrt[a] q, q , a, -(Sqrt[a] q) \MATHvStrich Out[3]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 5 4\MATHvStrich a q 1 + n \MATHvStrich \MATHloEck ---, a q , -Sqrt[a], Sqrt[a] \MATHroEck b \goodbreakpoint% In[4]:= \%/.phTausche[1,4,b] \goodbreakpoint% \MATHluEck -n \MATHruEck \MATHvStrich b, q , a, -(Sqrt[a] q), Sqrt[a] q \MATHvStrich Out[4]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 5 4\MATHvStrich 1 + n a q \MATHvStrich \MATHloEck a q , -Sqrt[a], Sqrt[a], --- \MATHroEck b \endMATH \Seealso phPerm, phOrdne, ph, W, PQSort, Ers, PosListe. \Name PosListe \Description Function that provides a list of subexpressions of \hbox{\tt Expr} together with the respective positions in \hbox{\tt Expr}. This helps to use controlled application of rules or functions by means of \hbox{\tt Ers}. \Usage PosListe[Expr]. \Example \MATH In[1]:= pq[a,n]/pq[q,n]*SUM[pq[b,k]/pq[c,k+1]*q\MATHhoch k,\MATHlbrace k,0,Infinity\MATHrbrace ] \goodbreakpoint% \MATHinfty k \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q (b; q) \MATHbackslash k ( \MATHgroesser -----------) (a; q) / (c; q) n \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck 1 + k k=0 Out[1]= -------------------------- (q; q) n \goodbreakpoint% In[2]:= PosListe[\%] \goodbreakpoint% \MATHinfty k \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q (b; q) \MATHbackslash k 1 Out[2]= \MATHlbrace \MATHlbrace \MATHgroesser -----------, \MATHlbrace % \MATHlbrace 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace (a; q) , % \MATHlbrace \MATHlbrace 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace % -------, \MATHlbrace \MATHlbrace 3\MATHrbrace \MATHrbrace \MATHrbrace % \MATHrbrace / (c; q) n (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % 1 + k n k=0 \goodbreakpoint% In[3]:= PosListe[\%\%,2] \goodbreakpoint% Out[3]= \MATHlbrace \MATHlbrace -1, \MATHlbrace \MATHlbrace 3, 2\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace a, \MATHlbrace \MATHlbrace 2, 1\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace n, \MATHlbrace \MATHlbrace 2, 2\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace q, \MATHlbrace \MATHlbrace 2, 3\MATHrbrace % \MATHrbrace \MATHrbrace , \goodbreakpoint% k q (b; q) k \MATHgroesser \MATHlbrace \MATHlbrace k, 0, \MATHinfty \MATHrbrace , % \MATHlbrace \MATHlbrace 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace % -----------, \MATHlbrace \MATHlbrace 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , % \MATHlbrace (q; q) , \MATHlbrace \MATHlbrace 3, 1\MATHrbrace \MATHrbrace % \MATHrbrace \MATHrbrace (c; q) n 1 + k \goodbreakpoint% In[4]:= PosListe[\%\%\%,3] \goodbreakpoint% Out[4]= \MATHlbrace \MATHlbrace 0, \MATHlbrace \MATHlbrace 1, 2, % 2\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace k, \MATHlbrace \MATHlbrace % 1, 2, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace n, \MATHlbrace % \MATHlbrace 3, 1, 2\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% k \MATHgroesser \MATHlbrace q, \MATHlbrace \MATHlbrace 3, 1, 1\MATHrbrace % , \MATHlbrace 3, 1, 3\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace q , % \MATHlbrace \MATHlbrace 1, 1, 1\MATHrbrace \MATHrbrace \MATHrbrace , \MATHlbrace % \MATHinfty , \MATHlbrace \MATHlbrace 1, 2, 3\MATHrbrace \MATHrbrace \MATHrbrace , \goodbreakpoint% 1 \MATHgroesser \MATHlbrace (b; q) , \MATHlbrace \MATHlbrace 1, 1, 2\MATHrbrace % \MATHrbrace \MATHrbrace , \MATHlbrace -----------, \MATHlbrace \MATHlbrace 1, % 1, 3\MATHrbrace \MATHrbrace \MATHrbrace \MATHrbrace k (c; q) 1 + k \endMATH \Seealso Ers, Subst. \Name pq \Description \hbox{\tt pq[x,n,q]} is the $q$-factorial symbol $(x;q)_n$. \hbox{\tt pq[List1,List2,n,q]} is also provided as the usual abbreviation for the quotient of $q$-factorial symbols (see \cite{\GaRaAA, (1.2.41)}). In both cases the parameter \hbox{\tt q} is optional. It will be set equal \hbox{\tt q} if it is omitted. \Usage pq[x,n,q] \leavevmode\hphantom{Usa}\rm or: \tt pq[x,n] \leavevmode\hphantom{Usa}\rm or: \tt pq[List1,List2,n,q] \leavevmode\hphantom{Usa}\rm or: \tt pq[List1,List2,n]. \Example \MATH In[1]:= pq[a,n] \goodbreakpoint% Out[1]= (a; q) n \goodbreakpoint% In[2]:= pq[a,n,q\MATHhoch 2] \goodbreakpoint% 2 Out[2]= (a; q ) n \goodbreakpoint% In[3]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,2*m] \goodbreakpoint% (a, b; q) 2 m Out[3]= ------------ (c, d; q) 2 m \goodbreakpoint% In[4]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,2*m,1/q] \goodbreakpoint% 1 (a, b; -) q 2 m Out[4]= ------------ 1 (c, d; -) q 2 m \endMATH \Seealso pqinf, Binomialq, Binomialpq, Multinomialpq, Multinomialq, Factorialq, Factorialpq, PQ, pqaufl, pqzerl, pqzus,\linebreak phFormat. \Name PQ \Description Is a switch that activates automatic evaluating of $q$-factorial symbols \hbox{\tt pq} and basic hypergeometric series \hbox{\tt ph}, \hbox{\tt Ph}, \hbox{\tt ps}, or makes it inactive, respectively. By default automatic evaluating is inactive. \Usage PQ. \Example \MATH In[1]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[2]:= pq[a,5] \goodbreakpoint% Out[2]= (a; q) 5 \goodbreakpoint% In[3]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich a, b \MATHvStrich Out[3]= \MATHphi \MATHvStrich ; q, z \MATHvStrich 2 1\MATHvStrich c \MATHvStrich \MATHloEck \MATHroEck \goodbreakpoint% In[4]:= ph[\MATHlbrace q\MATHhoch -n,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich Out[4]= \MATHphi \MATHvStrich q , b; q, z \MATHvStrich 2 1\MATHvStrich \MATHvStrich \MATHloEck c \MATHroEck \goodbreakpoint% In[5]:= ph[\MATHlbrace q\MATHhoch -3,b\MATHrbrace ,\MATHlbrace c\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -3 \MATHvStrich Out[5]= \MATHphi \MATHvStrich q , b; q, z \MATHvStrich 2 1\MATHvStrich \MATHvStrich \MATHloEck c \MATHroEck \goodbreakpoint% In[6]:= PQ \goodbreakpoint% In[7]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is active. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \goodbreakpoint% In[8]:= pq[a,5] \goodbreakpoint% 2 3 4 Out[8]= (1 - a) (1 - a q) (1 - a q ) (1 - a q ) (1 - a q ) \goodbreakpoint% In[9]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z] A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% \MATHinfty k % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q) (b; q) \MATHbackslash k k Out[9]= \MATHgroesser ------------------ / (c; q) (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k k=0 \goodbreakpoint% In[10]:= ph[% \MATHlbrace q\MATHhoch -n,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z] Is n a nonnegative integer? [y|n]: y A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: j \goodbreakpoint% n j -n % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (b; q) (q ; q) \MATHbackslash j j Out[10]= \MATHgroesser -------------------- / (c; q) (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck j j j=0 \goodbreakpoint% In[11]:= ph[% \MATHlbrace q\MATHhoch -3,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q,z] A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: s \goodbreakpoint% -3 -3 -2 2 (1 - b) (1 - q ) z (1 - b) (1 - q ) (1 - q ) (1 - b q) z Out[11]= 1 + ------------------- + ---------------------------------------- + (1 - c) (1 - q) 2 (1 - c) (1 - q) (1 - c q) (1 - q ) -3 -2 1 2 3 (1 - b) (1 - q ) (1 - q ) (1 - -) (1 - b q) (1 - b q ) z q \MATHgroesser ----------------------------------------------------------- 2 2 3 (1 - c) (1 - q) (1 - c q) (1 - q ) (1 - c q ) (1 - q ) \goodbreakpoint% In[12]:= PQ \goodbreakpoint% In[13]:= hypqAttributes \goodbreakpoint% Automatic evaluation of pq and ph is inactive. Automatic cancelling in ph is active. The output of TeXForm can be used with AmS-TeX. TeXForm uses W[] for very well-poised basic hypergeometric series. \endMATH \Seealso ph, Ph, ps, pq, hypqAttributes. \Name pqaufl \Description Rule that writes $(x;q)_n$ as the defining product $\prod _{i=0} ^{n-1}(1-xq^{i})$, if $n$ is an integer. \Usage Expr/.pqaufl. \Example \MATH In[1]:= pq[a,-3]/pq[b,2]*pq[c,1] \goodbreakpoint% (a; q) (c; q) -3 1 Out[1]= ---------------- (b; q) 2 \goodbreakpoint% In[2]:= \%/.pqaufl \goodbreakpoint% 1 - c Out[2]= ------------------------------------------- a a a (1 - b) (1 - --) (1 - --) (1 - -) (1 - b q) 3 2 q q q \goodbreakpoint% In[3]:= ph[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,e,f% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich d, e, f % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= \%/.C01 \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich q, a q, b q, c q % \MATHvStrich z \MATHphi % \MATHvStrich ; q, q z % \MATHvStrich (a; q) (b; q) (c; q) 4 4% \MATHvStrich 2 % \MATHvStrich 1 1 1 % \MATHloEck q , d q, e q, f q % \MATHroEck Out[4]= 1 - ---------------------------------------------------------- (d; q) (e; q) (f; q) (q; q) 1 1 1 1 \goodbreakpoint% In[5]:= \%/.pqaufl \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich q, a q, b q, c q % \MATHvStrich (1 - a) (1 - b) (1 - c) z \MATHphi % \MATHvStrich ; q, q z % \MATHvStrich 4 4% \MATHvStrich 2 % \MATHvStrich % \MATHloEck q , d q, e q, f q % \MATHroEck Out[5]= 1 - ---------------------------------------------------------- (1 - d) (1 - e) (1 - f) (1 - q) \endMATH \Seealso pqzerl, pqzus, pq, pqinfzerl, pqinfzus, Ers, PosListe. \Name pqinf \Description \hbox{\tt pqinf[x,q]} is the infinite $q$-factorial symbol $(x;q)_\infty$. \hbox{\tt pqinf[List1,List2,q]} is also provided as the usual abbreviation for the quotient of infinite $q$-factorial symbols (see \cite{\GaRaAA, (1.2.42)}). In both cases the parameter \hbox{\tt q} is optional. If it is omitted it is set equal \hbox{\tt q}. \Usage pqinf[x,q] \leavevmode\hphantom{Usa}\rm or: \tt pqinf[x] \leavevmode\hphantom{Usa}\rm or: \tt pqinf[List1,List2,q] \leavevmode\hphantom{Usa}\rm or: \tt pqinf[List1,List2]. \Example \MATH In[1]:= pqinf[a\MATHhoch 2*q] \goodbreakpoint% 2 Out[1]= (a q;q) \MATHinfty \goodbreakpoint% In[2]:= pqinf[a\MATHhoch 2*q,q\MATHhoch 3] \goodbreakpoint% 2 3 Out[2]= (a q;q ) \MATHinfty \goodbreakpoint% In[3]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ] \goodbreakpoint% (a, b; q) \MATHinfty Out[3]= ---------- (c, d; q) \MATHinfty \goodbreakpoint% In[4]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,1/q] \goodbreakpoint% 1 (a, b; -) q \MATHinfty Out[4]= ---------- 1 (c, d; -) q \MATHinfty \endMATH \Seealso pq, PQ, pqinfzerl, pqinfzus, phFormat. \Name pqinfzerl \Description Rule that splits \hbox{\tt pqinf[List1,List2,q]} into a quotient of products of infinite $q$-factorial symbols. \Usage Expr/.pqinfzerl. \Example \MATH In[1]:= pqinf[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ] \goodbreakpoint% (a, b; q) \MATHinfty Out[1]= ---------- (c, d; q) \MATHinfty \goodbreakpoint% In[2]:= \%/.pqinfzerl \goodbreakpoint% (a;q) (b;q) \MATHinfty \MATHinfty Out[2]= ------------- (c;q) (d;q) \MATHinfty \MATHinfty \endMATH \Seealso pqaufl, pqzerl, pqzus, pqinf, pqinfzus, Ers, PosListe. \Name pqinfzus \Description Rule that collects several infinite $q$-factorial symbols \hbox{\tt pqinf[x$_{\text {\tt i}}$,q]} to an expression\linebreak \hbox{\tt pqinf[List1,List2,q]}. The parameter \hbox{\tt q} is optional. It is set equal \hbox{\tt q} if it is omitted. \Usage Expr/.pqinfzus[q] \leavevmode\hphantom{Usa}\rm or: \tt Expr/.pqinfzus[]. \Example \MATH In[1]:= pqinf[a]*pqinf[b]/pqinf[c]/pqinf[d] \goodbreakpoint% (a;q) (b;q) \MATHinfty \MATHinfty Out[1]= ------------- (c;q) (d;q) \MATHinfty \MATHinfty \goodbreakpoint% In[2]:= \%/.pqinfzus[] \goodbreakpoint% (a, b; q) \MATHinfty Out[2]= ---------- (c, d; q) \MATHinfty \goodbreakpoint% In[3]:= pqinf[a,q]*pqinf[b,q\MATHhoch 2]/pqinf[c,q\MATHhoch 2]/pqinf[d,q] \goodbreakpoint% 2 (a;q) (b;q ) \MATHinfty \MATHinfty Out[3]= -------------- 2 (c;q ) (d;q) \MATHinfty \MATHinfty \goodbreakpoint% In[4]:= \%/.pqinfzus[] \goodbreakpoint% (a; q) 2 \MATHinfty (b;q ) ------- \MATHinfty (d; q) \MATHinfty Out[4]= --------------- 2 (c;q ) \MATHinfty \goodbreakpoint% In[5]:= \%/.pqinfzus[q\MATHhoch 2] \goodbreakpoint% 2 (a; q) (b; q ) \MATHinfty \MATHinfty Out[5]= ------- -------- (d; q) 2 \MATHinfty (c; q ) \MATHinfty \endMATH \Seealso pqaufl, pqzerl, pqzus, pqinf, pqinfzerl, Ers, PosListe. \Name PQSort \Description Rule that orders the parameters of basic hypergeometric series \hbox{\tt ph[List1,List2,q,z]}, \hbox{\tt Ph[\dots]}, \hbox{\tt ps[\dots]}, \hbox{\tt W[\dots]}, of ``multiple" upper $q$-factorials \hbox{\tt pq[List1,List2,n,q]}, and of ``multiple" infinite $q$-factorials \hbox{\tt pqinf[List1,List2,q]} in a standard order. For instance, this function can be used for a quick test if two expressions agree. It is recommended to apply \hbox{\tt pqinfzus[q]} and \hbox{\tt pqzus[n,q]} first. \Usage Expr/.PQSort. \Example \MATH In[1]:= pq[% \MATHlbrace c,c/a,b% \MATHrbrace ,% \MATHlbrace a/b/c,b*q% \MATHrbrace ,n,q\MATHhoch 2]*pqinf[% \MATHlbrace b/a,c% \MATHrbrace ,% \MATHlbrace a*b*c,1/b% \MATHrbrace ,q]* ph[% \MATHlbrace b/c,b*a,a% \MATHrbrace ,% \MATHlbrace b,a% \MATHrbrace ,1/q,z] \goodbreakpoint% b c 2 (-, c; q) % \MATHluEck b % \MATHruEck (c, -, b; q ) a \MATHinfty % \MATHvStrich -, a b 1 % \MATHvStrich a n Out[1]= -------------- \MATHphi % \MATHvStrich c ; -, z % \MATHvStrich --------------- 1 2 1% \MATHvStrich q % \MATHvStrich a 2 (a b c, -; q) % \MATHloEck b % \MATHroEck (---, b q; q ) b \MATHinfty b c n \goodbreakpoint% In[2]:= \%/.PQSort \goodbreakpoint% b c 2 (-, c; q) % \MATHluEck b % \MATHruEck (b, c, -; q ) a \MATHinfty % \MATHvStrich a b, - 1 % \MATHvStrich a n Out[2]= -------------- \MATHphi % \MATHvStrich c; -, z % \MATHvStrich --------------- 1 2 1% \MATHvStrich q % \MATHvStrich a 2 (-, a b c; q) % \MATHloEck b % \MATHroEck (---, b q; q ) b \MATHinfty b c n \endMATH \Seealso SimplifyPQ, SUMExpand, phEinf, phOrdne, phPerm, phTausche, ph, Ph, ps, W, pq, pqinf. \Name pqzerl \Description Rule that splits \hbox{\tt pq[List1,List2,n,q]} into a quotient of products of $q$-factorial symbols. \Usage Expr/.pqzerl. \Example \MATH In[1]:= pq[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace c,d\MATHrbrace ,m,q\MATHhoch 2] \goodbreakpoint% 2 (a, b; q ) m Out[1]= ----------- 2 (c, d; q ) m \goodbreakpoint% In[2]:= \%/.pqzerl \goodbreakpoint% 2 2 (a; q ) (b; q ) m m Out[2]= ----------------- 2 2 (c; q ) (d; q ) m m \endMATH \Seealso pqaufl, pqzus, pq, pqinfzerl, pqinfzus, Ers, PosListe. \Name pqzus \Description Rule that collects several $q$-factorial symbols \hbox{\tt pq[x$_{\text {\tt i}}$,n,q]} to an expression \hbox{\tt pq[List1,List2,n,q]}. The parameter \hbox{\tt q} is optional. It is set equal \hbox{\tt q} if it is omitted. \Usage Expr/.pqzus[n,q] \leavevmode\hphantom{Usa}\rm or: \tt Expr/.pqzus[n]. \Example \MATH In[1]:= pq[a,n]*pq[b,m,q\MATHhoch 2]/pq[c,n]/pq[d,m,q\MATHhoch 2] \goodbreakpoint% 2 (a; q) (b; q ) n m Out[1]= ---------------- 2 (c; q) (d; q ) n m \goodbreakpoint% In[2]:= \%/.pqzus[n,q] \goodbreakpoint% (a; q) 2 n (b; q ) ------- m (c; q) n Out[2]= ---------------- 2 (d; q ) m \goodbreakpoint% In[3]:= \%/.pqzus[m,q\MATHhoch 2] \goodbreakpoint% 2 (a; q) (b; q ) n m Out[3]= ------- -------- (c; q) 2 n (d; q ) m \endMATH \Seealso pqaufl, pqzerl, pq, pqinfzus, pqinfzerl, Ers, PosListe. \Name ps \Description \hbox{\tt ps[List1,List2,q,z]} is the bilateral basic hypergeometric series with upper parameters \hbox{\tt List}1, lower parameters \hbox{\tt List}2, base \hbox{\tt q}, and argument \hbox{\tt z}. \Usage ps[List1,List2,q,z]. \Example \MATH In[1]:= ps[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,e,f% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich d, e, f % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso SListe, TListe, SUMRegeln, SUMps, psSUM, ph, Ph, pq, pqinf, phCancel, phOrdne, phPerm,\linebreak phTausche, PQ, phFormat. \Name psEinf \Description Rule that inactivates automatic cancelling in \hbox{\tt ps[]} and then adds a parameter which has to be entered on request to the upper and lower parameters of \hbox{\tt ps[]}. \Usage Expr/.psEinf. \Example \MATH In[1]:= ps[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,e,f% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich d, e, f % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.psEinf Add the parameter: A \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich A, a, b, c % \MATHvStrich Out[2]= ps % \MATHvStrich ; q, z % \MATHvStrich 4 4% \MATHvStrich A, d, e, f % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso phCancel, psOrdne, psPerm, PQSort, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name psinv \Description Rule that transforms a bilateral basic hypergeometric series \hbox{\tt ps[List1,List2,q,z]} with base \hbox{\tt q} into a bilateral basic hypergeometric series \hbox{\tt ps[\dots,1/q,..]} with base 1/\hbox{\tt q}. \Usage Expr/.psinv. \Example \MATH In[1]:= ps[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c,d% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 2 2% \MATHvStrich c, d % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.psinv \goodbreakpoint% 1 1 % \MATHluEck -, - % \MATHruEck % \MATHvStrich a b 1 a b z % \MATHvStrich Out[2]= ps % \MATHvStrich ; -, ----- % \MATHvStrich 2 2% \MATHvStrich 1 1 q c d % \MATHvStrich % \MATHloEck -, - % \MATHroEck c d \goodbreakpoint% In[3]:= ps[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c,d,0% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b % \MATHvStrich Out[3]= ps % \MATHvStrich ; q, z % \MATHvStrich 2 3% \MATHvStrich c, d, 0 % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= \%/.psinv \goodbreakpoint% 1 1 % \MATHluEck -, -, 0 % \MATHruEck % \MATHvStrich a b 1 a b z % \MATHvStrich Out[4]= ps % \MATHvStrich ; -, ----- % \MATHvStrich 3 2% \MATHvStrich 1 1 q c d % \MATHvStrich % \MATHloEck -, - % \MATHroEck c d \endMATH \Seealso SUMRegeln, Ers, PosListe, phinv, Phinv. \Name psOrdne \Description Rule that tries to order the parameters of a bilateral basic hypergeometric series in ``well-poised" order. If there is an upper parameter of the form $q^{-n}$, where $n$ is a nonnegative integer, then it is put at the very last place in the upper list. \Usage Expr/.psOrdne. \Example \MATH In[1]:= ps[% \MATHlbrace q\MATHhoch -n,b,q*Sqrt[a],-q*Sqrt[a]% \MATHrbrace ,% \MATHlbrace a*q/b,-Sqrt[a],a*q\MATHhoch (1+n), Sqrt[a]% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck -n % \MATHruEck % \MATHvStrich q , b, Sqrt[a] q, -(Sqrt[a] q) % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 4 4% \MATHvStrich a q 1 + n % \MATHvStrich % \MATHloEck ---, -Sqrt[a], a q , Sqrt[a] % \MATHroEck b \goodbreakpoint% In[2]:= \%/.psOrdne \goodbreakpoint% % \MATHluEck -n % \MATHruEck % \MATHvStrich Sqrt[a] q, -(Sqrt[a] q), b, q % \MATHvStrich Out[2]= ps % \MATHvStrich ; q, z % \MATHvStrich 4 4% \MATHvStrich a q 1 + n % \MATHvStrich % \MATHloEck Sqrt[a], -Sqrt[a], ---, a q % \MATHroEck b \endMATH \Seealso psEinf, psPerm, ps, PQSort, Ers, PosListe. \Name psPerm \Description Rule for permuting parameters in bilateral basic hypergeometric series. \Usage Expr/.psPerm[$\langle$Permutation$\rangle$,x].\newline \rm {\tt x} can be {\tt u}, {\tt l}, {\tt b}. {\tt u} causes a permutation of upper parameters, {\tt l} causes a permutation of lower parameters, {\tt b} causes a simultaneous permutation of respective upper and lower parameters. \hbox{\tt Permutation} must be a sequence of positive numbers forming a permutation. The effect is that the new parameter at position {\tt i} is the old parameter from position \hbox{\tt Permutation[i]}. \Example \MATH In[1]:= ps[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace e,f,g% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%1/.psPerm[3,2,1,u] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich c, b, a % \MATHvStrich Out[2]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[3]:= \%1/.psPerm[3,2,1,l] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[3]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich g, f, e % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[4]:= \%1/.psPerm[3,2,1,b] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich c, b, a % \MATHvStrich Out[4]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich g, f, e % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso psOrdne, ps, PQSort, Ers, PosListe, phPerm, PhPerm. \Name psph \Description Rule that transforms a \hbox{\tt ps[]} into a sum of two \hbox{\tt ph[]}'s. \vskip6pt \leavevmode\hphantom{Description: } $$\multline \hskip2cm {}_r \psi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q,z\right] \to\sum _{n=-\infty} ^{m-1}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}} {\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom n2}\right)^{s-r+1}z^n\\ +\sum _{n=m} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}} {\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom n2}\right)^{s-r+1}z^n. \endmultline$$ \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.psph. \Example \MATH In[1]:= ps[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace e,f,g% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.psph Split at: 0 \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c, q % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; q, z % \MATHvStrich + 4 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck 2 2 2 q q q % \MATHluEck --, --, --, q % \MATHruEck (a, b, c; q) % \MATHvStrich e f g e f g % \MATHvStrich -1 \MATHphi % \MATHvStrich ; q, ------- % \MATHvStrich -------------- 4 3% \MATHvStrich 2 2 2 a b c z % \MATHvStrich (e, f, g; q) % \MATHloEck q q q % \MATHroEck -1 --, --, -- a b c \MATHgroesser ----------------------------------------------- z \goodbreakpoint% In[3]:= \%1/.psph Split at: 4 \goodbreakpoint% 1 1 1 ----, ----, ----, q % \MATHluEck 2 2 2 % \MATHruEck (a, b, c; q) 3 % \MATHvStrich e q f q g q e f g % \MATHvStrich 3 Out[3]= z \MATHphi % \MATHvStrich ; q, ------- % \MATHvStrich ------------- + 4 3% \MATHvStrich 1 1 1 a b c z % \MATHvStrich (e, f, g; q) % \MATHloEck ----, ----, ---- % \MATHroEck 3 2 2 2 a q b q c q % \MATHluEck 4 4 4 % \MATHruEck (a, b, c; q) 4 % \MATHvStrich a q , b q , c q , q % \MATHvStrich 4 \MATHgroesser z \MATHphi % \MATHvStrich ; q, z % \MATHvStrich ------------- 4 3% \MATHvStrich 4 4 4 % \MATHvStrich (e, f, g; q) % \MATHloEck e q , f q , g q % \MATHroEck 4 \endMATH \Seealso ps, ph, phps, Ers, PosListe. \Name psShift \Description Rule that shifts the summation index in a bilateral basic hypergeometric series. \vskip6pt \leavevmode\hphantom{Description: } $\dsize {}_r \psi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q,z\right] \to z^m\left((-1)^mq^{\binom m2}\right)^{s-r} \frac {\prod _{i=1} ^{r}(a_i;q)_m} {\prod _{i=1} ^{s}(b_i;q)_m} {}_r \psi_s\!\left[\matrix a_1q^m,\dots,a_rq^m\\ b_1q^m,\dots,b_sq^m\endmatrix; q,zq^{m(s-r)}\right]$. \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.psShift. \Example \MATH In[1]:= ps[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace e,f,g% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 3 3% \MATHvStrich e, f, g % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.psShift shift by: 5 \goodbreakpoint% (a, b, c; q) % \MATHluEck 5 5 5 % \MATHruEck 5 5 % \MATHvStrich a q , b q , c q % \MATHvStrich Out[2]= z ------------- ps % \MATHvStrich ; q, z % \MATHvStrich (e, f, g; q) 3 3% \MATHvStrich 5 5 5 % \MATHvStrich 5 % \MATHloEck e q , f q , g q % \MATHroEck \endMATH \Seealso psEinf, psPerm, ps, PSort, Ers, PosListe. \Name psSUM \Description Rule that transforms a \hbox{\tt ps[]} into a \hbox{\tt SUM[]}. \Usage Expr/.psSUM. \Example \MATH In[1]:= ps[% \MATHlbrace a,b,c,q\MATHhoch -n% \MATHrbrace ,% \MATHlbrace e,f,q\MATHhoch m% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck -n % \MATHruEck % \MATHvStrich a, b, c, q % \MATHvStrich Out[1]= ps % \MATHvStrich ; q, z % \MATHvStrich 4 3% \MATHvStrich m % \MATHvStrich % \MATHloEck e, f, q % \MATHroEck \goodbreakpoint% In[2]:= \%/.psSUM Is n a nonnegative integer? [y|n]: n Is m a nonnegative integer? [y|n]: n A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% 2 \MATHinfty k/2 - k /2 k -n % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck q z (a; q) (b; q) (c; q) (q ; q) \MATHbackslash k k k k Out[2]= \MATHgroesser ------------------------------------------------ / k m % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck (-1) (e; q) (f; q) (q ; q) k=-\MATHinfty k k k \goodbreakpoint% In[3]:= \%1/.psSUM Is n a nonnegative integer? [y|n]: y Is m a nonnegative integer? [y|n]: y A basic hypergeometric series is converted into a sum. Enter a variable for the summation index: k \goodbreakpoint% 2 n k/2 - k /2 k -n % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck q z (a; q) (b; q) (c; q) (q ; q) \MATHbackslash k k k k Out[3]= \MATHgroesser ------------------------------------------------ / k m % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck (-1) (e; q) (f; q) (q ; q) k=1 - m k k k \endMATH \Seealso ps, SUM, SUMps, Ers, PosListe. \Name RS \Description \hbox{\tt RS} is the right-hand side in \hbox{\tt Gleichung}. \Usage RS. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= RS \goodbreakpoint% n c a (-; q) a n Out[2]= ---------- (c; q) n \goodbreakpoint% In[3]:= Add[1] \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[3]= 1 + \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == 1 + % ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[4]:= RS \goodbreakpoint% n c a (-; q) a n Out[4]= 1 + ---------- (c; q) n \goodbreakpoint% In[5]:= RS=1/pq[q,m] \goodbreakpoint% 1 Out[5]= ------- (q; q) m \goodbreakpoint% In[6]:= Gleichung \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich 1 Out[6]= 1 + \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ------- 2 1\MATHvStrich \MATHvStrich (q; q) \MATHloEck c \MATHroEck m \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers,\linebreak Subst. \Name S1001 \Description Summation formula (\cite{\GaRaAA}, (1.3.2); Appendix (II.3)) in form of a rule. $$ {}_1\phi _0\!\left [ \matrix \let\over/ a\\ \let\over/ -\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ a z;q)_\infty}} {{( \let\over/ z;q)_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S0110 \Description Summation formula (\cite{\GaRaAA}, (1.6.1); Appendix (II.28)) in form of a rule. $$ {}_0\psi _1\!\left [ \matrix -\\ 0\endmatrix ;q,z\right ] \longrightarrow (q,q/z,z;q)_\infty $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S1101 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.6(ii); Appendix (II.5)) in form of a rule. $$ {}_1\phi _1\!\left [ \matrix \let\over/ a\\ \let\over/ c\endmatrix ;q,{c\over a}\right ] \longrightarrow \frac {{( \let\over/ {c\over a};q)_\infty}} {{( \let\over/ c;q)_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S1102 \Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9); $b\to\infty$) in form of a rule. $$ {}_1\phi _1\!\left [ \matrix \let\over/ a\\ \let\over/ 0\endmatrix ;q,-q\right ] \longrightarrow {{( \let\over/ -q;q)_\infty}} {{( \let\over/ aq;q^2)_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S1110 \Description Summation formula (\cite{\GaRaAA}, (5.2.1); Appendix (II.29)) in form of a rule. $$ {} _{1} \psi _{1} \! \left [ \matrix \let \over / a\\ \let \over / b\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow { {(\let \over / q, {b\over a}, a z, {q\over {a z}} ;q) _\infty} \over {(\let \over / b, {q\over a}, z, {b\over {a z}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2101 \Description Summation formula (\cite{\GaRaAA}, (1.5.3); Appendix (II.6)) in form of a rule. $$ {}_2\phi _1\!\left [ \matrix \let\over/ a,{q^{-n}}\\ \let\over/ c\endmatrix ;q,q\right ] \longrightarrow {{{a^n} {(\let\over/ {c\over a};q)}_{n}}\over {{(\let\over/ c;q)}_{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2102 \Description Summation formula (\cite{\GaRaAA}, (1.5.2); Appendix (II.7)) in form of a rule. $$ {}_2\phi _1\!\left [ \matrix \let\over/ a,{q^{-n}}\\ \let\over/ c\endmatrix ;q,{{c {q^n}}\over a}\right ] \longrightarrow {{{(\let\over/ {c\over a};q)}_{n}}\over {{(\let\over/ c;q)}_{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2103 \Description Summation formula (\cite{\GaRaAA}, (1.5.1); Appendix (II.8)) in form of a rule. $$ {}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,{c\over {a b}}\right ] \longrightarrow \frac {{( \let\over/ {c\over a},{c\over b};q)_\infty}} {{( \let\over/ c,{c\over {a b}};q)_\infty}} $$ \Example \MATH In[1]:= ph[\MATHlbrace a,b\MATHrbrace ,\MATHlbrace q\MATHhoch 2*a*b\MATHrbrace % ,q,q\MATHhoch 2] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich a, b 2 \MATHvStrich Out[1]= \MATHphi \MATHvStrich ; q, q \MATHvStrich 2 1\MATHvStrich 2 \MATHvStrich \MATHloEck a b q \MATHroEck \goodbreakpoint% In[2]:= \%/.S2103 \goodbreakpoint% 2 2 (b q , a q ; q) \MATHinfty Out[2]= ---------------- 2 2 (a b q , q ; q) \MATHinfty \endMATH \Seealso S3201, SListe, SumListe, Ers, PosListe. \Name S2104 \Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9)) in form of a rule. $$ {}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ {{a q}\over b}\endmatrix ;q,-{q\over b}\right ] \longrightarrow \frac {{( \let\over/ -q;q)_\infty}} {{( \let\over/ -{q\over b},{{a q}\over b};q)_\infty}} {(\let\over/ a q,{{a {q^2}}\over {{b^2}}};{q^2})}_{\infty} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2105 \Description Summation formula (\cite{\GaRaAA}, Ex~1.6(i)) in form of a rule. $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / {q^{-n}}, {q^{1 - n}}\\ \let \over / b^2 q\endmatrix ;q^2, {\displaystyle q^2} \right ] \longrightarrow q^{-\binom n2} {{({\let \over / b^2}; q^2) _{n} }\over { ({\let \over / b^2}; q) _{n} }} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2106 \Description Summation formula (\cite{\GaRaAA}, Ex~1.7) in form of a rule. $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, a q\\ \let \over / a\endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{\left( 1 + a z \right) ({\let \over / a^2 q z}; q) _{\infty} }\over {({\let \over / z}; q) _{\infty} }} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2107 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.8) in form of a rule. $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, {{a^2}\over b}\\ \let \over / b\endmatrix ;q^2, {\displaystyle {{b q}\over {a^2}}} \right ] \longrightarrow \frac {1} {2} {{\left( { {(\let \over / -{b\over a} ;q) _\infty} \over {(\let \over / -a ;q) _\infty} } + { {(\let \over / {b\over a} ;q) _\infty} \over {(\let \over / a ;q) _\infty} } \right) { {(\let \over / a^2, q ;q^2) _\infty} \over {(\let \over / b, {{b q}\over {a^2}} ;q^2) _\infty} }}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2161 \Description Summation formula (\cite{\GaRaAA}, (2.10.13); Appendix (II.23)) in form of a rule. $$ {}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,q\right ] \longrightarrow \frac {{( \let\over/ {q\over c},{{a b q}\over c};q)_\infty}} {{( \let\over/ {{a q}\over c},{{b q}\over c};q)_\infty}} - \frac {{( \let\over/ {q\over c},a,b;q)_\infty}} {{( \let\over/ {c\over q},{{a q}\over c},{{b q}\over c};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ {{a q}\over c},{{b q}\over c}\\ \let\over/ {{{q^2}}\over c}\endmatrix ;q,q\right ] $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2201 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(i); Appendix (II.10)) in form of a rule. $$ {}_2\phi _2\!\left [ \matrix \let\over/ a,{q\over a}\\ \let\over/ -q,b\endmatrix ;q,-b\right ] \longrightarrow {{(\let\over/ a b,{{b q}\over a};{q^2})}_{\infty} \over {{(\let\over/ b;q)}_{\infty}}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2202 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(ii); Appendix (II.11)) in form of a rule. $$ {}_2\phi _2\!\left [ \matrix \let\over/ {a^2},{b^2}\\ \let\over/ a b {\sqrt{q}},- a b {\sqrt{q}} \endmatrix ;q,-q\right ] \longrightarrow \frac {{( \let\over/ {a^2} q,{b^2} q;{q^2})_\infty}} {{( \let\over/ q,{a^2} {b^2} q;{q^2})_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S2210 \Description Summation formula (\cite{\GaRaAA}, (5.3.4); Appendix (II.30)) in form of a rule. $$ {} _{2} \psi _{2} \! \left [ \matrix \let \over / b, c\\ \let \over / {{a q}\over b}, {{a q}\over c}\endmatrix ;q, {\displaystyle -{{a q}\over {b c}}} \right ] \longrightarrow { {(\let \over / {{a q}\over {b c}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {q\over b}, {q\over c}, -{{a q}\over {b c}} ;q) _\infty} } ({\let \over / {{a {q^2}}\over {{b^2}}}, {{a {q^2}}\over {{c^2}}}, {q^2}, a q, {q\over a}}; {q^2}) _{\infty} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S3201 \Description Summation formula (\cite{\GaRaAA}, (1.7.2); Appendix (II.12)) in form of a rule. $$ {}_3\phi _2\!\left [ \matrix \let\over/ a,b,{q^{-n}}\\ \let\over/ c,{{a b {q^{1 - n}}}\over c}\endmatrix ;q,q\right ] \longrightarrow {{{(\let\over/ {c\over a};q)}_{n} {(\let\over/ {c\over b};q)}_{n}}\over {{(\let\over/ c;q)}_{n} {(\let\over/ {c\over {a b}};q)}_{n}}} $$ where $n$ is a nonnegative integer. \Example \MATH In[1]:= ph[\MATHlbrace a,b,q\MATHhoch -n\MATHrbrace ,\MATHlbrace % a*b/q\MATHhoch n,q\MATHrbrace ,q,q] \goodbreakpoint% -n \MATHluEck a, b, q \MATHruEck \MATHvStrich \MATHvStrich Out[1]= \MATHphi \MATHvStrich a b ; q, q \MATHvStrich 3 2\MATHvStrich ---, q \MATHvStrich \MATHloEck n \MATHroEck q \goodbreakpoint% In[2]:= \%/.S3201 Is n a nonnegative integer? [y|n]: y \goodbreakpoint% a b (--; q) (--; q) n n n n q q Out[2]= ------------------- -n a b (q ; q) (---; q) n n n q \goodbreakpoint% In[3]:= \%\%/.S3201 Is n a nonnegative integer? [y|n]: n \goodbreakpoint% -n \MATHluEck a, b, q \MATHruEck \MATHvStrich \MATHvStrich Out[3]= \MATHphi \MATHvStrich a b ; q, q \MATHvStrich 3 2\MATHvStrich ---, q \MATHvStrich \MATHloEck n \MATHroEck q \goodbreakpoint% In[4]:= ph[\MATHlbrace a,q\MATHhoch -m,q\MATHhoch -n\MATHrbrace ,\MATHlbrace % a/q\MATHhoch (n+m),q\MATHrbrace ,q,q] \goodbreakpoint% \MATHluEck -m -n \MATHruEck \MATHvStrich a, q , q \MATHvStrich Out[4]= \MATHphi \MATHvStrich ; q, q \MATHvStrich 3 2\MATHvStrich -m - n \MATHvStrich \MATHloEck a q , q \MATHroEck \goodbreakpoint% In[5]:= \%/.S3201 Is n a nonnegative integer? [y|n]: n Is m a nonnegative integer? [y|n]: y \goodbreakpoint% a -m - n (--; q) (q ; q) m m m q Out[5]= ------------------------- -m -m - n (q ; q) (a q ; q) m m \endMATH \Seealso S2103, SListe, SumListe, Ers, PosListe. \Name S3202 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.7; Appendix (II.15)) in form of a rule. $$ {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-2 n}},b,c\\ \let\over/ {{{q^{1 - 2 n}}}\over b},{{{q^{1 - 2 n}}}\over c}\endmatrix ;q,{{{q^{2 - n}}}\over {b c}}\right ] \longrightarrow {{{(\let\over/ b;q)}_{n} {(\let\over/ c;q)}_{n} {(\let\over/ b c;q)}_{2 n} {(\let\over/ q;q)}_{2 n}}\over {{(\let\over/ b;q)}_{2 n} {(\let\over/ c;q)}_{2 n} {(\let\over/ b c;q)}_{n} {(\let\over/ q;q)}_{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S3203 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.1) in form of a rule. $$ {} _{3} W _{2} ({\displaystyle a; -}; q, {\displaystyle t}) \longrightarrow \left( 1 - a q t^2 \right) { {(\let \over / a q^2 t ;q) _\infty} \over {(\let \over / t ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S3204 \Description Summation formula (\cite{\GaRaAA}, Ex.~3.9) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, { \lambda } q, b\\ \let \over / { \lambda }, {{{{{ \lambda }}^2} q}\over b}\endmatrix ;q, {\displaystyle {{{{{ \lambda }}^2}}\over {a b^2}}} \right ] \longrightarrow {{\left( 1 - { \lambda } + {{{{ \lambda } \over b} \left( 1 - {{{ \lambda }}\over a} \right) }} \right)} \over {\left( 1 - { \lambda } \right) \left( 1 + {{{ \lambda }}\over b} \right) }} {{ { {(\let \over / {{{{{ \lambda }}^2}}\over {b^2}}, {{{{{ \lambda }}^2} q}\over {a b}} ;q) _\infty} \over {(\let \over / {{{{{ \lambda }}^2} q}\over b}, {{{{{ \lambda }}^2}}\over {a b^2}} ;q) _\infty} }} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S3261 \Description Summation formula (\cite{\GaRaAA}, (2.10.12); Appendix (II.24)) in form of a rule. $$\multline {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/ e,{{a b c q}\over e}\endmatrix ;q,q\right ] \\\longrightarrow \frac {{( \let\over/ {q\over e},{{b c q}\over e},{{a c q}\over e},{{a b q}\over e};q)_\infty}} {{( \let\over/ {{a q}\over e},{{b q}\over e},{{c q}\over e},{{a b c q}\over e};q)_\infty}} - \frac {{( \let\over/ {q\over e},a,b,c,{{a b c {q^2}}\over {{e^2}}};q)_\infty}} {{( \let\over/ {e\over q},{{a q}\over e},{{b q}\over e},{{c q}\over e},{{a b c q}\over e};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {{a q}\over e},{{b q}\over e},{{c q}\over e}\\ \let\over/ {{{q^2}}\over e},{{a b c {q^2}}\over {{e^2}}}\endmatrix ;q,q\right ] \endmultline$$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S3310 \Description Summation formula (\cite{\GaRaAA}, Ex.~5.18(i); Appendix (II.31)) in form of a rule. $$ {} _{3} \psi _{3} \! \left [ \matrix \let \over / b, c, d\\ \let \over / {q\over b}, {q\over c}, {q\over d}\endmatrix ;q, {\displaystyle {q\over {b c d}}} \right ] \longrightarrow { {(\let \over / q, {q\over {b c}}, {q\over {b d}}, {q\over {c d}} ;q) _\infty} \over {(\let \over / {q\over b}, {q\over c}, {q\over d}, {q\over {b c d}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4301 \Description Summation formula (\cite{\GaRaAA}, (2.7.2); Appendix (II.13)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a,- {\sqrt{a}} q ,b,c\\ \let\over/ -{\sqrt{a}},{{a q}\over b},{{a q}\over c}\endmatrix ;q,{{{\sqrt{a}} q}\over {b c}}\right ] \longrightarrow \frac {{( \let\over/ a q,{{{\sqrt{a}} q}\over b},{{{\sqrt{a}} q}\over c},{{a q}\over {b c}};q)_\infty}} {{( \let\over/ {{a q}\over b},{{a q}\over c},{\sqrt{a}} q,{{{\sqrt{a}} q}\over {b c}};q)_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4302 \Description Summation formula (\cite{\GaRaAA}, (2.7.2), terminating form; Appendix (II.14)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a,- {\sqrt{a}} q ,b,{q^{-n}}\\ \let\over/ -{\sqrt{a}},{{a q}\over b},a {q^{1 + n}}\endmatrix ;q,{{{\sqrt{a}} {q^{1 + n}}}\over b}\right ] \longrightarrow {{{(\let\over/ a q;q)}_{n} {(\let\over/ {{{\sqrt{a}} q}\over b};q)}_{n}}\over {{(\let\over/ {\sqrt{a}} q;q)}_{n} {(\let\over/ {{a q}\over b};q)}_{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4303 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.8; Appendix (II.17)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a {q^n},c,-c,{q^{-n}}\\ \let\over/ {\sqrt{a}} {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}} ,{c^2}\endmatrix ;q,q\right ] \longrightarrow \cases 0&\text {if $n$ is odd}\\ \displaystyle c^n {{(\let\over/ q;{q^2})}_{{n\over 2}} {(\let\over/ {{a q}\over {{c^2}}};{q^2})}_{{n\over 2}}}\over {\displaystyle{(\let\over/ a q;{q^2})}_{{n\over 2}} {(\let\over/ {c^2} q;{q^2})}_{{n\over 2}}}& \text {if $n$ is even}\endcases $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4304 \Description Summation formula (Andrews [1976a] in \cite{\GaRaAA}, Appendix (II.19), corrected) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ {q^{1 + n}},c,-c,{q^{-n}}\\ \let\over/ e,{{{c^2} q}\over e},-q\endmatrix ;q,q\right ] \longrightarrow q^{n(n+1)/2}\, { {(\let\over/ {e\over {{q^n}}},e {q^{1 + n}},{{{c^2} {q^{1 - n}}}\over e},{{{c^2} {q^{2 + n}}}\over e};{q^2})}_{\infty}\over {{(\let\over/ e,{{{c^2} q} \over e};q)}_{\infty}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4305 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.6) in form of a rule. $$ {} _{4} \phi _{3} \! \left [ \matrix \let \over / {q^{-n}}, b, c, -{{{q^{1 - n}}}\over {b c}}\\ \let \over / {{{q^{1 - n}}}\over b}, {{{q^{1 - n}}}\over c}, - b c \endmatrix ;q, {\displaystyle q} \right ] \longrightarrow \cases 0&\text {if $n$ is odd}\\ \displaystyle {{({\let \over / b c}; q) _{n}}\over {({\let \over / b, c}; q) _{n}}} {{({\let \over / q, b^2, c^2}; q^2) _{\frac {n} {2}}}\over {({\let \over / b^2 c^2}; q^2) _{\frac {n} {2}}}} &\text {if $n$ is even}\endcases $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4306 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.14(i)) in form of a rule. $$ {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, b, {q^{-n}}\\ \let \over / {\sqrt{a}}, {{a q}\over b}, b^2 {q^{1 - n}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow {{({\let \over / {a\over {b^2}}, {1\over b}, -{{{\sqrt{a}} q}\over b}}; q) _{n}}\over {({\let \over / {b^{-2}}, {{a q}\over b}, -{{{\sqrt{a}}}\over b}}; q) _{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4307 \Description Summation formula (\cite{\GaRaAA}, (2.7.2)) in form of a rule. $$ {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, b, c\\ \let \over / {\sqrt{a}}, {{a q}\over b}, {{a q}\over c}\endmatrix ;q, {\displaystyle -{{{\sqrt{a}} q}\over {b c}}} \right ] \longrightarrow { {(\let \over / a q, {{a q}\over {b c}}, -{{{\sqrt{a}} q}\over b}, -{{{\sqrt{a}} q}\over c} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, - {\sqrt{a}} q , -{{{\sqrt{a}} q}\over {b c}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4308 \Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to \sqrt{aq}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA}, (2.6.2)) in form of a rule. $$ {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, - {\sqrt{a}} q , b, {q^{-n}}\\ \let \over / -{\sqrt{a}}, {{a q}\over b}, {b^2} {q^{1 - n}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow {{({\let \over / {a\over {{b^2}}}, {1\over b}, {{{\sqrt{a}} q}\over b}}; q) _{n}} \over {({\let \over / {b^{-2}}, {{a q}\over b}, {{{\sqrt{a}}}\over b}}; q) _{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4361 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.9) in form of a rule.\NoBlackBoxes $$\multline {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, -b, {{a q}\over {c^2}}\\ \let \over / {{a q}\over c}, -{{a q}\over c}, b^2\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / {q\over {b^2}}, -q ;q) _\infty} \over {(\let \over / {{a q}\over {b^2}}, - a q ;q) _\infty} } { {(\let \over / a^2 q^2, {{a q^2}\over {b^2}}, {{a q^2}\over {c^2}}, {{a^2 q^2}\over {b^2 c^2}} ;q^2) _\infty} \over {(\let \over / {{q^2}\over {b^2}}, {{a^2 q^2}\over {c^2}}, a q^2, {{a q^2}\over {b^2 c^2}} ;q^2) _\infty} }\\ - { {(\let \over / {q\over {b^2}}, a, b, -b, {{a q}\over {c^2}}, {{a q^2}\over {b^2 c}}, -{{a q^2}\over {b^2 c}} ;q) _\infty} \over {(\let \over / {{b^2}\over q}, {q\over b}, -{q\over b}, {{a q}\over {b^2}}, {{a q}\over c}, -{{a q}\over c}, {{a q^2}\over {b^2 c^2}} ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over / {q\over b}, -{q\over b}, {{a q}\over {b^2}}, {{a q^2}\over {b^2 c^2}}\\ \let \over / {{a q^2}\over {b^2 c}}, -{{a q^2}\over {b^2 c}}, {{q^2}\over {b^2}}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S4410 \Description Summation formula (\cite{\GaRaAA}, (5.3.3); Appendix (II.32)) in form of a rule. $$ {} _{4} \psi _{4} \! \left [ \matrix \let \over / - {\sqrt{a}} q , b, c, d\\ \let \over / -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d}\endmatrix ;q, {\displaystyle {{{a^{{3\over 2}}} q}\over {b c d}}} \right ] \longrightarrow { {(\let \over / a q, {{a q}\over {b c}}, {{a q}\over {b d}}, {{a q}\over {c d}}, {{{\sqrt{a}} q}\over b}, {{{\sqrt{a}} q}\over c}, {{{\sqrt{a}} q}\over d}, q, {q\over a} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {q\over b}, {q\over c}, {q\over d}, {\sqrt{a}} q, {q\over {{\sqrt{a}}}}, {{{a^{{3\over 2}}} q}\over {b c d}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S5401 \Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to \sqrt{aq}$) in form of a rule. $$ {} _{5} W _{4} ({\displaystyle a; b, c}; q, {\displaystyle {{{\sqrt{a q}} }\over {b c}}}) \longrightarrow { {(\let \over / a q, {{a q}\over {b c}}, {{{\sqrt{a q}} }\over b}, {{{\sqrt{a q}} }\over c} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {\sqrt{a q}} , {{{\sqrt{a q}} }\over {b c}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S5402 \Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to q\sqrt{a}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA}, (2.6.2)) in form of a rule. $$ {} _{5} \phi _{4} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, - {\sqrt{a}} q , b, {q^{-n}}\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {b^2} {q^{2 - n}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow {{({\let \over / {a\over {{b^2} q}}, {1\over {b q}}}; q) _{n}} \over {({\let \over / {1\over {{b^2} q}}, {{a q}\over b}}; q) _{n}}} {{({\let \over / {{a q}\over {{b^2}}}}; {q^2}) _{n}} \over {({\let \over / {a\over {{b^2} q}}}; {q^2}) _{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S5501 \Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to\infty$) in form of a rule. $$ {} _{5} \phi _{5} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, - {\sqrt{a}} q , b, c\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, 0\endmatrix ;q, {\displaystyle {{a q}\over {b c}}} \right ] \longrightarrow { {(\let \over / a q, {{a q}\over {b c}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S6501 \Description Summation formula (\cite{\GaRaAA}, (2.7.1); Appendix (II.20)) in form of a rule. \BlackBoxes $$ {}_6W _5(a;b,c,d;q,{{a q}\over {b c d}}) \longrightarrow \frac {{( \let\over/ a q,{{a q}\over {b c}},{{a q}\over {b d}},{{a q}\over {c d}};q)_\infty}} {{( \let\over/ {{a q}\over b},{{a q}\over c},{{a q}\over d},{{a q}\over {b c d}};q)_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S6502 \Description Summation formula (\cite{\GaRaAA}, (2.4.2); Appendix (II.21)) in form of a rule. $$ {}_6W _5(a;b,c,{q^{-n}};q,{{a {q^{1 + n}}}\over {b c}}) \longrightarrow {{{(\let\over/ a q;q)}_{n} {(\let\over/ {{a q}\over {b c}};q)}_{n}}\over {{(\let\over/ {{a q}\over b};q)}_{n} {(\let\over/ {{a q}\over c};q)}_{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S6610 \Description Summation formula (\cite{\GaRaAA}, (5.3.1); Appendix (II.33)) in form of a rule. $$ {} _{6} \psi _{6} \! \left [ \matrix \let \over / {\sqrt{a}} q, - {\sqrt{a}} q , b, c, d, e\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}\endmatrix ;q, {\displaystyle {{{a^2} q}\over {b c d e}}} \right ] \longrightarrow { {(\let \over / a q, {{a q}\over {b c}}, {{a q}\over {b d}}, {{a q}\over {b e}}, {{a q}\over {c d}}, {{a q}\over {c e}}, {{a q}\over {d e}}, q, {q\over a} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {q\over b}, {q\over c}, {q\over d}, {q\over e}, {{{a^2} q}\over {b c d e}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S8701 \Description Summation formula (\cite{\GaRaAA}, (2.6.2); Appendix (II.22)) in form of a rule. $$ {}_8\phi _7\!\left [ \matrix \let\over/ a,{\sqrt{a}} q,- {\sqrt{a}} q ,b,c,d,{a^2 q^{1+n} \over b c d },{q^{-n}}\\ \let\over/ {\sqrt{a}},-{\sqrt{a}}, {{a q}\over b},{{a q}\over c},{{a q}\over d},{{b c d}\over {a {q^n}}},a {q^{1 + n}}\endmatrix ;q,q\right ] \longrightarrow {{{(\let\over/ a q;q)}_{n} {(\let\over/ {{a q}\over {b c}};q)}_{n} {(\let\over/ {{a q}\over {b d}};q)}_{n} {(\let\over/ {{a q}\over {c d}};q)}_{n}}\over {{(\let\over/ {{a q}\over b};q)}_{n} {(\let\over/ {{a q}\over c};q)}_{n} {(\let\over/ {{a q}\over d};q)}_{n} {(\let\over/ {{a q}\over {b c d}};q)}_{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S8702 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(i); Appendix (II.16)) in form of a rule. $$ {}_8W _7(-{{c {\sqrt{a b}}}\over {{\sqrt{q}}}};a,b,c,-c,-{{{\sqrt{a b q}} }\over c};q,{{c {\sqrt{q}}}\over {{\sqrt{a b}} }}) \longrightarrow \frac {{( \let\over/ - c {\sqrt{a b q}} ,-{{c {\sqrt{q}}}\over {{\sqrt{a b}} }};q)_\infty}} {{( \let\over/ -{{c {\sqrt{b q}} }\over {{\sqrt{a}}}},-{{c {\sqrt{a q}} }\over {{\sqrt{b}}}};q)_\infty}} \frac {{( \let\over/ a q,b q,{{{c^2} q}\over a},{{{c^2} q}\over b};{q^2})_\infty}} {{( \let\over/ q,a b q,{c^2} q,{{{c^2} q}\over {a b}};{q^2})_\infty}} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S8703 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(ii); Appendix (II.18)) in form of a rule. $$ {}_8W _7(-c;a,{q\over a},c,-d,-{q\over d};q,c) \longrightarrow \frac {{( \let\over/ -c,- c q ;q)_\infty}} {{( \let\over/ c d,{{c q}\over d},- a c ,-{{c q}\over a};q)_\infty}} {(\let\over/ a c d,{{a c q}\over d},{{c d q}\over a},{{c {q^2}}\over {a d}};{q^2})}_{\infty} $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S8704 \Description Summation formula (\cite{\GaRaAA}, Ex.~3.10) in form of a rule. $$ {} _{8} W _{7} ({\displaystyle -\lambda ; {\sqrt{\lambda }} q, - {\sqrt{\lambda }} q , a, b, -b}; q, {\displaystyle {{ \lambda}\over {a {b^2}}}}) \longrightarrow { {(\let \over / { \lambda } q, {{{ \lambda }}\over a}, - { \lambda } q , {{{ \lambda }}\over {b^2}}, {{{ \lambda } q}\over {a b}}, -{{{ \lambda } q}\over {a b}} ;q) _\infty} \over {(\let \over / { \lambda }, {{{ \lambda } q}\over a}, -{{{ \lambda } q}\over a}, {{{ \lambda } q}\over b}, -{{{ \lambda } q}\over b}, {{{ \lambda }}\over {a b^2}} ;q) _\infty} } $$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S8761 \Description Summation formula (\cite{\GaRaAA}, (2.11.7); Appendix (II.25)) in form of a rule. $$\multline {}_8W _7(a;b,c,d,e,{{{a^2} q}\over {b c d e}};q,q) \longrightarrow \frac {{( \let\over/ a q,{b\over a},{{a q}\over {c d}},{{a q}\over {c e}},{{b d e}\over a},{{a q}\over {d e}},{{b c e}\over a},{{b c d}\over a};q)_\infty}} {{( \let\over/ {{a q}\over c},{{a q}\over d},{{a q}\over e},{{b c d e}\over a},{{b c}\over a},{{b d}\over a},{{b e}\over a},{{a q}\over {c d e}};q)_\infty}} \\ - \frac {{( \let\over/ a q,{b\over a},c,d,e,{{{a^2} q}\over {b c d e}},{{b q}\over c},{{b q}\over d},{{b q}\over e},{{{b^2} c d e}\over {{a^2}}};q)_\infty}} {{( \let\over/ {a\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{b c d e}\over a},{{b c}\over a},{{b d}\over a},{{b e}\over a},{{a q}\over {c d e}},{{{b^2} q}\over a};q)_\infty}} \\ {}_8W _7({{{b^2}}\over a};b,{{b c}\over a},{{b d}\over a},{{b e}\over a},{{a q}\over {c d e}};q,q) \endmultline$$ \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name S10901 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.12) in form of a rule. $$ {} _{10} W _{9} ({\displaystyle a; {\sqrt{b}}, -{\sqrt{b}}, {\sqrt{b q}} , - {\sqrt{b q}} , {a\over b}, {{a^2 {q^{1 + n}}}\over b}, {q^{-n}}}; q, {\displaystyle q}) \longrightarrow {{({\let \over / a q, {{a^2 q}\over {b^2}}}; q) _{n}}\over {({\let \over / {{a q}\over b}, {{a^2 q}\over b}}; q) _{n}}} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, SListe, SumListe, Ers, PosListe. \Name SchreibeZahl \Description Variable that counts the number of expressions already written by using \hbox{\tt TeXMat}. Can be reset by defining a new value. \Usage SchreibeZahl=n\MATHtief\ Integer. \Example \MATH In[1]:= SchreibeZahl \goodbreakpoint% Out[1]= 0 \goodbreakpoint% In[2]:= TeXMat[pq[a,n],file] \goodbreakpoint% In[3]:= !type file.m A[1]:= pq[a, n, q] \goodbreakpoint% In[3]:= !type file.tex A[1]:= (\MATHlbrace \MATHbackslash let \MATHbackslash % over / a\MATHrbrace ; q) \MATHtief \MATHlbrace n\MATHrbrace \goodbreakpoint% In[3]:= SchreibeZahl=4 \goodbreakpoint% Out[3]= 4 \goodbreakpoint% In[4]:= TeXMat[pq[a,n],file] \goodbreakpoint% In[5]:= !type file.m A[1]:= pq[a, n, q] A[5]:= pq[a, n, q] \goodbreakpoint% In[5]:= !type file.tex A[1]:= (\MATHlbrace \MATHbackslash let \MATHbackslash % over / a\MATHrbrace ; q) \MATHtief \MATHlbrace n\MATHrbrace A[5]:= (\MATHlbrace \MATHbackslash let \MATHbackslash % over / a\MATHrbrace ; q) \MATHtief \MATHlbrace n\MATHrbrace \endMATH \Seealso TeXMat. \Name Sgl0110 \Description Summation formula (\cite{\GaRaAA}, (1.6.1); Appendix (II.28)) in form of an equation. It is the same summation as that in \hbox{\tt S0110}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl1001 \Description Summation formula (\cite{\GaRaAA}, (1.3.2); Appendix (II.3)) in form of an equation. It is the same summation as that in \hbox{\tt S1001}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl1101 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.6(ii); Appendix (II.5)) in form of an equation. It is the same summation as that in \hbox{\tt S1101}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl1102 \Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9); $b\to\infty$) in form of an equation. It is the same summation as that in \hbox{\tt S1102}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl1110 \Description Summation formula (\cite{\GaRaAA}, (5.2.1); Appendix (II.29)) in form of an equation. It is the same summation as that in \hbox{\tt S1110}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2101 \Description Summation formula (\cite{\GaRaAA}, (1.5.3); Appendix (II.6)) in form of an equation. It is the same summation as that in \hbox{\tt S2101}. \Example \MATH In[1]:= \MATHlbrace a,c,q\MATHrbrace \goodbreakpoint% Out[1]= \MATHlbrace a, c, q\MATHrbrace \goodbreakpoint% In[2]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[3]:= Sgl2101 Do you want to set values for the equation? [y|n]: y a=a\MATHhoch 2 c=1/b n=n Do you want to set a value for q in the equation? [y|n]: y q=q\MATHhoch 2 \goodbreakpoint% 2 n 1 2 a (----; q ) \MATHluEck 2 -2 n \MATHruEck 2 n \MATHvStrich a , q 2 2 \MATHvStrich a b Out[3]= \MATHphi \MATHvStrich ; q , q \MATHvStrich == % ---------------- 2 1\MATHvStrich 1 \MATHvStrich 1 2 \MATHloEck - \MATHroEck (-; q ) b b n \goodbreakpoint% In[4]:= \MATHlbrace a,c,q\MATHrbrace \goodbreakpoint% Out[4]= \MATHlbrace a, c, q\MATHrbrace \goodbreakpoint% In[5]:= a=b\MATHhoch 2 \goodbreakpoint% 2 Out[5]= b \goodbreakpoint% In[6]:= q=p \goodbreakpoint% Out[6]= p \goodbreakpoint% In[7]:= Sgl2101 Some variables have a value. Should the variables \MATHlbrace a, c, n\MATHrbrace be cleared? Do you want to set values for the equation (v)? [y|n|yv|nv]: nv a=a c=c n=n q has a value. Should q be cleared? Do you want to set a value for q in the equation (v)? [y|n|yv|nv]: yv q=p\MATHhoch 2 \goodbreakpoint% 2 n c 2 b (--; p ) \MATHluEck \MATHruEck 2 n \MATHvStrich 2 -2 n 2 2 \MATHvStrich b Out[7]= \MATHphi \MATHvStrich b , p ; p , p \MATHvStrich == % -------------- 2 1\MATHvStrich \MATHvStrich 2 \MATHloEck c \MATHroEck (c; p ) n \goodbreakpoint% In[8]:= \MATHlbrace a,c,q\MATHrbrace \goodbreakpoint% 2 Out[8]= \MATHlbrace b , c, q\MATHrbrace \goodbreakpoint% In[9]:= Sgl2101 Some variables have a value. Should the variables \MATHlbrace a, c, n\MATHrbrace be cleared? Do you want to set values for the equation (v)? [y|n|yv|nv]: y Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[9]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[10]:= \MATHlbrace a,c,q\MATHrbrace \goodbreakpoint% Out[10]= \MATHlbrace a, c, q\MATHrbrace \endMATH \Seealso SumListe\$gl, Gleichung. \Name Sgl2102 \Description Summation formula (\cite{\GaRaAA}, (1.5.2); Appendix (II.7)) in form of an equation. It is the same summation as that in \hbox{\tt S2102}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2103 \Description Summation formula (\cite{\GaRaAA}, (1.5.1); Appendix (II.8)) in form of an equation. It is the same summation as that in \hbox{\tt S2103}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2104 \Description Summation formula (\cite{\GaRaAA}, (1.8.1); Appendix (II.9)) in form of an equation. It is the same summation as that in \hbox{\tt S2104}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2105 \Description Summation formula (\cite{\GaRaAA}, Ex~1.6(i)) in form of an equation. It is the same summation as that in \hbox{\tt S2105}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2106 \Description Summation formula (\cite{\GaRaAA}, Ex~1.7) in form of an equation. It is the same summation as that in \hbox{\tt S2106}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2107 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.8) in form of an equation. It is the same summation as that in \hbox{\tt S2107}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2161 \Description Summation formula (\cite{\GaRaAA}, (2.10.13); Appendix (II.23)) in form of an equation. It is the same summation as that in \hbox{\tt S2161}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2201 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(i); Appendix (II.10)) in form of an equation. It is the same summation as that in \hbox{\tt S2201}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2202 \Description Summation formula (\cite{\GaRaAA}, Ex.~1.19(ii); Appendix (II.11)) in form of an equation. It is the same summation as that in \hbox{\tt S2202}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl2210 \Description Summation formula (\cite{\GaRaAA}, (5.3.4); Appendix (II.30)) in form of an equation. It is the same summation as that in \hbox{\tt S2210}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl3201 \Description Summation formula (\cite{\GaRaAA}, (1.7.2); Appendix (II.12)) in form of an equation. It is the same summation as that in \hbox{\tt S3201}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl3202 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.7; Appendix (II.15)) in form of an equation. It is the same summation as that in \hbox{\tt S3202}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl3203 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.1) in form of an equation. It is the same summation as that in \hbox{\tt S3203}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl3204 \Description Summation formula (\cite{\GaRaAA}, Ex.~3.9) in form of an equation. It is the same summation as that in \hbox{\tt S3204}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl3261 \Description Summation formula (\cite{\GaRaAA}, (2.10.12); Appendix (II.24)) in form of an equation. It is the same summation as that in \hbox{\tt S3261}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl3310 \Description Summation formula (\cite{\GaRaAA}, Ex.~5.18(i); Appendix (II.31)) in form of an equation. It is the same summation as that in \hbox{\tt S3310}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4301 \Description Summation formula (\cite{\GaRaAA}, (2.7.2); Appendix (II.13)) in form of an equation. It is the same summation as that in \hbox{\tt S4301}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4302 \Description Summation formula (\cite{\GaRaAA}, (2.7.2), terminating form; Appendix (II.14)) in form of an equation. It is the same summation as that in \hbox{\tt S4302}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4303 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.8; Appendix (II.17)) in form of an equation. It is the same summation as that in \hbox{\tt S4303}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4304 \Description Summation formula (Andrews [1976a] in \cite{\GaRaAA}, Appendix (II.19)) in form of an equation. It is the same summation as that in \hbox{\tt S4304}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4305 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.6) in form of an equation. It is the same summation as that in \hbox{\tt S4305}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4306 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.14(i)) in form of an equation. It is the same summation as that in \hbox{\tt S4306}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4307 \Description Summation formula (\cite{\GaRaAA}, (2.7.2)) in form of an equation. It is the same summation as that in \hbox{\tt S4307}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4308 \Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to \sqrt{aq}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA}, (2.6.2)) in form of an equation. It is the same summation as that in \hbox{\tt S4308}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4361 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.9) in form of an equation. It is the same summation as that in \hbox{\tt S4361}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl4410 \Description Summation formula (\cite{\GaRaAA}, (5.3.3); Appendix (II.32)) in form of an equation. It is the same summation as that in \hbox{\tt S4410}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl5401 \Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to \sqrt{aq}$) in form of an equation. It is the same summation as that in \hbox{\tt S5401}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl5402 \Description Summation formula (\cite{\GaRaAA}, (2.8.3), $c\to q\sqrt{a}$, $d\to -q\sqrt a$, sum the $_8\phi_7$ by \cite{\GaRaAA}, (2.6.2)) in form of an equation. It is the same summation as that in \hbox{\tt S5402}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl5501 \Description Summation formula (\cite{\GaRaAA}, (2.7.1), $d\to\infty$) in form of an equation. It is the same summation as that in \hbox{\tt S5501}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl6501 \Description Summation formula (\cite{\GaRaAA}, (2.7.1); Appendix (II.20)) in form of an equation. It is the same summation as that in \hbox{\tt S6501}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl6502 \Description Summation formula (\cite{\GaRaAA}, (2.4.2); Appendix (II.21)) in form of an equation. It is the same summation as that in \hbox{\tt S6502}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl6610 \Description Summation formula (\cite{\GaRaAA}, (5.3.1); Appendix (II.33)) in form of an equation. It is the same summation as that in \hbox{\tt S6610}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl8701 \Description Summation formula (\cite{\GaRaAA}, (2.6.2); Appendix (II.22)) in form of an equation. It is the same summation as that in \hbox{\tt S8701}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl8702 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(i); Appendix (II.16)) in form of an equation. It is the same summation as that in \hbox{\tt S8702}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl8703 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.17(ii); Appendix (II.18)) in form of an equation. It is the same summation as that in \hbox{\tt S8703}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl8704 \Description Summation formula (\cite{\GaRaAA}, Ex.~3.10) in form of an equation. It is the same summation as that in \hbox{\tt S8704}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl8761 \Description Summation formula (\cite{\GaRaAA}, (2.11.7); Appendix (II.25)) in form of an equation. It is the same summation as that in \hbox{\tt S8761}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name Sgl10901 \Description Summation formula (\cite{\GaRaAA}, Ex.~2.12) in form of an equation. It is the same summation as that in \hbox{\tt S10901}. \Seealso Sgl2101, SumListe\$gl, Gleichung. \Name SimplifyPQ \Description Rule that simplifies arguments in \hbox{\tt pq}, \hbox{\tt pqinf}, \hbox{\tt ph}, \hbox{\tt Ph}, \hbox{\tt ps}, \hbox{\tt W}, \hbox{\tt SUM}, and expands exponents in powers. \Usage Expr/.SimplifyPQ. \Example \MATH In[1]:= pq[a\MATHhoch 2+a,(k-l)*2]/pqinf[1-f\MATHhoch 2,p]*ph[% \MATHlbrace a\MATHhoch (1/2-(1-s)/2),-n% \MATHrbrace ,% \MATHlbrace a\MATHhoch (s/2)% \MATHrbrace ,q, (t-1)/(1-t)] \goodbreakpoint% % \MATHluEck 1/2 + (-1 + s)/2 % \MATHruEck % \MATHvStrich a , -n -1 + t % \MATHvStrich 2 \MATHphi % \MATHvStrich ; q, ------ % \MATHvStrich (a + a ; q) 2 1% \MATHvStrich s/2 1 - t % \MATHvStrich 2 (k - l) % \MATHloEck a % \MATHroEck Out[1]= ------------------------------------------------------------ 2 (1 - f ;p) \MATHinfty \goodbreakpoint% In[2]:= \%/.SimplifyPQ \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -n % \MATHvStrich \MATHphi % \MATHvStrich ; q, -1 % \MATHvStrich (a (1 + a); q) 1 0% \MATHvStrich - % \MATHvStrich 2 k - 2 l % \MATHloEck % \MATHroEck Out[2]= ---------------------------------------- ((1 - f) (1 + f);p) \MATHinfty \goodbreakpoint% In[3]:= Simplify[\%1] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -n % \MATHvStrich 2 \MATHphi % \MATHvStrich ; q, -1 % \MATHvStrich (a + a ; q) 1 0% \MATHvStrich - % \MATHvStrich 2 (k - l) % \MATHloEck % \MATHroEck Out[3]= ------------------------------------- 2 (1 - f ;p) \MATHinfty \endMATH \Seealso Expandq, MinusOne, SUMExpand, PQSort. \Name SListe \Description Rule that gives for a basic hypergeometric series a list of applicable summation formulas. Each entry of this list has the format \hbox{\tt $\{$ArgumentPermutations,S$\langle$number$\rangle$$\}$}, where \hbox{\tt ArgumentPermutations} is a sequence of reorderings of the parameters of the basic hypergeometric series (given in terms of \hbox{\tt phPerm} and \hbox{\tt phTausche}) and \hbox{\tt S$\langle$number$\rangle$} is the name of the summation in form of a rule which can be applied subsequently. You should be aware that \hbox{\tt SListe} automatically applies \hbox{\tt phOrdne} before checking which summation could be applied. \vskip6pt \hangafter1 \hangindent10pt\rm \underbar{Important Note}: If the value returned by \hbox{\tt SListe} is the empty set this does {\it not} mean that no summation can be applied. You always must remember that the list of summations included in this package is a list of {\it basic} summations. There are numerous special cases of these summations which are not contained in this list as a separate summation. The examples below should illustrate these remarks. \Usage Expr/.SListe. \Example \MATH In[1]:= ph[% \MATHlbrace a,b% \MATHrbrace ,% \MATHlbrace c% \MATHrbrace ,q\MATHhoch 2,c/a/b] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b 2 c % \MATHvStrich Out[1]= \MATHphi % \MATHvStrich ; q , --- % \MATHvStrich 2 1% \MATHvStrich c a b % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% In[2]:= \%/.SListe \goodbreakpoint% Be sure to apply "phOrdne" before using the following information! \goodbreakpoint% Out[2]= % \MATHlbrace % \MATHlbrace S2103% \MATHrbrace % \MATHrbrace \goodbreakpoint% In[3]:= ph[% \MATHlbrace q\MATHhoch -n,-c,c,q\MATHhoch (1+n)% \MATHrbrace ,% \MATHlbrace -q,e,c\MATHhoch 2*q/e% \MATHrbrace ,q,q] \goodbreakpoint% -n 1 + n % \MATHluEck q , -c, c, q % \MATHruEck % \MATHvStrich % \MATHvStrich Out[3]= \MATHphi % \MATHvStrich 2 ; q, q % \MATHvStrich 4 3% \MATHvStrich c q % \MATHvStrich % \MATHloEck -q, e, ---- % \MATHroEck e \goodbreakpoint% In[4]:= \%/.SListe Is n a nonnegative integer? [y|n]: y Is -1 - n a nonnegative integer? [y|n]: n \goodbreakpoint% Be sure to apply "phOrdne" before using the following information! \goodbreakpoint% Out[4]= % \MATHlbrace % \MATHlbrace phPerm[3,1,2,4,u], phTausche[1,3,l], S4304% \MATHrbrace % \MATHrbrace \goodbreakpoint% In[5]:= \%3/.phOrdne/.phPerm[3,1,2,4,u]/.phTausche[1,3,l]/.S4304 Is n a nonnegative integer? [y|n]: y \goodbreakpoint% 2 1 - n 2 2 + n 1 c q c q e 1 + n 2 Out[5]= ------------- (---------, ---------, --, e q ; q ) 2 e e n \MATHinfty c q q (----, e; q) e \MATHinfty \endMATH \vskip10pt\noindent Now we consider two examples illustrating the note above. Though none of the implemented summations can be applied, both series are special cases of a $q$-analogue of Dixon's sum. This fact is also observed by using this package. \vskip10pt \MATH In[6]:= ph[% \MATHlbrace a,-q*Sqrt[a],q\MATHhoch -n% \MATHrbrace ,% \MATHlbrace -Sqrt[a],q*a*q\MATHhoch n% \MATHrbrace ,q,-Sqrt[q]*q\MATHhoch n] \goodbreakpoint% % \MATHluEck -n % \MATHruEck % \MATHvStrich a, -(Sqrt[a] q), q 1/2 + n % \MATHvStrich Out[6]= \MATHphi % \MATHvStrich ; q, -q % \MATHvStrich 3 2% \MATHvStrich 1 + n % \MATHvStrich % \MATHloEck -Sqrt[a], a q % \MATHroEck \goodbreakpoint% In[7]:= \%/.SListe \goodbreakpoint% Out[7]= % \MATHlbrace % \MATHrbrace \goodbreakpoint% In[8]:= Sgl4301 Do you want to set values for the equation? [y|n]: y a=a b=q\MATHhoch -n c=-Sqrt[a*q] Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% % \MATHluEck -n % \MATHruEck % \MATHvStrich a, -(Sqrt[a] q), q 1/2 + n % \MATHvStrich Out[8]= \MATHphi % \MATHvStrich ; q, -q % \MATHvStrich == 3 2% \MATHvStrich 1 + n % \MATHvStrich % \MATHloEck -Sqrt[a], a q % \MATHroEck 1 + n 1/2 + n (a q, Sqrt[a] q , -Sqrt[q], -(Sqrt[a] q ); q) \MATHinfty \MATHgroesser -------------------------------------------------------- 1 + n 1/2 + n (a q , -(Sqrt[a] Sqrt[q]), Sqrt[a] q, -q ; q) \MATHinfty \goodbreakpoint% In[9]:= ph[% \MATHlbrace -Sqrt[a]*q,q,c% \MATHrbrace ,% \MATHlbrace -Sqrt[a],a*q/c% \MATHrbrace ,q,Sqrt[a]/c] \goodbreakpoint% % \MATHluEck -(Sqrt[a] q), q, c % \MATHruEck % \MATHvStrich Sqrt[a] % \MATHvStrich Out[9]= \MATHphi % \MATHvStrich a q ; q, ------- % \MATHvStrich 3 2% \MATHvStrich -Sqrt[a], --- c % \MATHvStrich % \MATHloEck c % \MATHroEck \goodbreakpoint% In[10]:= \%/.SListe \goodbreakpoint% Out[10]= % \MATHlbrace % \MATHrbrace \goodbreakpoint% In[11]:= \%\%/.phEinf Add the parameter: a \goodbreakpoint% % \MATHluEck a, -(Sqrt[a] q), q, c % \MATHruEck % \MATHvStrich Sqrt[a] % \MATHvStrich Out[11]= \MATHphi % \MATHvStrich a q ; q, ------- % \MATHvStrich 4 3% \MATHvStrich a, -Sqrt[a], --- c % \MATHvStrich % \MATHloEck c % \MATHroEck \goodbreakpoint% In[12]:= \%/.SListe \goodbreakpoint% Be sure to apply "phOrdne" before using the following information! \goodbreakpoint% Out[12]= % \MATHlbrace % \MATHlbrace S4301% \MATHrbrace % \MATHrbrace \endMATH \Seealso TListe, phPerm, phTausche, SumListe. \Name Sub \Description Function that subtracts \hbox{\tt Expr} from \hbox{\tt Gleichung}. \Usage Sub[Expr]. \Example \MATH In[1]:= Sgl2101 Do you want to set values for the equation? [y|n]: n Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% n c \MATHluEck \MATHruEck a (-; q) \MATHvStrich -n \MATHvStrich a n Out[1]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich == ---------- 2 1\MATHvStrich \MATHvStrich (c; q) \MATHloEck c \MATHroEck n \goodbreakpoint% In[2]:= Sub[pq[a,n]/pq[c/a,n]] \goodbreakpoint% n c \MATHluEck \MATHruEck (a; q) (a; q) a (-; q) \MATHvStrich -n \MATHvStrich n n a n Out[2]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich - ------- == -(-------) + ---------- 2 1\MATHvStrich \MATHvStrich c c (c; q) \MATHloEck c \MATHroEck (-; q) (-; q) n a n a n \goodbreakpoint% In[3]:= Gleichung \goodbreakpoint% n c \MATHluEck \MATHruEck (a; q) (a; q) a (-; q) \MATHvStrich -n \MATHvStrich n n a n Out[3]= \MATHphi \MATHvStrich a, q ; q, q \MATHvStrich - ------- == -(-------) + ---------- 2 1\MATHvStrich \MATHvStrich c c (c; q) \MATHloEck c \MATHroEck (-; q) (-; q) n a n a n \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, Mal, Add, Div, Hoch, GlTausche, Ers. \Name Subst \Description Function that substitutes \hbox{\tt RS} instead of \hbox{\tt LS} at position \hbox{\tt Position} in \hbox{\tt Expr}. The parameters \hbox{\tt LS} and \hbox{\tt RS} are optional. If they are omitted, the right-hand side ``\hbox{\tt RS}" of \hbox{\tt Gleichung} is substituted instead of the left-hand side ``\hbox{\tt LS}" of \hbox{\tt Gleichung}. \Usage Subst[Expr,Position,LS,RS]. \Example \MATH In[1]:= SUM[pq[a,k]/pq[b,k]/pq[q,k],% \MATHlbrace k,0,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck (a; q) \MATHbackslash k Out[1]= \MATHgroesser --------------- / (b; q) (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k k=0 \goodbreakpoint% In[2]:= Subst[\%,% \MATHlbrace 1% \MATHrbrace ,pq[a,k]/pq[b,k],ph[% \MATHlbrace b/a,q\MATHhoch -k% \MATHrbrace ,% \MATHlbrace b% \MATHrbrace ,q,a*q\MATHhoch k]] \goodbreakpoint% % \MATHluEck b -k % \MATHruEck % \MATHvStrich -, q k % \MATHvStrich \MATHinfty \MATHphi % \MATHvStrich a ; q, a q % \MATHvStrich % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck 2 1% \MATHvStrich % \MATHvStrich \MATHbackslash % \MATHloEck b % \MATHroEck Out[2]= \MATHgroesser ---------------------- / (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k=0 \goodbreakpoint% In[3]:= Sgl2102 Do you want to set values for the equation? [y|n]: y a=b/a c=b n=k Do you want to set a value for q in the equation? [y|n]: n \goodbreakpoint% % \MATHluEck b -k % \MATHruEck (a; q) % \MATHvStrich -, q k % \MATHvStrich k Out[3]= \MATHphi % \MATHvStrich a ; q, a q % \MATHvStrich == ------- 2 1% \MATHvStrich % \MATHvStrich (b; q) % \MATHloEck b % \MATHroEck k \goodbreakpoint% In[4]:= GlTausche \goodbreakpoint% (a; q) % \MATHluEck b -k % \MATHruEck k % \MATHvStrich -, q k % \MATHvStrich Out[4]= ------- == \MATHphi % \MATHvStrich a ; q, a q % \MATHvStrich (b; q) 2 1% \MATHvStrich % \MATHvStrich k % \MATHloEck b % \MATHroEck \goodbreakpoint% In[5]:= Gleichung \goodbreakpoint% (a; q) % \MATHluEck b -k % \MATHruEck k % \MATHvStrich -, q k % \MATHvStrich Out[5]= ------- == \MATHphi % \MATHvStrich a ; q, a q % \MATHvStrich (b; q) 2 1% \MATHvStrich % \MATHvStrich k % \MATHloEck b % \MATHroEck \goodbreakpoint% In[6]:= Subst[\%1,% \MATHlbrace 1% \MATHrbrace ] \goodbreakpoint% % \MATHluEck b -k % \MATHruEck % \MATHvStrich -, q k % \MATHvStrich \MATHinfty \MATHphi % \MATHvStrich a ; q, a q % \MATHvStrich % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck 2 1% \MATHvStrich % \MATHvStrich \MATHbackslash % \MATHloEck b % \MATHroEck Out[6]= \MATHgroesser ---------------------- / (q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k k=0 \endMATH \Seealso Gleichung, SumListe\$gl, TransListe\$gl, LS, RS, GlTausche, Ers, PosListe. \Name SUM \Description This is HYPQ's internal object for entering sums. It should be used instead of Mathematica's \hbox{\tt Sum}. \Usage SUM[Summand,{summation-index,lower-bound,upper-bound}]. \Example See the examples for \hbox{\tt SUMph} and \hbox{\tt SUMInfinity}. \Seealso SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMSammle, SUMTausche. \Name SUMErw1 \Description Rule that extends a {\tt SUM[]} at the top. \vskip6pt \leavevmode\hphantom{Description: } $\sum\limits _{k=l} ^{n} \text {Expr} \to \sum\limits _{k=l} ^{n+m} \text{Expr}-\sum\limits _{k=n+1} ^{n+m} \text{Expr}$. \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.SUMErw1. \Example \MATH In[1]:= SUM[a[k],% \MATHlbrace k,0,N% \MATHrbrace ] \goodbreakpoint% N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[1]= \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 \goodbreakpoint% In[2]:= \%/.SUMErw1 top-extend by: 3 \goodbreakpoint% 3 + N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[2]= -a[1 + N] - a[2 + N] - a[3 + N] + \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 \goodbreakpoint% In[3]:= \%\%/.SUMErw1 top-extend by: M \goodbreakpoint% M + N M + N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash \MATHbackslash Out[3]= \MATHgroesser a[k] - \MATHgroesser a[k] / / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 k=1 + N \endMATH \Seealso SUM, SUMErw2, SUMZerl, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name SUMErw2 \Description Rule that extends a \hbox{\tt SUM[]} at the bottom. \vskip6pt \leavevmode\hphantom{Description: } $ \sum\limits _{k=l} ^{n} \text{Expr} \to \sum\limits _{k=l-m} ^{n} \text{Expr}-\sum\limits _{k=l-m} ^{l-1} \text{Expr}$. \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.SUMErw2. \Example \MATH In[1]:= SUM[a[k],% \MATHlbrace k,0,N% \MATHrbrace ] \goodbreakpoint% N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[1]= \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 \goodbreakpoint% In[2]:= \%/.SUMErw2 bottom-extend by: 3 \goodbreakpoint% N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[2]= -a[-3] - a[-2] - a[-1] + \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=-3 \goodbreakpoint% In[3]:= \%\%/.SUMErw2 bottom-extend by: M \goodbreakpoint% -1 N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash \MATHbackslash Out[3]= -( \MATHgroesser a[k]) + \MATHgroesser a[k] / / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=-M k=-M \endMATH \Seealso SUM, SUMErw1, SUMZerl, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name SUMExpand \Description Rule that expands \hbox{\tt SUM}s. \Usage Expr/.SUMExpand. \Example \MATH In[1]:= SUM[(x[k]-y[k])\MATHhoch 2,% \MATHlbrace k,0,(m+n)/2% \MATHrbrace ] \goodbreakpoint% m + n ----- 2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck 2 Out[1]= \MATHbackslash (x[k] - y[k]) \MATHgroesser / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 \goodbreakpoint% In[2]:= \%/.SUMExpand \goodbreakpoint% m n m n m n - + - - + - - + - 2 2 2 2 2 2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck 2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck 2 Out[2]= \MATHbackslash x[k] + \MATHbackslash -2 x[k] y[k] + \MATHbackslash y[k] \MATHgroesser \MATHgroesser \MATHgroesser / / / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 k=0 k=0 \endMATH \Seealso SUM, SimplifyP, Expandq, MinusOne, PSort. \Name SUMInfinity \Description Rule that changes the upper bound of a \hbox{\tt SUM[]} to \hbox{\tt Infinity}. \Usage Expr/.SUMInfinity. \Example \MATH In[1]:= SUM[pq[q\MATHhoch -n,k]/pq[q,k]*q\MATHhoch k,\MATHlbrace k,0,n\MATHrbrace ] \goodbreakpoint% n k -n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q (q ; q) \MATHbackslash k Out[1]= \MATHgroesser ------------ / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck k k=0 \goodbreakpoint% In[2]:= \%/.SUMInfinity \goodbreakpoint% \MATHinfty k -n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q (q ; q) \MATHbackslash k Out[2]= \MATHgroesser ------------ / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck k k=0 \endMATH \Seealso SUM, SUMph, SUMPh, SUMps, Ers, PosListe. \Name SumListe \Description List of all summation formulas. \Usage SumListe. \Seealso SumListe\$gl, SListe. \Name SumListe\$gl \Description List of all summation formulas. \Usage SumListe\$gl. \Seealso SumListe. \Name SUMPh \Description Rule that transforms a \hbox{\tt SUM[]} into multibasic hypergeometric notation, if possible. If the upper bound is not \hbox{\tt Infinity} you have to apply \hbox{\tt SUMInfinity} first (if allowed). \Usage Expr/.SUMPh. \Example \MATH In[1]:= SUM[pq[a,n]*pq[b,n,q\MATHhoch 2]/pq[c,n,q\MATHhoch 2]/pq[q,n]*z\MATHhoch n,% \MATHlbrace n,0,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty n 2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q) (b; q ) \MATHbackslash n n Out[1]= \MATHgroesser ------------------- / 2 % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck (c; q ) (q; q) n=0 n n \goodbreakpoint% In[2]:= \%/.SUMPh \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a: b 2 % \MATHvStrich Out[2]= \MATHphi % \MATHvStrich ; q, q ; z % \MATHvStrich % \MATHvStrich -: c % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso SUM, Ph, SUMRegeln, SUMph, SUMInfinity, PhSUM, Phph, phPh, Ers, PosListe. \Name SUMph \Description Rule that transforms a \hbox{\tt SUM[]} into basic hypergeometric notation, if possible. If the upper bound is not \hbox{\tt Infinity} you have to apply \hbox{\tt SUMInfinity} first (if allowed). \Usage Expr/.SUMph. \Example \MATH In[1]:= SUM[pq[q\MATHhoch -n,k]/pq[q,k]*q\MATHhoch k,\MATHlbrace k,0,Infinity\MATHrbrace ] \goodbreakpoint% \MATHinfty k -n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q (q ; q) \MATHbackslash k Out[1]= \MATHgroesser ------------ / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck k k=0 \goodbreakpoint% In[2]:= \%/.SUMph \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich Out[2]= \MATHphi \MATHvStrich q ; q, q \MATHvStrich 1 0\MATHvStrich \MATHvStrich \MATHloEck - \MATHroEck \goodbreakpoint% In[3]:= SUM[(1-q\MATHhoch (k+2))*pq[% \MATHlbrace q\MATHhoch -n,a% \MATHrbrace ,% \MATHlbrace b,c,q% \MATHrbrace ,k+1]*q\MATHhoch (k*(k+1)/2), % \MATHlbrace k,0,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty -n % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck (q , a; q) \MATHbackslash (k (1 + k))/2 2 + k 1 + k Out[3]= \MATHgroesser q (1 - q ) ----------------- / (b, c, q; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck 1 + k k=0 \goodbreakpoint% In[4]:= \%/.SUMph \goodbreakpoint% % \MATHluEck 1 - n 3 % \MATHruEck % \MATHvStrich a q, q , q, q % \MATHvStrich -n (1 - q) (1 + q) \MATHphi % \MATHvStrich ; q, -q % \MATHvStrich (a; q) (q ; q) 4 4% \MATHvStrich 2 2 % \MATHvStrich 1 1 % \MATHloEck b q, c q, q , q % \MATHroEck Out[4]= ------------------------------------------------------------------ (b; q) (c; q) (q; q) 1 1 1 \endMATH \Seealso SUM, ph, SUMRegeln, SUMps, SUMInfinity, phSUM, Ers, PosListe. \Name SUMps \Description Rule that transforms a bilateral \hbox{\tt SUM[]} into basic hypergeometric notation, if possible. If the upper bound is not \hbox{\tt Infinity} you have to apply \hbox{\tt SUMInfinity} first (if allowed). If the lower bound is not \hbox{\tt Infinity} then \hbox{\tt SUMph} is applied. \Usage Expr/.SUMps. \Example \MATH In[1]:= SUM[pq[a,n]/pq[b,n]*z\MATHhoch n,% \MATHlbrace n,-Infinity,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty n % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck z (a; q) \MATHbackslash n Out[1]= \MATHgroesser ---------- / (b; q) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck n n=-\MATHinfty \goodbreakpoint% In[2]:= \%/.SUMps \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a % \MATHvStrich Out[2]= ps % \MATHvStrich ; q, z % \MATHvStrich 1 1% \MATHvStrich b % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH \Seealso SUM, ps, SUMRegeln, SUMph, SUMInfinity, psSUM, Ers, PosListe. \Name SUMRegeln \Description Rule that transforms the expressions in a \hbox{\tt SUM[]} into a form that could also be expressed in basic hypergeometric notation. This is useful, if you want to convert a \hbox{\tt SUM[]} into basic hypergeometric notation but without using the \hbox{\tt ph[]}-notation. In particular, expressions of the form $(-1)^{dk}$, where $d$ is an integer and $k$ is the summation index, will simplify. \Usage Expr/.SUMRegeln. \Example \MATH In[1]:= SUM[Binomialq[n,i]*Binomialq[m,k-i]*q\MATHhoch ((n-i)*(k-i)),\MATHlbrace % i,0,Infinity\MATHrbrace ] \goodbreakpoint% \MATHinfty \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck % \MATHluEck \MATHruEck \MATHluEck \MATHruEck \MATHbackslash (-i + k) (-i + n) \MATHvStrich m % \MATHvStrich \MATHvStrich n \MATHvStrich Out[1]= \MATHgroesser q \MATHvStrich % \MATHvStrich \MATHvStrich \MATHvStrich / \MATHvStrich -i + k \MATHvStrich % \MATHvStrich i \MATHvStrich \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % \MATHloEck \MATHroEck \MATHloEck \MATHroEck i=0 q q \goodbreakpoint% In[2]:= \%/.SUMRegeln \goodbreakpoint% \MATHinfty i -k -n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck % q (q ; q) (q ; q) k n \MATHbackslash i i 1 - k + m q ( \MATHgroesser ------------------------) (q ; q) / 1 - k + m k \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck % (q; q) (q ; q) i=0 i i Out[2]= ----------------------------------------------------- (q; q) k \endMATH \Seealso SUM, ph, ps, SUMph, SUMps, phSUM, psSUM, MinusOne, Ers, PosListe. \Name SUMSammle \Description Rule that causes all terms of an expression \hbox{\tt Expr}, which involves a \hbox{\tt SUM[]} to be put into the \hbox{\tt SUM[]}. \Usage Expr/.SUMSammle. \Example \MATH In[1]:= pq[a,n]/pq[b,m]*(-1)\MATHhoch n*SUM[1/pq[q,k],\MATHlbrace k,0,Infinity\MATHrbrace ] \goodbreakpoint% \MATHinfty \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck n \MATHbackslash 1 (-1) ( \MATHgroesser -------) (a; q) / (q; q) n \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck k k=0 Out[1]= ---------------------------- (b; q) m \goodbreakpoint% In[2]:= \%/.SUMSammle \goodbreakpoint% \MATHinfty n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (-1) (a; q) \MATHbackslash n Out[2]= \MATHgroesser --------------- / (b; q) (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck m k k=0 \endMATH \Seealso SUM, SUMRegeln, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, pqzus, pqinfzus, Ers,\linebreak PosListe. \Name SUMShift \Description Rule that shifts the index in a \hbox{\tt SUM[]}. \vskip6pt \leavevmode\hphantom{Description: } $ \sum\limits _{k=l} ^{n} \text{Expr}(k) \to \sum\limits _{k=l-m} ^{n-m} \text{Expr}(k+m)$. \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.SUMShift. \Example \MATH In[1]:= SUM[a[k],% \MATHlbrace k,3,N% \MATHrbrace ] \goodbreakpoint% N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[1]= \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=3 \goodbreakpoint% In[2]:= \%/.SUMShift shift summation index by: 3 \goodbreakpoint% -3 + N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[2]= \MATHgroesser a[3 + k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 \endMATH \Seealso SUM, SUMErw1, SUMErw2, SUMTausche, SUMZerl, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name SUMTausche \Description Rule that exchanges summations. You should apply \hbox{\tt SUMSammle} before applying \hbox{\tt SUMTausche}. \vskip6pt \leavevmode\hphantom{Description: } $ \sum\limits _{k_1=l_1} ^{n_1}\sum\limits _{k_2=l_2} ^{n_2}\text {Expr} \to \sum\limits _{k_2=l_2} ^{n_2}\sum\limits _{k_1=l_1} ^{n_1}\text {Expr}$. \Usage Expr/.SUMTausche. \Example \MATH In[1]:= SUM[SUM[Binomialq[n,k+l],% \MATHlbrace k,0,n1% \MATHrbrace ],% \MATHlbrace l,0,n2% \MATHrbrace ] \goodbreakpoint% n2 n1 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHruEck \MATHbackslash \MATHbackslash % \MATHvStrich n % \MATHvStrich Out[1]= \MATHgroesser \MATHgroesser % \MATHvStrich % \MATHvStrich / / % \MATHvStrich k + l % \MATHvStrich % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHroEck l=0 k=0 q \goodbreakpoint% In[2]:= \%/.SUMTausche \goodbreakpoint% n1 n2 % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHruEck \MATHbackslash \MATHbackslash % \MATHvStrich n % \MATHvStrich Out[2]= \MATHgroesser \MATHgroesser % \MATHvStrich % \MATHvStrich / / % \MATHvStrich k + l % \MATHvStrich % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHroEck k=0 l=0 q \endMATH \Seealso SUM, SUMErw1, SUMErw2, SUMSammle, SUMShift, SUMZerl, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name SUMUmkehr \Description Rule that reverses the order of summation. \hbox{\tt SUMUmkehr} applies to \hbox{\tt SUM[]} as well as \hbox{\tt ph[]}. \Usage Expr/.SUMUmkehr. \Example \MATH In[1]:= ph[\MATHlbrace a,q\MATHhoch -n\MATHrbrace ,\MATHlbrace b\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -n \MATHvStrich Out[1]= \MATHphi \MATHvStrich a, q ; q, z \MATHvStrich 2 1\MATHvStrich \MATHvStrich \MATHloEck b \MATHroEck \goodbreakpoint% In[2]:= \%/.SUMUmkehr Is n a nonnegative integer? [y|n]: y \goodbreakpoint% 1 - n -n q \MATHluEck q , ------ 1 + n \MATHruEck (a; q) n n \MATHvStrich b b q \MATHvStrich n (-1) z \MATHphi \MATHvStrich ; q, -------- \MATHvStrich ------- 2 1\MATHvStrich 1 - n a z \MATHvStrich (b; q) \MATHloEck q \MATHroEck n ------ a Out[2]= ------------------------------------------------ (n (1 + n))/2 q \goodbreakpoint% In[3]:= ph[\MATHlbrace q\MATHhoch -m,q\MATHhoch -n\MATHrbrace ,\MATHlbrace b\MATHrbrace ,q,z] \goodbreakpoint% \MATHluEck \MATHruEck \MATHvStrich -m -n \MATHvStrich Out[3]= \MATHphi \MATHvStrich q , q ; q, z \MATHvStrich 2 1\MATHvStrich \MATHvStrich \MATHloEck b \MATHroEck \goodbreakpoint% In[4]:= \%/.SUMUmkehr Is m a nonnegative integer? [y|n]: n Is n a nonnegative integer? [y|n]: y \goodbreakpoint% 1 - n -m \MATHluEck -n q 1 + m + n \MATHruEck (q ; q) n n \MATHvStrich q , ------ b q \MATHvStrich n (-1) z \MATHphi \MATHvStrich b ; q, ------------ \MATHvStrich --------- 2 1\MATHvStrich z \MATHvStrich (b; q) \MATHloEck 1 + m - n \MATHroEck n q Out[4]= ------------------------------------------------------ (n (1 + n))/2 q \goodbreakpoint% In[5]:= SUM[pq[q\MATHhoch -n,k]/pq[q,k],\MATHlbrace k,0,Infinity\MATHrbrace ] \goodbreakpoint% \MATHinfty -n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck (q ; q) \MATHbackslash k Out[5]= \MATHgroesser --------- / (q; q) \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck k k=0 \goodbreakpoint% In[6]:= \%/.SUMUmkehr Is n a nonnegative integer? [y|n]: y \goodbreakpoint% n k + k n -n \MATHluEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHruEck q (q ; q) \MATHbackslash k -n ( \MATHgroesser ------------------) (q ; q) / (q; q) n \MATHloEck \MATHhStrich \MATHhStrich \MATHhStrich \MATHroEck k k=0 Out[6]= ----------------------------------- (q; q) n \endMATH \Seealso SUM, SUMErw1, SUMErw2, SUMZerl, SUMShift, SUMTausche, SUMRegeln, Ers, PosListe. \Name SUMZerl \Description Rule that splits a \hbox{\tt SUM[]} into two parts. \vskip6pt \leavevmode\hphantom{Description: } $ \sum\limits _{k=l} ^{n} \text{Expr} \to \sum\limits _{k=l} ^{l+m-1} \text{Expr} +\sum\limits _{k=l+m} ^{n} \text{Expr}$. \vskip6pt \leavevmode\hskip10pt The parameter \hbox{\tt m} has to be entered on request. \Usage Expr/.SUMZerl. \Example \MATH In[1]:= SUM[a[k],% \MATHlbrace k,0,N% \MATHrbrace ] \goodbreakpoint% N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[1]= \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 \goodbreakpoint% In[2]:= \%/.SUMZerl bottom-split by: 3 \goodbreakpoint% N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash Out[2]= a[0] + a[1] + a[2] + \MATHgroesser a[k] / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=3 \goodbreakpoint% In[3]:= \%\%/.SUMZerl bottom-split by: M \goodbreakpoint% -1 + M N % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck \MATHbackslash \MATHbackslash Out[3]= \MATHgroesser a[k] + \MATHgroesser a[k] / / % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck k=0 k=M \endMATH \Seealso SUM, SUMErw1, SUMErw2, SUMShift, SUMTausche, SUMRegeln, SUMUmkehr, Ers, PosListe. \Name T2101 \Description Transformation formula (\cite{\GaRaAA}, (1.4.1); Appendix (III.1)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ b,a z;q)_\infty}} {{( \let\over/ c,z;q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ {c\over b},z\\ \let\over/ a z\endmatrix ;q,b\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2102 \Description Transformation formula (\cite{\GaRaAA}, (1.4.5); Appendix (III.2)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ {c\over b},b z;q)_\infty}} {{( \let\over/ c,z;q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ {{a b z}\over c},b\\ \let\over/ b z\endmatrix ;q,{c\over b}\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2103 \Description Transformation formula (\cite{\GaRaAA}, (1.4.6); Appendix (III.3)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ {{a b z}\over c};q)_\infty}} {{( \let\over/ z;q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ {c\over a},{c\over b}\\ \let\over/ c\endmatrix ;q,{{a b z}\over c}\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2104 \Description Transformation formula (\cite{\GaRaAA}, (1.5.4); Appendix (III.4)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ a z;q)_\infty}} {{( \let\over/ z;q)_\infty}} {}_2\phi _2\!\left [ \matrix \let\over/ a,{c\over b}\\ \let\over/ c,a z\endmatrix ;q,b z\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2105 \Description Transformation formula (\cite{\GaRaAA}, (3.2.4); Appendix (III.5)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ {{a b z}\over c};q)_\infty}} {{( \let\over/ {{b z}\over c};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ a,{c\over b},0\\ \let\over/ c,{{c q}\over {b z}}\endmatrix ;q,q\right ] $$ provided the $_3\phi_2$ is terminating. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2106 \Description Transformation formula (\cite{\GaRaAA}, (1.5.6); Appendix (III.6)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ b,{q^{-n}}\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow {{b^n} {z^n}\over {q^n}} {{(\let\over/ {c\over b};q)}_{n}\over{{(\let\over/ c;q)}_{n}}} {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},{q\over z},{{{q^{1 - n}}}\over c}\\ \let\over/ {{b {q^{1 - n}}}\over c},0\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2107 \Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii); Appendix (III.7)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ b,{q^{-n}}\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow {{{(\let\over/ {c\over b};q)}_{n}}\over {{(\let\over/ c;q)}_{n}}} {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},b,{{b z}\over {c {q^n}}}\\ \let\over/ {{b {q^{1 - n}}}\over c},0\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2108 \Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii); Appendix (III.8)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ b,{q^{-n}}\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow {{(\let\over/ {c\over b};q)}_{n}\over {{(\let\over/ c;q)}_{n}} } {b^n} {}_3\phi _1\!\left [ \matrix \let\over/ {q^{-n}},b,{q\over z}\\ \let\over/ {{b {q^{1 - n}}}\over c}\endmatrix ;q,{z\over c}\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2109 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.2, reversed) in form of a rule. $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, b\\ \let \over / {{b q}\over a}\endmatrix ;q, {\displaystyle t} \right ] \longrightarrow { {(\let \over / b, a t ;q) _\infty} \over {(\let \over / t, {b\over a} ;q) _\infty} } {} _{4} W _{3} ({\displaystyle {t\over q}; {1\over a}}; q, {\displaystyle b}) $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2110 \Description Transformation formula (\cite{\GaRaAA}, (3.4.7)) in form of a rule. $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, b\\ \let \over / {{a q}\over b}\endmatrix ;q, {\displaystyle {{q x}\over {b^2}}} \right ] \longrightarrow { {(\let \over / {{q x}\over b}, {{a q x^2}\over {b^2}} ;q) _\infty} \over {(\let \over / {{a q x}\over b}, {{q x^2}\over {b^2}} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{a x}\over b}; x, {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a q}}, - {\sqrt{a q}} }; q, {\displaystyle {{q x}\over {{b^2}}}}) $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2111 \Description Transformation formula (\cite{\GaRaAA}, (3.5.4)) in form of a rule.\NoBlackBoxes $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, b^2\\ \let \over / {{a^2 q^2}\over {b^2}}\endmatrix ;q^2, {\displaystyle {{q^2 x^2}\over {{b^4}}}} \right ] \longrightarrow { {(\let \over / {{a q}\over {b^2}}, {{q x^2}\over {b^2}} ;q) _\infty} \over {(\let \over / {{a^2 q}\over {b^2}}, {{a q x^2}\over {b^2}} ;q) _\infty} } { {(\let \over / {{a^2 q x^2}\over {b^2}}, {{a^2 q^2 x^2}\over {{b^4}}} ;q^2) _\infty} \over {(\let \over / {{q x^2}\over {b^2}}, {{q^2 x^2}\over {{b^4}}} ;q^2) _\infty} } {} _{8} W _{7} ({\displaystyle {{a x^2}\over {b^2}}; a, x, -x, {{{\sqrt{q}} x}\over b}, -{{{\sqrt{q}} x}\over b}}; q, {\displaystyle {{a q}\over {b^2}}}) $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2112 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(i), reversed) in form of a rule.\BlackBoxes $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, a q\\ \let \over / b^2 q\endmatrix ;q^2, {\displaystyle z^2} \right ] \longrightarrow { {(\let \over / a z ;q) _\infty} \over {(\let \over / z ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, -b\\ \let \over / b^2, a z\endmatrix ;q, {\displaystyle -z} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2161 \Description Transformation formula (\cite{\GaRaAA}, (3.3.5); Appendix (III.31)) in form of a rule. $$\multline {}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ {{a b z}\over c},{q\over c};q)_\infty}} {{( \let\over/ {{a z}\over c},{q\over a};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ {c\over a},{{c q}\over {a b z}}\\ \let\over/ {{c q}\over {a z}}\endmatrix ;q,{{b q}\over c}\right ] \\- \frac {{( \let\over/ b,{q\over c},{c\over a},{{a z}\over q},{{{q^2}}\over {a z}};q)_\infty}} {{( \let\over/ {c\over q},{{b q}\over c},{q\over a},{{a z}\over c},{{c q}\over {a z}};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ {{a q}\over c},{{b q}\over c}\\ \let\over/ {{{q^2}}\over c}\endmatrix ;q,z\right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2162 \Description Transformation formula (\cite{\GaRaAA}, Appendix (III.32)) in form of a rule. $${}_2\phi _1\!\left [ \matrix \let\over/ a,b\\ \let\over/ c\endmatrix ;q,z\right ] \longrightarrow \frac {{( \let\over/ b,{c\over a},a z,{q\over {a z}};q)_\infty}} {{( \let\over/ c,{b\over a},z,{q\over z};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ a,{{a q}\over c}\\ \let\over/ {{a q}\over b}\endmatrix ;q,{{c q}\over {a b z}}\right ] + \frac {{( \let\over/ a,{c\over b},b z,{q\over {b z}};q)_\infty}} {{( \let\over/ c,{a\over b},z,{q\over z};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ b,{{b q}\over c}\\ \let\over/ {{b q}\over a}\endmatrix ;q,{{c q}\over {a b z}}\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2163 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.8) in form of a rule. $$ {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, b\\ \let \over / c\endmatrix ;q, {\displaystyle x} \right ] \longrightarrow { {(\let \over / b, {c\over a}, a x ;q) _\infty} \over {(\let \over / {b\over a}, c, x ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {c\over b}, 0\\ \let \over / {{a q}\over b}, a x\endmatrix ;q, {\displaystyle q} \right ] + { {(\let \over / a, {c\over b}, b x ;q) _\infty} \over {(\let \over / {a\over b}, c, x ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / b, {c\over a}, 0\\ \let \over / {{b q}\over a}, b x\endmatrix ;q, {\displaystyle q} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2201 \Description Transformation formula (\cite{\GaRaAA}, (1.5.4), reversed; Appendix (III.4), reversed) in form of a rule. $${}_2\phi _2\!\left [ \matrix \let\over/ a,b\\ \let\over/ c,d\endmatrix ;q,{{c d}\over {a b}}\right ] \longrightarrow \frac {{( \let\over/ {d\over a};q)_\infty}} {{( \let\over/ d;q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ a,{c\over b}\\ \let\over/ c\endmatrix ;q,{d\over a}\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T2202 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii), reversed) in form of a rule. $$ {} _{2} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2\\ \let \over / a^2 b^2 q, a^2 z^2\endmatrix ;q^2, {\displaystyle a^2 q z^2} \right ] \longrightarrow {{({\let \over / z}; q) _{\infty} ({\let \over / - a^2 z }; q) _{\infty} }\over {({\let \over / a^2 z^2}; q^2) _{\infty} }} {{ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, a b, - a b \\ \let \over / a^2 b^2, - a^2 z \endmatrix ;q, {\displaystyle z} \right ]}} $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3101 \Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(ii), reversed; Appendix (III.8), reversed) in form of a rule. $$ {}_3\phi _1\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/ {{b c z}\over {{q^n}}}\endmatrix ;q,z\right ] \longrightarrow {b^n} {{(\let\over/ {q\over {c z}};q)}_{n}\over {{(\let\over/ {q\over {b c z}};q)}_{n}}} {}_2\phi _1\!\left [ \matrix \let\over/ {q^{-n}},b\\ \let\over/ {q\over {c z}}\endmatrix ;q,{{q}\over c}\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3201 \Description Transformation formula (\cite{\GaRaAA}, (3.2.4), reversed; Appendix (III.5), reversed) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ a,b,0\\ \let\over/ c,d\endmatrix ;q,q\right ] \longrightarrow \frac {{( \let\over/ {q \over d};q)_\infty}} {{( \let\over/ {{a q} \over d};q)_\infty}} {}_2\phi _1\!\left [ \matrix \let\over/ a,{c\over b}\\ \let\over/ c\endmatrix ;q,{{b q}\over {d}}\right ] $$ provided the $_3\phi_2$ is terminating. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3202 \Description Transformation formula (\cite{\GaRaAA}, (1.5.6), reversed; Appendix (III.6), reversed) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/ d,0\endmatrix ;q,q\right ] \longrightarrow {{b^n} {c^n}\over {d^n}} { {(\let\over/ {{{q^{1 - n}}}\over c};q)}_{n}\over { {(\let\over/ {{{q^{1 - n}}}\over d};q)}_{n}} } {}_2\phi _1\!\left [ \matrix \let\over/ {q^{-n}},{d\over c}\\ \let\over/ {{{q^{1 - n}}}\over c}\endmatrix ;q,{q\over b}\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3203 \Description Transformation formula (\cite{\GaRaAA}, Ex.~1.15(iii), reversed; Appendix (III.7), reversed) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/ d,0\endmatrix ;q,q\right ] \longrightarrow {{(\let\over/ {{b {q^{1 - n}}}\over d};q)}_{n} \over {{(\let\over/ {{{q^{1 - n}}}\over d};q)}_{n}} } {}_2\phi _1\!\left [ \matrix \let\over/ {q^{-n}},b\\ \let\over/ {{b {q^{1 - n}}}\over d}\endmatrix ;q,{{c q}\over d}\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3204 \Description Transformation formula (\cite{\GaRaAA}, (3.2.7); Appendix (III.9)) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/ d,e\endmatrix ;q,{{d e}\over {a b c}}\right ] \longrightarrow \frac {{( \let\over/ {e\over a},{{d e}\over {b c}};q)_\infty}} {{( \let\over/ e,{{d e}\over {a b c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ a,{d\over b},{d\over c}\\ \let\over/ d,{{d e}\over {b c}}\endmatrix ;q,{e\over a}\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3205 \Description Transformation formula (\cite{\GaRaAA}, (3.2.10); Appendix (III.10)) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/ d,e\endmatrix ;q,{{d e}\over {a b c}}\right ] \longrightarrow \frac {{( \let\over/ b,{{d e}\over {a b}},{{d e}\over {b c}};q)_\infty}} {{( \let\over/ d,e,{{d e}\over {a b c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {d\over b},{e\over b},{{d e}\over {a b c}}\\ \let\over/ {{d e}\over {a b}},{{d e}\over {b c}}\endmatrix ;q,b\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3206 \Description Transformation formula (\cite{\GaRaAA}, (3.2.3); Appendix (III.11)) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/ d,e\endmatrix ;q,q\right ] \longrightarrow {{b^n} {c^n} \over {d^n} } {{(\let\over/ {{d e}\over {b c}};q)}_{n} \over {{(\let\over/ e;q)}_{n}} } {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},{d\over b},{d\over c}\\ \let\over/ d,{{d e}\over {b c}}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3207 \Description Transformation formula (\cite{\GaRaAA}, (3.2.2); Appendix (III.12)) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/ d,e\endmatrix ;q,q\right ] \longrightarrow {c^n}{ {(\let\over/ {e\over c};q)}_{n} \over {{(\let\over/ e;q)}_{n}}} {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},c,{d\over b}\\ \let\over/ d,{{c {q^{1 - n}}}\over e}\endmatrix ;q,{{b q}\over e}\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3208 \Description Transformation formula (\cite{\GaRaAA}, (3.2.5); Appendix (III.13)) in form of a rule. $${}_3\phi _2\!\left [ \matrix \let\over/ b,c,{q^{-n}}\\ \let\over/ d,e\endmatrix ;q,{{d e {q^n}}\over {b c}}\right ] \longrightarrow { {(\let\over/ {e\over c};q)}_{n}\over {{(\let\over/ e;q)}_{n}}} {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},c,{d\over b}\\ \let\over/ d,{{c {q^{1 - n}}}\over e}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3209 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.26; Appendix (III.14)) in form of a rule. $$ {}_3\phi _2\!\left [ \matrix \let\over/ {q^{-n}},b,c\\ \let\over/ {{{q^{1 - n}}}\over b},{{{q^{1 - n}}}\over c}\endmatrix ;q,{{{q^{1 - n}} z}\over {b c}}\right ] \longrightarrow \frac {{( \let\over/ {z\over {{q^n}}};q)_\infty}} {{( \let\over/ z;q)_\infty}} {}_5\phi _4\!\left [ \matrix \let\over/ {q^{{{-n}\over 2}}},-{q^{{{-n}\over 2}}},{q^{{1\over 2} - {n\over 2}}},-{q^{{1\over 2} - {n\over 2}}},{{{q^{1 - n}}}\over {b c}}\\ \let\over/ {{{q^{1 - n}}}\over b},{{{q^{1 - n}}}\over c},{z\over {{q^n}}},{q\over z}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3210 \Description Transformation formula (\cite{\GaRaAA}, (3.2.6)) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, {q^{-n}}\\ \let \over / e, d\endmatrix ;q, {\displaystyle {{d e q^n}\over {a b}}} \right ] \longrightarrow {\left( \frac {d e q^{n-1}} {a} \right) }^n {{({\let \over / {{a q^{1-n}}\over d}, {{a q^{1-n}}\over e}}; q) _{n}}\over {({\let \over / d, e}; q) _{n}}} {{ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {{a b q^{1-n}}\over {d e}}, {q^{-n}}\\ \let \over / {{a q^{1-n}}\over e}, {{a q^{1-n}}\over d}\endmatrix ;q, {\displaystyle {q\over b}} \right ] }} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3211 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.1) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z\\ \let \over / a b {\sqrt{q}}, - a b {\sqrt{q}} \endmatrix ;q, {\displaystyle q} \right ] \longrightarrow {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z^2\\ \let \over / a^2 b^2 q, 0\endmatrix ;q^2, {\displaystyle q^2} \right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3212 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.1, reversed) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z^2\\ \let \over / a^2 b^2 q, 0\endmatrix ;q^2, {\displaystyle q^2} \right ] \longrightarrow {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, b^2, z\\ \let \over / a b {\sqrt{q}}, - a b {\sqrt{q}} \endmatrix ;q, {\displaystyle q} \right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3213 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(i)) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, -b\\ \let \over / b^2, a z\endmatrix ;q, {\displaystyle -z} \right ] \longrightarrow { {(\let \over / z ;q) _\infty} \over {(\let \over / a z ;q) _\infty} } {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, a q\\ \let \over / b^2 q\endmatrix ;q^2, {\displaystyle z^2} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3214 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.2(ii)) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, -b\\ \let \over / b^2, - a z \endmatrix ;q, {\displaystyle z} \right ] \longrightarrow {{({\let \over / a z^2}; q^2) _{\infty} }\over {({\let \over / z}; q) _{\infty} ({\let \over / - a z }; q) _{\infty} }} {{ {} _{2} \phi _{2} \! \left [ \matrix \let \over / a, {{b^2}\over a}\\ \let \over / b^2 q, a z^2\endmatrix ;q^2, {\displaystyle a q z^2} \right ]}} $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3215 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.3) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {q\over a}, z\\ \let \over / c, -q\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / -1, -{{q z}\over c} ;q) _\infty} \over {(\let \over / -{q\over c}, -z ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {c\over a}, {{a c}\over q}, z^2\\ \let \over / c^2, 0\endmatrix ;q^2, {\displaystyle q^2} \right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3216 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.3, reversed) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / c, {{a^2}\over {c q}}, z^2\\ \let \over / a^2, 0\endmatrix ;q^2, {\displaystyle q^2} \right ] \longrightarrow { {(\let \over / -{q\over a}, -z ;q) _\infty} \over {(\let \over / -1, -{{q z}\over a} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {a\over c}, {{c q}\over a}, z\\ \let \over / a, -q\endmatrix ;q, {\displaystyle q} \right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3217 \Description Transformation formula (\cite{\GaRaAA}, (3.2.11)) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d, e\endmatrix ;q, {\displaystyle {{d e}\over {a b c}}} \right ] \longrightarrow { {(\let \over / {{d e}\over {a b}}, {{d e}\over {a c}} ;q) _\infty} \over {(\let \over / {{d e}\over a}, {{d e}\over {a b c}} ;q) _\infty} } {} _{7} \phi _{7} \! \left [ \matrix \let \over / {{d e}\over {a q}}, {{{\sqrt{d e q}}}\over {{\sqrt{a}}}}, -{{{\sqrt{d e q}}}\over {{\sqrt{a}}}}, {e\over a}, {d\over a}, b, c\\ \let \over / {{{\sqrt{d e}}}\over {{\sqrt{a q}}}}, -{{{\sqrt{d e}}}\over {{\sqrt{a q}}}}, d, e, {{d e}\over {a b}}, {{d e}\over {a c}}, 0\endmatrix ;q, {\displaystyle {{d e}\over {b c}}} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3261 \Description Transformation formula (\cite{\GaRaAA}, (3.3.3); Appendix (III.33)) in form of a rule. $$\multline {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/ d,e\endmatrix ;q,{{d e}\over {a b c}}\right ] \longrightarrow \frac {{( \let\over/ {e\over b},{e\over c},{{c q}\over a},{q\over d};q)_\infty}} {{( \let\over/ c,{{c q}\over d},{q\over a},{e\over {b c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ c,{d\over a},{{c q}\over e}\\ \let\over/ {{c q}\over a},{{b c q}\over e}\endmatrix ;q,{{b q}\over d}\right ]\\- \frac {{( \let\over/ {q\over d},{{e q}\over d},b,c,{d\over a},{{d e}\over {b c q}},{{b c {q^2}}\over {d e}};q)_\infty}} {{( \let\over/ {d\over q},e,{{b q}\over d},{{c q}\over d},{q\over a},{e\over {b c}},{{b c q}\over e};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {{a q}\over d},{{b q}\over d},{{c q}\over d}\\ \let\over/ {{{q^2}}\over d},{{e q}\over d}\endmatrix ;q,{{d e}\over {a b c}}\right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3262 \Description Transformation formula (\cite{\GaRaAA}, (3.3.1); Appendix (III.34)) in form of a rule. $$ {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/ d,e\endmatrix ;q,{{d e}\over {a b c}}\right ] \longrightarrow \frac {{( \let\over/ {e\over b},{e\over c};q)_\infty}} {{( \let\over/ e,{e\over {b c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {d\over a},b,c\\ \let\over/ d,{{b c q}\over e}\endmatrix ;q,q\right ] + \frac {{( \let\over/ {d\over a},b,c,{{d e}\over {b c}};q)_\infty}} {{( \let\over/ d,e,{{b c}\over e},{{d e}\over {a b c}};q)_\infty}} {}_3\phi _2\!\left [ \matrix \let\over/ {e\over b},{e\over c},{{d e}\over {a b c}}\\ \let\over/ {{d e}\over {b c}},{{e q}\over {b c}}\endmatrix ;q,q\right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3263 \Description Transformation formula (\cite{\GaRaAA}, (3.3.3), reversed; Appendix (III.33)) in form of a rule. $$\multline {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d, e\endmatrix ;q, {\displaystyle {{d e}\over {a b c}}} \right ] \longrightarrow { {(\let \over / {{d q}\over {b c}}, {e\over c}, a, b, {{a b q}\over {d e}}, {{d e}\over {a b}} ;q) _\infty} \over {(\let \over / {{a q}\over e}, {q\over c}, d, {{a b}\over d}, {{d e}\over {a b c}}, e ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {q\over b}, {{d e}\over {a b c}}, {d\over b}\\ \let \over / {{d q}\over {a b}}, {{d q}\over {b c}}\endmatrix ;q, {\displaystyle {{b q}\over e}} \right ] \\ + { {(\let \over / {{a q}\over c}, {d\over b}, {d\over a}, {q\over e} ;q) _\infty} \over {(\let \over / {{a q}\over e}, {q\over c}, d, {d\over {a b}} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{a q}\over d}, {e\over c}, a\\ \let \over / {{a b q}\over d}, {{a q}\over c}\endmatrix ;q, {\displaystyle {{b q}\over e}} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3264 \Description Transformation formula (\cite{\GaRaAA}, (3.3.1), reversed; Appendix (III.34)) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d, e\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / {{b c q}\over e}, {q\over e} ;q) _\infty} \over {(\let \over / {{c q}\over e}, {{b q}\over e} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {d\over a}, b, c\\ \let \over / d, {{b c q}\over e}\endmatrix ;q, {\displaystyle {{a q}\over e}} \right ] - { {(\let \over / {q\over e}, a, b, c, {{d q}\over e} ;q) _\infty} \over {(\let \over / {{c q}\over e}, {{b q}\over e}, d, {e\over q}, {{a q}\over e} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{c q}\over e}, {{b q}\over e}, {{a q}\over e}\\ \let \over / {{d q}\over e}, {{{q^2}}\over e}\endmatrix ;q, {\displaystyle q} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3265 \Description Transformation formula (\cite{\GaRaAA}, (3.4.1); Appendix (III.35)) in form of a rule. $$\multline {}_3\phi _2\!\left [ \matrix \let\over/ a,b,c\\ \let\over/ {{a q}\over b},{{a q}\over c}\endmatrix ;q,{{a q x}\over {b c}}\right ] \longrightarrow \frac {{( \let\over/ a x;q)_\infty}} {{( \let\over/ x;q)_\infty}} {}_5\phi _4\!\left [ \matrix \let\over/ {\sqrt{a}},-{\sqrt{a}},{\sqrt{a}} {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}} ,{{a q}\over {b c}}\\ \let\over/ {{a q}\over b},{{a q}\over c},a x,{q\over x}\endmatrix ;q,q\right ]\\+ \frac {{( \let\over/ a,{{a q}\over {b c}},{{a q x}\over b},{{a q x}\over c};q)_\infty}} {{( \let\over/ {{a q}\over b},{{a q}\over c},{{a q x}\over {b c}},{1\over x};q)_\infty}} {}_5\phi _4\!\left [ \matrix \let\over/ {\sqrt{a}} x,- {\sqrt{a}} x ,{\sqrt{a}} {\sqrt{q}} x,- {\sqrt{a}} {\sqrt{q}} x ,{{a q x}\over {b c}}\\ \let\over/ {{a q x}\over b},{{a q x}\over c},q x,a {x^2}\endmatrix ;q,q\right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3266 \Description Transformation formula (\cite{\GaRaAA}, (3.5.2)) in form of a rule. $$\multline {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, x^2, y^2\\ \let \over / a^2 b^2, b^2 x^2 y^2\endmatrix ;q^2, {\displaystyle b^2 q} \right ] \longrightarrow { {(\let \over / b^2 ;q^2) _\infty} \over {(\let \over / a^2 b^2 ;q^2) _\infty} } { {(\let \over / a, - a b^2 ;q) _\infty} \over {(\let \over / -1, b^2 ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / -a, - b x , b x, - b y , b y\\ \let \over / -q, - a b^2 , - b x y , b x y\endmatrix ;q, {\displaystyle q} \right ]\\ + { {(\let \over / b^2 ;q^2) _\infty} \over {(\let \over / a^2 b^2 ;q^2) _\infty} } { {(\let \over / -a, a b^2 ;q) _\infty} \over {(\let \over / -1, b^2 ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / a, b x, - b x , b y, - b y \\ \let \over / -q, a b^2, b x y, - b x y \endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3267 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.6) in form of a rule. $$\multline {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d, e\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / {q\over e}, {{a b q}\over e}, {{a c q}\over e}, {d\over a} ;q) _\infty} \over {(\let \over / d, {{a q}\over e}, {{b q}\over e}, {{c q}\over e} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {{a q}\over e}, {{a b c q}\over {d e}}\\ \let \over / {{a b q}\over e}, {{a c q}\over e}\endmatrix ;q, {\displaystyle {d\over a}} \right ]\\ - { {(\let \over / {q\over e}, a, b, c, {{d q}\over e} ;q) _\infty} \over {(\let \over / {e\over q}, {{a q}\over e}, {{b q}\over e}, {{c q}\over e}, d ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{a q}\over e}, {{b q}\over e}, {{c q}\over e}\\ \let \over / {{q^2}\over e}, {{d q}\over e}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3268 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.6, reversed) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, c\\ \let \over / d, e\endmatrix ;q, {\displaystyle {{d e}\over {a b c}}} \right ] \longrightarrow { {(\let \over / {{d e}\over {b c}}, b, {d\over a}, {e\over a} ;q) _\infty} \over {(\let \over / {b\over a}, d, e, {{d e}\over {a b c}} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {d\over b}, {e\over b}\\ \let \over / {{d e}\over {b c}}, {{a q}\over b}\endmatrix ;q, {\displaystyle q} \right ] + { {(\let \over / a, {d\over b}, {e\over b}, {{d e}\over {a c}} ;q) _\infty} \over {(\let \over / d, e, {{d e}\over {a b c}}, {a\over b} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / b, {d\over a}, {e\over a}\\ \let \over / {{b q}\over a}, {{d e}\over {a c}}\endmatrix ;q, {\displaystyle q} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T3269 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.8, reversed) in form of a rule. $$ {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, b, 0\\ \let \over / c, x\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / {q\over c}, {{a b q}\over c}, {x\over a} ;q) _\infty} \over {(\let \over / {{a q}\over c}, {{b q}\over c}, x ;q) _\infty} } {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, {{a q}\over c}\\ \let \over / {{a b q}\over c}\endmatrix ;q, {\displaystyle {x\over a}} \right ] - { {(\let \over / a, b, {{q x}\over c}, {q\over c} ;q) _\infty} \over {(\let \over / {c\over q}, {{a q}\over c}, {{b q}\over c}, x ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{a q}\over c}, {{b q}\over c}, 0\\ \let \over / {{q^2}\over c}, {{q x}\over c}\endmatrix ;q, {\displaystyle q} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4201 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.4, reversed) in form of a rule. $$ {} _{4} \phi _{2} \! \left [ \matrix \let \over / a, b, -b, {q^{-n}}\\ \let \over / {{a {q^{1 - n}}}\over d}, b^2\endmatrix ;q, {\displaystyle -{q\over d}} \right ] \longrightarrow a^{-n} {{({\let \over / d}; q) _{n}}\over {({\let \over / {d\over a}}; q) _{n}}} {{{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, a q, {q^{1 - n}}, {q^{-n}}\\ \let \over / d, d q, b^2 q\endmatrix ;q^2, {\displaystyle q^2} \right ] }} $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4301 \Description Transformation formula (\cite{\GaRaAA}, (2.10.4); Appendix (III.15)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/ e,f,{{a b c {q^{1 - n}}}\over {e f}}\endmatrix ;q,q\right ] \longrightarrow {a^n} { {{(\let\over/ {e\over a},{f\over a};q)}_{n}}\over{{(\let\over/ e,f;q)}_{n}}} {}_4\phi _3\!\left [ \matrix \let\over/ {q^{-n}},a,{{a c {q^{1 - n}}}\over {e f}},{{a b {q^{1 - n}}}\over {e f}}\\ \let\over/ {{a b c {q^{1 - n}}}\over {e f}},{{a {q^{1 - n}}}\over e},{{a {q^{1 - n}}}\over f}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4302 \Description Transformation formula (\cite{\GaRaAA}, (3.2.9); Appendix (III.16)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/ e,f,{{a b c {q^{1 - n}}}\over {e f}}\endmatrix ;q,q\right ] \longrightarrow {{{(\let\over/ a,{{e f}\over {a b}},{{e f}\over {a c}};q)}_{n}}\over{{(\let\over/ e,f,{{e f}\over {a b c}};q)}_{n}} } {}_4\phi _3\!\left [ \matrix \let\over/ {q^{-n}},{e\over a},{f\over a},{{e f}\over {a b c}}\\ \let\over/ {{e f}\over {a b}},{{e f}\over {a c}},{{{q^{1 - n}}}\over a}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4303 \Description Transformation formula (\cite{\GaRaAA}, (2.5.1), reversed; Appendix (III.19)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/ e,f,{{a b c {q^{1 - n}}}\over {e f}}\endmatrix ;q,q\right ] \longrightarrow {{{(\let\over/ {{e f}\over {a b}},{{e f}\over {a c}};q)}_{n}}\over{{(\let\over/ {{e f}\over a},{{e f}\over {a b c}};q)}_{n}} } {}_8W _7({{e f}\over {a q}};{f\over a},{e\over a},b,c,{q^{-n}};q,{{e f {q^n}}\over {b c}}) $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4304 \Description Transformation formula (\cite{\GaRaAA}, (2.10.7); Appendix (III.20)) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ a,b,c,{q^{-n}}\\ \let\over/ d, e,{{a b c {q^{1 - n}}}\over {d e}}\endmatrix ;q,q\right ] \longrightarrow \frac {{( \let\over/ {{d e {q^n}}\over c},{{d e {q^n}}\over b},{{d e {q^n}}\over a},{{d e {q^n}}\over {a b c}};q)_\infty}} {{( \let\over/ {{d e {q^n}}\over {b c}},{{d e {q^n}}\over {a c}},{{d e {q^n}}\over {a b}},d e {q^n};q)_\infty}} {}_8W _7(d e {q^{-1 + n}};a,b,c,d {q^n},e {q^n};q,{{d e}\over {a b c}}) $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4305 \Description Transformation formula (\cite{\GaRaAA}, (3.10.13); Appendix (III.21)) in form of a rule. $${}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},c,d\\ \let\over/ a b {\sqrt{q}},- a b {\sqrt{q}} ,- c d \endmatrix ;q,q\right ] \longrightarrow {}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},{c^2},{d^2}\\ \let\over/ {a^2} {b^2} q,- c d ,- c d q \endmatrix ;{q^2},{q^2}\right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4306 \Description Transformation formula (\cite{\GaRaAA}, (3.10.13), reversed; Appendix (III.21), reversed) in form of a rule. $$ {}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},{c^2},{d^2}\\ \let\over/ {a^2} {b^2} q,- c d ,- c d q \endmatrix ;{q^2},{q^2}\right ] \longrightarrow {}_4\phi _3\!\left [ \matrix \let\over/ {a^2},{b^2},c,d\\ \let\over/ a b {\sqrt{q}},- a b {\sqrt{q}} ,- c d \endmatrix ;q,q\right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4307 \Description Transformation formula (\cite{\GaRaAA}, (8.8.3); Appendix (III.22)) in form of a rule. $$ {{{}_4\phi _3\!\left [ \matrix \let\over/ a,b,a b z,{{a b}\over z}\\ \let\over/ a b {\sqrt{q}},- a b {\sqrt{q}} ,- a b \endmatrix ;q,q\right ]}^2} \longrightarrow {}_5\phi _4\!\left [ \matrix \let\over/ {a^2},{b^2},a b,a b z,{{a b}\over z}\\ \let\over/ a b {\sqrt{q}},- a b {\sqrt{q}} ,- a b ,{a^2} {b^2}\endmatrix ;q,q\right ] $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4308 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.2) in form of a rule. $$ {} _{4} W _{3} ({\displaystyle a; b}; q, {\displaystyle t}) \longrightarrow { {(\let \over / a q, b t ;q) _\infty} \over {(\let \over / t, {{a q}\over b} ;q) _\infty} } {} _{2} \phi _{1} \! \left [ \matrix \let \over / {1\over b}, t\\ \let \over / b q t\endmatrix ;q, {\displaystyle a q} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4309 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(i)) in form of a rule. $$\multline {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, c, d\\ \let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}\endmatrix ;q, {\displaystyle {{a^2 {q^3}}\over {b^2 c^2 d^2}}} \right ] \\\longrightarrow { {(\let \over / {{a q^2}\over {b c d}}, {{{a^3} {q^3}}\over {b^2 c^2 d^2}} ;q) _\infty} \over {(\let \over / {{a^2 q^2}\over {b c d}}, {{a^2 {q^3}}\over {b^2 c^2 d^2}} ;q) _\infty} } {} _{10} W _{9} ({\displaystyle {{{a^2} q}\over {b c d}}; {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a q}}, - {\sqrt{a q}} , {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over {b c}}}; q, {\displaystyle {{a {q^2}}\over {b c d}}}) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4310 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii)) in form of a rule. $$ {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, c, d\\ \let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}\endmatrix ;q, {\displaystyle -{{a q^2}\over {b c d}}} \right ] \longrightarrow { {(\let \over / a q, -q, {{{a^{{3\over 2}}} q^2}\over {b c d}}, -{{{a^{{3\over 2}}} q^2}\over {b c d}} ;q) _\infty} \over {(\let \over / {{a^2 q^2}\over {b c d}}, -{{a q^2}\over {b c d}}, {\sqrt{a}} q, - {\sqrt{a}} q ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{{a^2} q}\over {b c d}}; {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over {b c}}}; q, {\displaystyle -q}) $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4311 \Description Transformation formula (\cite{\GaRaAA}, (3.4.8)) in form of a rule. $$ {} _{4} W _{3} ({\displaystyle a; b}; q, {\displaystyle {x\over {b^2 q}}}) \longrightarrow { {(\let \over / {{a x^2}\over {b^2}}, {x\over {b q}} ;q) _\infty} \over {(\let \over / {{a q x}\over b}, {{x^2}\over {b^2 q}} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{a x}\over b}; {\sqrt{a q}}, - {\sqrt{a q}} , {\sqrt{a}} q, - {\sqrt{a}} q , x}; q, {\displaystyle {x\over {{b^2} q}}})$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4312 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.4) in form of a rule. $$ {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, a q, {q^{1 - n}}, {q^{-n}}\\ \let \over / d, d q, b^2 q\endmatrix ;q^2, {\displaystyle q^2} \right ] \longrightarrow {a^n} {{({\let \over / {d\over a}}; q) _{n}}\over {({\let \over / d}; q) _{n}}} {} _{4} \phi _{2} \! \left [ \matrix \let \over / a, b, -b, {q^{-n}}\\ \let \over / {{a {q^{1 - n}}}\over d}, b^2\endmatrix ;q, {\displaystyle -{q\over d}} \right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4313 \Description Transformation formula (\cite{\GaRaAA}, Ex.~8.15) in form of a rule. $$\multline {} _{4} \phi _{3} \! \left [ \matrix a, b, c, d\\ {b q}/{a}, {c q}/{a}, {d q}/{a}\endmatrix ;q, {\displaystyle \frac{q^ 2}{a^ 2}} \right ] \longrightarrow \frac {({a}/{d}, {b q}/{d}, {c q}/{d}, {a b c}/{d} ;q) _\infty} {( {q}/{d}, {a b}/{d}, {a c}/{d}, {b c q}/{d} ;q) _\infty}\\ \times {} _{12} W _{11} \left({\displaystyle \frac{b c}{d}; \frac{{\sqrt{b c q}}}{{\sqrt{a d}}}, - \frac{{\sqrt{b c q}}}{{\sqrt{a d}}} , \frac{{\sqrt{b c}} q}{{\sqrt{a d}}}, - \frac{{\sqrt{b c}} q}{{\sqrt{a d}}} , \frac{a b}{d}, \frac{a c}{d}, a, b, c}; q, {\displaystyle \frac{q}{a}}\right) \endmultline$$ provided at least one of $a,b,c$ is of the form $q^-n$, $n=0,1,2,\dots$. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4361 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16) in form of a rule. $$\multline {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, - {\sqrt{a}} q , b, c\\ \let \over / -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}\endmatrix ;q, {\displaystyle x} \right ] \longrightarrow \left( 1 - {{b c x}\over {{\sqrt{a}} q}} \right) { {(\let \over / b c x ;q) _\infty} \over {(\let \over / {{b c x}\over {a q}} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}}, - {\sqrt{a}} q , {\sqrt{a q}}, -{\sqrt{a q}}, {{a q}\over {b c}}\\ \let \over / {{a q}\over b}, {{a q}\over c}, {{a q^2}\over {b c x}}, b c x\endmatrix ;q, {\displaystyle q} \right ]\\ + \left( 1 - {\sqrt{a}} \right) { {(\let \over / a q, {{a q}\over {b c}}, c x, b x ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, x, {{a q}\over {b c x}} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{b c x}\over {{\sqrt{a}} q}}, -{{b c x}\over {{\sqrt{a}}}}, {{b c x}\over {{\sqrt{a q}}}}, -{{b c x}\over {{\sqrt{a q}}}}, x\\ \let \over / c x, b x, {{b c x}\over a}, {{b^2 c^2 x^2}\over {a q}}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T4362 \Description Transformation formula (\cite{\GaRaAA}, (2.10.10), reversed; Appendix (III.36)) in form of a rule. $$\multline {} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, c, d\\ \let \over / e, f, {{a b c d q}\over {e f}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / {{e f}\over {a b}}, {{e f}\over {a c}}, {{e f}\over {a d}}, {{e f}\over {a b c d}} ;q) _\infty} \over {(\let \over / {{e f}\over a}, {{e f}\over {a b c}}, {{e f}\over {a b d}}, {{e f}\over {a c d}} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{e f}\over {a q}}; {f\over a}, {e\over a}, b, c, d}; q, {\displaystyle {{e f}\over {b c d}}}) \\- { {(\let \over / {{e f}\over {a b c d}}, a, b, c, d, {{{e^2} f}\over {a b c d}}, {{e {f^2}}\over {a b c d}} ;q) _\infty} \over {(\let \over / {{e f}\over {a b c}}, {{e f}\over {a b d}}, {{e f}\over {a c d}}, e, f, {{e f}\over {b c d}}, {{a b c d}\over {e f}} ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over / {{e f}\over {a b c}}, {{e f}\over {a b d}}, {{e f}\over {a c d}}, {{e f}\over {b c d}}\\ \let \over / {{{e^2} f}\over {a b c d}}, {{e {f^2}}\over {a b c d}}, {{e f q}\over {a b c d}}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5401 \Description Transformation formula (\cite{\GaRaAA}, (2.8.3); Appendix (III.25)) in form of a rule. $$\multline {}_5\phi _4\!\left [ \matrix \let\over/ a,b,c,d,{q^{-n}}\\ \let\over/ {{a q}\over b},{{a q}\over c},{{a q}\over d},{{{b^2} {c^2} {d^2} {q^{-2 - n}}}\over {{a^2}}}\endmatrix ;q,q\right ] \longrightarrow { {{(\let\over/ {{a {q^2}}\over {b c d}},{{{a^3} {q^3}}\over {{b^2} {c^2} {d^2}}};q)}_{n}}\over{{(\let\over/ {{{a^2} {q^2}}\over {b c d}},{{{a^2} {q^3}}\over {{b^2} {c^2} {d^2}}};q)}_{n}}}\\ {}_{12}W _{11}({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b d}},{{a q}\over {b c}},{\sqrt{a}},-{\sqrt{a}},{\sqrt{a}} {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}} ,{{{a^3} {q^{3 + n}}}\over {{b^2} {c^2} {d^2}}},{q^{-n}};q,q) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5402 \Description Transformation formula (\cite{\GaRaAA}, (2.8.4); Appendix (III.26)) in form of a rule. $$\multline {}_5\phi _4\!\left [ \matrix \let\over/ {q^{-n}},b,c,d,e\\ \let\over/ {{{q^{1 - n}}}\over b},{{{q^{1 - n}}}\over c},{{{q^{1 - n}}}\over d},{b^2} {c^2} {d^2} e {q^{-2 + 2 n}}\endmatrix ;q,q\right ] \longrightarrow { {{(\let\over/ {{{q^{2 - 2 n}}}\over {b c d e}},{{{q^{3 - 3 n}}}\over {{b^2} {c^2} {d^2}}};q)}_{n}}\over{{(\let\over/ {{{q^{2 - 2 n}}}\over {b c d}},{{{q^{3 - 3 n}}}\over {{b^2} {c^2} {d^2} e}};q)}_{n}} }\\ {}_{12}W _{11}({{{q^{1 - 2 n}}}\over {b c d}};{{{q^{1 - n}}}\over {c d}},{{{q^{1 - n}}}\over {b d}},{{{q^{1 - n}}}\over {b c}},{q^{{{-n}\over 2}}},-{q^{{{-n}\over 2}}},{q^{{1\over 2} - {n\over 2}}},-{q^{{1\over 2} - {n\over 2}}},e,{{{q^{3 - 3 n}}}\over {{b^2} {c^2} {d^2} e}};q,q) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5403 \Description Transformation formula (\cite{\GaRaAA}, (3.10.4), reversed) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / x^2, y^2, a, a q, {q^{-2 n}}\\ \let \over / a^2, a b, a b q, {{{q^{2 - 2 n}} x^2 y^2}\over {a^2 b^2}}\endmatrix ;q^2, {\displaystyle q^2} \right ] \\ \longrightarrow {{({\let \over / {{a^2 b^2}\over {x^2}}, {{a^2 b^2}\over {y^2}}}; q^2) _{n}}\over {({\let \over / a^2 b^2, {{a^2 b^2}\over {x^2 y^2}}}; q^2) _{n}}} {} _{10} W _{9} ({\displaystyle -{{a b}\over q}; b, x, -x, y, -y, -{q^{-n}}, {q^{-n}}}; q, {\displaystyle {{{a^3} {b^2} {q^{2 n}}}\over {{x^2} {y^2}}}}) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5404 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.26, Appendix (III.14), reversed) in form of a rule. $$ {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{q^{1 - n}}}\over {b c}}, -{q^{{1\over 2} - {n\over 2}}}, {q^{{1\over 2} - {n\over 2}}}, -{q^{{{-n}\over 2}}}, {q^{{{-n}\over 2}}}\\ \let \over / {{{q^{1 - n}}}\over b}, {{{q^{1 - n}}}\over c}, {z\over {{q^n}}}, {q\over z}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / z ;q) _\infty} \over {(\let \over / {z\over {{q^n}}} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {q^{-n}}, b, c\\ \let \over / {{{q^{1 - n}}}\over b}, {{{q^{1 - n}}}\over c}\endmatrix ;q, {\displaystyle {{{q^{1 - n}} z}\over {b c}}} \right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5405 \Description Transformation formula (\cite{\GaRaAA}, (8.8.3), Appendix (III.22), reversed) in form of a rule. $$ {} _{5} \phi _{4} \! \left [ \matrix \let \over / {a^2}, {b^2}, a b, z, {{{a^2} {b^2}}\over z}\\ \let \over / a b {\sqrt{q}}, - a b {\sqrt{q}} , - a b , {a^2} {b^2}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow {{{} _{4} \phi _{3} \! \left [ \matrix \let \over / a, b, z, {{{a^2} {b^2}}\over z}\\ \let \over / a b {\sqrt{q}}, - a b {\sqrt{q}} , - a b \endmatrix ;q, {\displaystyle q} \right ]}^2} $$ provided both series terminate. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5461 \Description Transformation formula (\cite{\GaRaAA}, (3.4.4)) in form of a rule. $$\multline {} _{5} W _{4} ({\displaystyle a; b, c}; q, {\displaystyle {{{\sqrt{a q}} x}\over {b c}}}) \longrightarrow \left( 1 - x^2 \right) { {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x ;q) _\infty} \over {(\let \over / {x\over {{\sqrt{a q}}}} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a q}}, -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q , {{a q}\over {b c}}\\ \let \over / {{a q}\over b}, {{a q}\over c}, {\sqrt{a}} {q^{{3\over 2}}} x, {{{\sqrt{a}} {q^{{3\over 2}}}}\over x}\endmatrix ;q, {\displaystyle q} \right ]\\ + { {(\let \over / a q, {{a q}\over {b c}}, {{{\sqrt{a q}} x}\over b}, {{{\sqrt{a q}} x}\over c} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{{\sqrt{a q}} x}\over {b c}}, {{{\sqrt{a q}}}\over x} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / x, -x, {\sqrt{q}} x, - {\sqrt{q}} x , {{{\sqrt{a q}} x}\over {b c}}\\ \let \over / {{{\sqrt{a q}} x}\over b}, {{{\sqrt{a q}} x}\over c}, {{{\sqrt{q}} x}\over {{\sqrt{a}}}}, q x^2\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5462 \Description Transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, first form) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a q}}, -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q , {{b c}\over {a q}}\\ \let \over / b, c, {\sqrt{a}} {q^{{3\over 2}}} x, {{{\sqrt{a}} {q^{{3\over 2}}}}\over x}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow \frac {1} {{1 - x^2}} {{{ {(\let \over / {x\over {{\sqrt{a q}}}} ;q) _\infty} \over {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x ;q) _\infty} } {} _{5} W _{4} ({\displaystyle a; {{a q}\over b}, {{a q}\over c}}; q, {\displaystyle {{b c x}\over {{{\left( a q \right) }^{{3\over 2}}}}}})}} \\- \frac {1} {{1 - x^2}} {{{ {(\let \over / {x\over {{\sqrt{a q}}}}, a q, {{b c}\over {a q}}, {{b x}\over {{\sqrt{a q}}}}, {{c x}\over {{\sqrt{a q}}}} ;q) _\infty} \over {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x, b, c, {{b c x}\over {{{\left( a q \right) }^{{3\over 2}}}}}, {{{\sqrt{a q}}}\over x} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / x, -x, {\sqrt{q}} x, - {\sqrt{q}} x , {{b c x}\over {{{\left( a q \right) }^{{3\over 2}}}}}\\ \let \over / {{b x}\over {{\sqrt{a q}}}}, {{c x}\over {{\sqrt{a q}}}}, {{{\sqrt{q}} x}\over {{\sqrt{a}}}}, q x^2\endmatrix ;q, {\displaystyle q} \right ]}} \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5463 \Description Transformation formula (\cite{\GaRaAA}, (3.4.4), reversed, second form) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / x, -x, {\sqrt{q}} x, - {\sqrt{q}} x , {{b c}\over {{\sqrt{a q}} x}}\\ \let \over / b, c, {{{\sqrt{q}} x}\over {{\sqrt{a}}}}, q x^2\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow { {(\let \over / {{b {\sqrt{a q}}}\over x}, {{c {\sqrt{a q}}}\over x}, {{b c}\over {{\sqrt{a q}} x}}, {{{\sqrt{a q}}}\over x} ;q) _\infty} \over {(\let \over / a q, {{b c}\over {x^2}}, b, c ;q) _\infty} } {} _{5} W _{4} ({\displaystyle a; {{{\sqrt{a q}} x}\over b}, {{{\sqrt{a q}} x}\over c}}; q, {\displaystyle {{b c}\over {{\sqrt{a q}} x}}}) \\ - \left( 1 - x^2 \right) { {(\let \over / {\sqrt{a}} {q^{{3\over 2}}} x, {{b {\sqrt{a q}}}\over x}, {{c {\sqrt{a q}}}\over x}, {{b c}\over {{\sqrt{a q}} x}}, {{{\sqrt{a q}}}\over x} ;q) _\infty} \over {(\let \over / {x\over {{\sqrt{a q}}}}, a q, {{b c}\over {x^2}}, b, c ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a q}}, -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q , {{b c}\over {x^2}}\\ \let \over / {{b {\sqrt{a q}}}\over x}, {{c {\sqrt{a q}}}\over x}, {\sqrt{a}} {q^{{3\over 2}}} x, {{{\sqrt{a}} {q^{{3\over 2}}}}\over x}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5464 \Description Transformation formula (\cite{\GaRaAA}, (3.5.2), reversed) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / a, x, -x, y, -y\\ \let \over / -q, a b^2, {{x y}\over b}, -{{x y}\over b}\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow { {(\let \over / a^2 b^2 ;q^2) _\infty} \over {(\let \over / b^2 ;q^2) _\infty} } { {(\let \over / -1, b^2 ;q) _\infty} \over {(\let \over / -a, a b^2 ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a^2, {{x^2}\over {b^2}}, {{y^2}\over {b^2}}\\ \let \over / a^2 b^2, {{x^2 y^2}\over {b^2}}\endmatrix ;q^2, {\displaystyle b^2 q} \right ] - { {(\let \over / a, - a b^2 ;q) _\infty} \over {(\let \over / -a, a b^2 ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / -a, -x, x, -y, y\\ \let \over / -q, - a b^2 , -{{x y}\over b}, {{x y}\over b}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5465 \Description Transformation formula (\cite{\GaRaAA}, (3.5.7), reversed) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}}, -{\sqrt{a}}, c, d, e\\ \let \over / {\sqrt{a b}}, -{\sqrt{a b}}, a, {{c d e q}\over {a b}}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow - { {(\let \over / a b q, {{{a^3} {b^3}}\over {c^2 d^2 e^2}} ;q^2) _\infty} \over {(\let \over / a q, {{{a^3} b^2}\over {c^2 d^2 e^2}} ;q^2) _\infty} } { {(\let \over / c, d, e, {{a^2 b}\over {c d e}}, {{a b}\over {c d e}} ;q) _\infty} \over {(\let \over / {{c d e}\over {a b}}, a b, {{a b}\over {c d}}, {{a b}\over {c e}}, {{a b}\over {d e}} ;q) _\infty} } \\ {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{a^{{3\over 2}}} b}\over {c d e}}, -{{{a^{{3\over 2}}} b}\over {c d e}}, {{a b}\over {c d}}, {{a b}\over {c e}}, {{a b}\over {d e}}\\ \let \over / {{{{\left( a b \right) }^{{3\over 2}}}}\over {c d e}}, -{{{{\left( a b \right) }^{{3\over 2}}}}\over {c d e}}, {{a^2 b}\over {c d e}}, {{a b q}\over {c d e}}\endmatrix ;q, {\displaystyle q} \right ] \\ + { {(\let \over / {{a b}\over c}, {{a b}\over d}, {{a b}\over e}, {{a b}\over {c d e}} ;q) _\infty} \over {(\let \over / a b, {{a b}\over {c d}}, {{a b}\over {c e}}, {{a b}\over {d e}} ;q) _\infty} } {} _{10} W _{9} ({\displaystyle {{a b}\over q}; b, c, c q, d, d q, e, e q}; {q^2}, {\displaystyle {{{a^3} {b^2}}\over {{c^2} {d^2} {e^2}}}})\endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5466 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, first form) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / a, - a q , a {\sqrt{q}}, - a {\sqrt{q}} , {{b c}\over {a^2 q}}\\ \let \over / b, c, x, {{a^2 q^2}\over x}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow \frac {1} {({1 - {{a q}\over x}})}{{{ {(\let \over / {q\over x} ;q) _\infty} \over {(\let \over / {{a^2 q^2}\over x} ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over / a^2, - a q , {{a^2 q}\over b}, {{a^2 q}\over c}\\ \let \over / -a, b, c\endmatrix ;q, {\displaystyle {{b c}\over {a^2 x}}} \right ]} } \\- \frac { (1 - a)} {(1 - {{a q}\over x})} {{ { {(\let \over / {q\over x}, a^2 q, {{b c}\over {a^2 q}}, {{b q}\over x}, {{c q}\over x} ;q) _\infty} \over {(\let \over / {{a^2 q^2}\over x}, b, c, {{b c}\over {a^2 x}}, {x\over q} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{a q}\over x}, -{{a q^2}\over x}, {{a {q^{{3\over 2}}}}\over x}, -{{a {q^{{3\over 2}}}}\over x}, {{b c}\over {a^2 x}}\\ \let \over / {{b q}\over x}, {{c q}\over x}, {{q^2}\over x}, {{a^2 {q^3}}\over {x^2}}\endmatrix ;q, {\displaystyle q} \right ] }} \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5467 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.16, reversed, second form) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / a, - a q , a {\sqrt{q}}, - a {\sqrt{q}} , {{b c x}\over {a^2 q^2}}\\ \let \over / b, c, x, a^2 q\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow \frac {1} {({1 - {{a q}\over x}})} {{{ {(\let \over / {{b q}\over x}, {{c q}\over x}, {{b c x}\over {a^2 q^2}}, {q\over x} ;q) _\infty} \over {(\let \over / {{a^2 {q^3}}\over {x^2}}, {{b c}\over {a^2 q}}, b, c ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over / {{a^2 q^2}\over {x^2}}, -{{a q^2}\over x}, {{a^2 q^2}\over {b x}}, {{a^2 q^2}\over {c x}}\\ \let \over / -{{a q}\over x}, {{b q}\over x}, {{c q}\over x}\endmatrix ;q, {\displaystyle {{b c x}\over {a^2 q^2}}} \right ]} }\\ - \frac {( 1 - a )} {(1 - {{a q}\over x})} { {(\let \over / {{a^2 q^2}\over x}, {{b q}\over x}, {{c q}\over x}, {{b c x}\over {a^2 q^2}}, {q\over x} ;q) _\infty} \over {(\let \over / {x\over q}, {{a^2 {q^3}}\over {x^2}}, {{b c}\over {a^2 q}}, b, c ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{a q}\over x}, -{{a q^2}\over x}, {{a {q^{{3\over 2}}}}\over x}, -{{a {q^{{3\over 2}}}}\over x}, {{b c}\over {a^2 q}}\\ \let \over / {{b q}\over x}, {{c q}\over x}, {{q^2}\over x}, {{a^2 q^2}\over x}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5468 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.25) in form of a rule. $$\align {} _{5} \phi &_{4} \! \left [ \matrix \let \over / a, b, c, d, e\\ \let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{b^2 c^2 d^2 e}\over {a^2 q^2}}\endmatrix ;q, {\displaystyle q} \right ] \\&\longrightarrow - { {(\let \over / a, b, c, d, e, {{a^2 q^3}\over {b^2 c^2 d^2 e}}, {{a^3 {q^4}}\over {b^3 c^2 d^2 e}}, {{a^3 {q^4}}\over {b^2 c^3 d^2 e}}, {{a^3 {q^4}}\over {b^2 c^2 d^3 e}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{b^2 c^2 d^2 e}\over {a^2 q^3}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d^2 e}}, {{a^2 q^3}\over {b^2 c^2 d e}}, {{a^2 q^3}\over {b^2 c^2 d^2}} ;q) _\infty} }\\&\hskip3cm {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{a^2 q^3}\over {b^2 c^2 d^2}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d^2 e}}, {{a^2 q^3}\over {b^2 c^2 d e}}\\ \let \over / {{a^2 {q^4}}\over {b^2 c^2 d^2 e}}, {{a^3 {q^4}}\over {b^3 c^2 d^2 e}}, {{a^3 {q^4}}\over {b^2 c^3 d^2 e}}, {{a^3 {q^4}}\over {b^2 c^2 d^3 e}}\endmatrix ;q, {\displaystyle q} \right ] \\&\hskip.3cm + { {(\let \over / {{a q^2}\over {b c d}}, {{a^2 q^2}\over {b c d e}}, {{a^3 q^3}\over {b^2 c^2 d^2}}, {{a^2 q^3}\over {b^2 c^2 d^2 e}} ;q) _\infty} \over {(\let \over / {{a^2 q^2}\over {b c d}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c^2 d^2}}, {{a q^2}\over {b c d e}} ;q) _\infty} }\\&\hskip3cm {} _{12} W _{11} ({\displaystyle {{a^2 q}\over {b c d}}; {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a}} {\sqrt{q}}, - {\sqrt{a}} {\sqrt{q}} , {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over {b c}}, e, {{a^3 q^3}\over {b^2 c^2 d^2 e}}}; q, {\displaystyle q})\\&\hskip.3cm + { {(\let \over / a, e, {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over {b c}}, {{a^2 q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d e}}, {{a^2 q^3}\over {b c^2 d e}}, {{a^2 q^3}\over {b c d^2 e}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d^2 e}}, {{a^2 q^3}\over {b^2 c^2 d e}}, {{a^2 q^3}\over {b^2 c^2 d^2}}, {{b c d e}\over {a q^2}} ;q) _\infty} } \\&\hskip.5cm {{(\let \over / {{{a^5} {q^7}}\over {{b^4} {c^4} {d^4} e^2}}; q)_{\infty}}\over {(\let \over / {{{a^4} {q^6}}\over {b^3 c^3 d^3 e^2 }};q)_{\infty} }} {} _{12} W _{11} ({\displaystyle {{{a^4} {q^5}}\over {b^3 c^3 d^3 e^2}}; {{{a^{{3\over 2}}} q^2}\over {b c d e}}, -{{{a^{{3\over 2}}} q^2}\over {b c d e}}, {{{a^{{3\over 2}}} {q^{{5\over 2}}}}\over {b c d e}}, -{{{a^{{3\over 2}}} {q^{{5\over 2}}}}\over {b c d e}}, {{a q^2}\over {b c d}}, {{a^3 q^3}\over {b^2 c^2 d^2 e}}, {{a^2 q^3}\over {b c^2 d^2 e}}, {{a^2 q^3}\over {b^2 c d^2 e}}, {{a^2 q^3}\over {b^2 c^2 d e}}}; q, {\displaystyle q}) \endalign$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T5469 \Description Transformation formula (\cite{\GaRaAA}, (3.4.1), reversed; Appendix (III.35)) in form of a rule. $$\multline {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a}} {\sqrt{q}}, - {\sqrt{a}} {\sqrt{q}} , {{b c}\over {a q}}\\ \let \over / b, c, a x, {q\over x}\endmatrix ;q, {\displaystyle q} \right ] \longrightarrow { {(\let \over / x ;q) _\infty} \over {(\let \over / a x ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / a, {{a q}\over b}, {{a q}\over c}\\ \let \over / b, c\endmatrix ;q, {\displaystyle {{b c x}\over {a q}}} \right ] \\ - { {(\let \over / x, a, {{b c}\over {a q}}, b x, c x ;q) _\infty} \over {(\let \over / a x, b, c, {{b c x}\over {a q}}, {1\over x} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {\sqrt{a}} x, - {\sqrt{a}} x , {\sqrt{a}} {\sqrt{q}} x, - {\sqrt{a}} {\sqrt{q}} x , {{b c x}\over {a q}}\\ \let \over / b x, c x, q x, a {x^2}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T6501 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.14(ii)) in form of a rule. $$\multline {} _{6} \phi _{5} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, b, c, d, {q^{-n}}\\ \let \over / {\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{b^2 c^2 d^2 {q^{-1 - n}}}\over {a^2}}\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow {{({\let \over / {{a q}\over {b c d}}, {{{a^3} q^2}\over {b^2 c^2 d^2}}, -{{{a^{{3\over 2}}} q^2}\over {b c d}}}; q) _{n}}\over {({\let \over / {{a^2 q^2}\over {b c d}}, {{a^2 q^2}\over {b^2 c^2 d^2}}, -{{{a^{{3\over 2}}} q}\over {b c d}}}; q) _{n}}} {} _{12} W _{11} ({\displaystyle {{{a^2} q}\over {b c d}}; {{a q}\over {c d}}, {{a q}\over {b d}}, {{a q}\over {b c}}, {\sqrt{a}} q, -{\sqrt{a}}, {\sqrt{a q}}, - {\sqrt{a q}} , {{{a^3} {q^{2 + n}}}\over {{b^2} {c^2} {d^2}}}, {q^{-n}}}; q, {\displaystyle q}) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T7601 \Description Transformation formula (\cite{\GaRaAA}, (2.8.5); Appendix (III.27)) in form of a rule. $$\multline {}_7\phi _6\!\left [ \matrix \let\over/ a,{\sqrt{a}} q,- {\sqrt{a}} q ,b,c,d,{q^{-n}}\\ \let\over/ {\sqrt{a}},-{\sqrt{a}},{{a q}\over b},{{a q}\over c},{{a q}\over d},{{{b^2} {c^2} {d^2}}\over {{a^2} {q^n}}}\endmatrix ;q,q\right ] \longrightarrow {{{\left(\let\over/ 1 - {{{a^3} {q^{1 + 2 n}}}\over {{b^2} {c^2} {d^2}}} \right)}\over {\let\over/ (1 - {{{a^3} q}\over {{b^2} {c^2} {d^2}}})}} { {{(\let\over/ {a\over {b c d}},{{{a^3} q}\over {{b^2} {c^2} {d^2}}};q)}_{n}}\over{{(\let\over/ {{{a^2} {q^2}}\over {b c d}},{{{a^2} q}\over {{b^2} {c^2} {d^2}}};q)}_{n}}}}\\ {}_{12}W _{11}({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b d}},{{a q}\over {b c}},{\sqrt{a}} {\sqrt{q}},- {\sqrt{a}} {\sqrt{q}} ,{\sqrt{a}} q,- {\sqrt{a}} q ,{{{a^3} {q^{1 + n}}}\over {{b^2} {c^2} {d^2}}},{q^{-n}};q,q) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T7701 \Description Transformation formula (\cite{\GaRaAA}, (3.2.11), reversed) in form of a rule. $$ {} _{7} \phi _{7} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, - {\sqrt{a}} q , b, c, d, e\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, 0\endmatrix ;q, {\displaystyle {{{a^2} {q^2}}\over {b c d e}}} \right ] \longrightarrow { {(\let \over / a q, {{a q}\over {d e}} ;q) _\infty} \over {(\let \over / {{a q}\over d}, {{a q}\over e} ;q) _\infty} } {} _{3} \phi _{2} \! \left [ \matrix \let \over / {{a q}\over {b c}}, d, e\\ \let \over / {{a q}\over b}, {{a q}\over c}\endmatrix ;q, {\displaystyle {{a q}\over {d e}}} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8701 \Description Transformation formula (\cite{\GaRaAA}, (2.10.10) terminated; Appendix (III.17)) in form of a rule. $$ {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow \frac {{( \let\over/ a q,{{a q}\over {d e}},{{a q}\over {d f}},{{a q}\over {e f}};q)_\infty}} {{( \let\over/ {{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over {d e f}};q)_\infty}} {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {b c}},d,e,f\\ \let\over/ {{a q}\over b},{{a q}\over c},{{d e f}\over a}\endmatrix ;q,q\right ], $$ provided the $_8\phi_7$ series converges and the $_4\phi_3$ series terminates. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8702 \Description Transformation formula (\cite{\GaRaAA}, (2.5.1); Appendix (III.18)) in form of a rule. $$ {}_8W _7(a;b,c,d,e,{q^{-n}};q,{{{a^2} {q^{2 + n}}}\over {b c d e}})\longrightarrow { {{(\let\over/ a q,{{a q}\over {d e}};q)}_{n}}\over{{(\let\over/ {{a q}\over d},{{a q}\over e};q)}_{n}} } {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {b c}},d,e,{q^{-n}}\\ \let\over/ {{a q}\over b},{{a q}\over c},{{d e}\over {a {q^n}}}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8703 \Description Transformation formula (\cite{\GaRaAA}, (2.10.7), reversed; Appendix (III.20), reversed) in form of a rule. $$ {}_8W _7(a;b,c,d,e,{{a {q^{1 + n}}}\over e};q,{{a {q^{1 - n}}}\over {b c d}}) \longrightarrow \frac {{( \let\over/ {{a q}\over {c d}},{{a q}\over {b d}},{{a q}\over {b c}},a q;q)_\infty}} {{( \let\over/ {{a q}\over d},{{a q}\over c},{{a q}\over b},{{a q}\over {b c d}};q)_\infty}} {}_4\phi _3\!\left [ \matrix \let\over/ b,c,d,{q^{-n}}\\ \let\over/ {{a q}\over e},{{b c d}\over a},{e\over {{q^n}}}\endmatrix ;q,q\right ] $$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8704 \Description Transformation formula (\cite{\GaRaAA}, (2.10.1); Appendix (III.23)) in form of a rule. $$ {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow \frac {{( \let\over/ a q,{{a q}\over {e f}},{{{a^2} {q^2}}\over {b c d e}},{{{a^2} {q^2}}\over {b c d f}};q)_\infty}} {{( \let\over/ {{a q}\over e},{{a q}\over f},{{{a^2} {q^2}}\over {b c d}},{{{a^2} {q^2}}\over {b c d e f}};q)_\infty}} {}_8W _7({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b d}},{{a q}\over {b c}},e,f;q,{{a q}\over {e f}}) $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8705 \Description Transformation formula (\cite{\GaRaAA}, (2.10.1), iterated; Appendix (III.24)) in form of a rule. $$\multline {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow \frac {{( \let\over/ a q,b,{{{a^2} {q^2}}\over {b d e f}},{{{a^2} {q^2}}\over {b c e f}},{{{a^2} {q^2}}\over {b c d f}},{{{a^2} {q^2}}\over {b c d e}};q)_\infty}} {{( \let\over/ {{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over f},{{{a^3} {q^3}}\over {{b^2} c d e f}},{{{a^2} {q^2}}\over {b c d e f}};q)_\infty}} \\{}_8W _7({{{a^3} {q^2}}\over {{b^2} c d e f}};{{a q}\over {b c}},{{a q}\over {b d}},{{a q}\over {b e}},{{a q}\over {b f}},{{{a^2} {q^2}}\over {b c d e f}};q,b) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8706 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(ii), reversed) in form of a rule. $$ {} _{8} W _{7} ({\displaystyle a; b, -b, c, d, {{a^2 q}\over {b^2 c d}}}; q, {\displaystyle -q}) \longrightarrow { {(\let \over / a q, -{{a q}\over {b^2}}, b q, - b q ;q) _\infty} \over {(\let \over / b^2 q, -q, {{a q}\over b}, -{{a q}\over b} ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over / b^2, {{b^2 c}\over a}, {{b^2 d}\over a}, {{a q}\over {c d}}\\ \let \over / {{a q}\over c}, {{a q}\over d}, {{b^2 c d}\over a}\endmatrix ;q, {\displaystyle -{{a q}\over {b^2}}} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8707 \Description Transformation formula (\cite{\GaRaAA}, (3.4.7), reversed) in form of a rule. $$ {} _{8} W _{7} ({\displaystyle b; {\sqrt{a}}, -{\sqrt{a}}, {\sqrt{a q}}, -{\sqrt{a q}}}, x; q, {\displaystyle {{b^2 q}\over {a^2 x}}}) \longrightarrow { {(\let \over / b q, {{b^2 q}\over {a^2}} ;q) _\infty} \over {(\let \over / {{b q}\over a}, {{b^2 q}\over a} ;q) _\infty} } {} _{2} \phi _{1} \! \left [ \matrix \let \over / a, {{a x}\over b}\\ \let \over / {{b q}\over x}\endmatrix ;q, {\displaystyle {{b^2 q}\over {a^2 x}}} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8708 \Description Transformation formula (\cite{\GaRaAA}, (3.4.8), reversed) in form of a rule. $$ {} _{8} W _{7} ({\displaystyle b; {\sqrt{a q}}, -{\sqrt{a q}}, {\sqrt{a}} q, - {\sqrt{a}} q , x}; q, {\displaystyle {{b^2}\over {a^2 q x}}}) \longrightarrow { {(\let \over / b q, {{b^2}\over {a^2 q}} ;q) _\infty} \over {(\let \over / {{b^2}\over a}, {b\over {a q}} ;q) _\infty} } {} _{4} W _{3} ({\displaystyle a; {{a x}\over b}}; q, {\displaystyle {{b^2}\over {a^2 q x}}}) $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8709 \Description Transformation formula (\cite{\GaRaAA}, (3.5.4), reversed) in form of a rule. $$ {} _{8} W _{7} ({\displaystyle b; x, -x, {{{\sqrt{b q}}}\over {{\sqrt{a}}}}, -{{{\sqrt{b q}}}\over {{\sqrt{a}}}}}, a; q, {\displaystyle {{b q}\over {{x^2}}}}) \longrightarrow { {(\let \over / {{b q}\over a}, {{b^2 q^2}\over {a^2 x^2}} ;q^2) _\infty} \over {(\let \over / a b q, {{b^2 q^2}\over {x^2}} ;q^2) _\infty} } { {(\let \over / {{a b q}\over {x^2}}, b q ;q) _\infty} \over {(\let \over / {{b q}\over {x^2}}, {{b q}\over a} ;q) _\infty} } {} _{2} \phi _{1} \! \left [ \matrix \let \over / a^2, {{a x^2}\over b}\\ \let \over / {{a b q^2}\over {x^2}}\endmatrix ;q^2, {\displaystyle {{b^2 q^2}\over {a^2 x^2}}} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8710 \Description Transformation formula (\cite{\GaRaAA}, (3.5.10)) in form of a rule. $$\multline {} _{8} W _{7} ({\displaystyle a; b, c, c q, d, d q}; q^2, {\displaystyle {{a^2 q^2}\over {b c^2 d^2}}}) \\\longrightarrow { {(\let \over / a q, {{a q}\over {b c}}, {{a q}\over {c d}}, -{{a q}\over {c d}}, {{a q}\over {{\sqrt{b}} d}}, -{{a q}\over {{\sqrt{b}} d}} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, -{{a q}\over d}, {{a q}\over {{\sqrt{b}} c d}}, -{{a q}\over {{\sqrt{b}} c d}} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle -{a\over d}; c, {\sqrt{b}}, -{\sqrt{b}}, {{{\sqrt{a q}}}\over d}, -{{{\sqrt{a q}}}\over d}}; q, {\displaystyle {{a q}\over {b c}}}) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8711 \Description Transformation formula (\cite{\GaRaAA}, (3.5.10), reversed) in form of a rule. $$\multline {} _{8} W _{7} ({\displaystyle a; b, -b, d, -d, c}; q, {\displaystyle {{a^2 q^2}\over {b^2 c d^2}}}) \\\longrightarrow { {(\let \over / {{a^2 q^2}\over {b^2 d^2}}, {{a^2 q^2}\over {c d^2}}, - a q , a q, -{{a q}\over {b c}}, {{a q}\over {b c}} ;q) _\infty} \over {(\let \over / {{a^2 q^2}\over {d^2}}, {{a^2 q^2}\over {b^2 c d^2}}, -{{a q}\over c}, {{a q}\over c}, -{{a q}\over b}, {{a q}\over b} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{a^2 q}\over {d^2}}; b^2, c, c q, -{{a q}\over {d^2}}, -{{a q^2}\over {d^2}}}; q^2, {\displaystyle {{a^2 q^2}\over {b^2 c^2}}}) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8761 \Description Transformation formula (\cite{\GaRaAA}, (2.10.10); Appendix (III.36)) in form of a rule. $$\multline {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \longrightarrow \frac {{( \let\over/ a q,{{a q}\over {d e}},{{a q}\over {d f}},{{a q}\over {e f}};q)_\infty}} {{( \let\over/ {{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over {d e f}};q)_\infty}} {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {b c}},d,e,f\\ \let\over/ {{a q}\over b},{{a q}\over c},{{d e f}\over a}\endmatrix ;q,q\right ] \\+ \frac {{( \let\over/ a q,{{a q}\over {b c}},d,e,f,{{{a^2} {q^2}}\over {b d e f}},{{{a^2} {q^2}}\over {c d e f}};q)_\infty}} {{( \let\over/ {{a q}\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over f},{{{a^2} {q^2}}\over {b c d e f}},{{d e f}\over {a q}};q)_\infty}} {}_4\phi _3\!\left [ \matrix \let\over/ {{a q}\over {d e}},{{a q}\over {d f}},{{a q}\over {e f}},{{{a^2} {q^2}}\over {b c d e f}}\\ \let\over/ {{{a^2} {q^2}}\over {b d e f}},{{{a^2} {q^2}}\over {c d e f}},{{a {q^2}}\over {d e f}}\endmatrix ;q,q\right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8762 \Description Transformation formula (\cite{\GaRaAA}, (2.11.1); Appendix (III.37)) in form of a rule. $$\multline {}_8W _7(a;b,c,d,e,f;q,{{{a^2} {q^2}}\over {b c d e f}}) \\\longrightarrow - \frac {{( \let\over/ a q,{b\over a},{{b q}\over c},{{b q}\over d},{{b q}\over e},{{b q}\over f},d,e,f,{{a q}\over {b c}}, {{b d e f}\over {a^2}}, {{a^2 q}\over {b d e f}};q)_\infty}} {{( \let\over/ {a\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over f},{{b d}\over a},{{b e}\over a}, {{b f}\over {a}}, {{d e f}\over {a}}, {{a q}\over {d e f}}, {{q}\over {c}}, {{b^2 q}\over {a}};q)_\infty}} \\ {}_8W _7({{{b^2}}\over a};b,{{b c}\over a},{{b d}\over a},{{b e}\over a},{{b f}\over a};q,{{{a^2} {q^2}}\over {b c d e f}}) \\+ \frac {{( \let\over/ a q,{{a q}\over {d e}},{{a q}\over {d f}},{{a q}\over {e f}},{{e q}\over c},{{f q}\over c}, {{b}\over {a}}, {{b e f}\over a};q)_\infty}} {{( \let\over/ {{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over {d e f}},{q\over c},{{e f q}\over c},{{b e}\over a},{{b f}\over a};q)_\infty}} \\ {}_8W _7({{e f}\over c};{{a q}\over {b c}},{{a q}\over {c d}},{{e f}\over a},e,f;q,{{b d}\over a}) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8763 \Description Transformation formula (\cite{\GaRaAA}, (2.11.1), reversed; Appendix (III.37)) in form of a rule. $$\multline {} _{8} W _{7} ({\displaystyle a; b, c, d, e, f}; q, {\displaystyle {{{a^2} {q^2}}\over {b c d e f}}}) \\ \longrightarrow { {(\let \over / {{c e f}\over a}, {{f q}\over d}, {{e q}\over d}, {c\over a}, {{a q}\over {e f}}, a q, {{a q}\over {b f}}, {{a q}\over {b e}} ;q) _\infty} \over {(\let \over / {{e f q}\over d}, {{c f}\over a}, {{c e}\over a}, {q\over d}, {{a q}\over f}, {{a q}\over e}, {{a q}\over {b e f}}, {{a q}\over b} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{e f}\over d}; {{a q}\over {b d}}, {{e f}\over a}, {{a q}\over {c d}}, e, f}; q, {\displaystyle {{b c}\over a}}) \\ + { {(\let \over / a q, {{{a^2} {q^2}}\over {b d e f}}, {{c q}\over b}, {{a {q^2}}\over {b d e}}, {{a {q^2}}\over {b d f}}, {{a q}\over {c d}}, e, f, b, {{{a^2} {q^2}}\over {b c e f}}, {{b c e f}\over {{a^2} q}} ;q) _\infty} \over {(\let \over / {{c f}\over a}, {{c e}\over a}, {q\over d}, {{a q}\over f}, {{a q}\over e}, {{a q}\over b}, {{b e f}\over {a q}}, {{a q}\over d}, {{{a^2} {q^2}}\over {b c d e f}}, {{a q}\over c}, {{{a^2} {q^3}}\over {{b^2} d e f}} ;q) _\infty} } \\ {} _{8} W _{7} ({\displaystyle {{{a^2} {q^2}}\over {{b^2} d e f}}; {{a q}\over {b d}}, {q\over b}, {{{a^2} {q^2}}\over {b c d e f}}, {{a q}\over {b f}}, {{a q}\over {b e}}}; q, {\displaystyle {{b c}\over a}}) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8764 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.15) in form of a rule. $$\multline {} _{8} W _{7} ({\displaystyle a^2; a b, a c, a d, a e, a f}; q, {\displaystyle {{q^2}\over {a b c d e f}}}) \\\longrightarrow -\frac {a} {b}{{ { {(\let \over / a^2 q, a c, {b\over a}, {c\over a}, {{b q}\over d}, {{b q}\over e}, {{b q}\over f}, {q\over {b d}}, {q\over {b e}}, {q\over {b f}} ;q) _\infty} \over {(\let \over / {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {q\over {a d}}, {q\over {a e}}, {q\over {a f}}, b^2 q, b c, {a\over b}, {c\over b} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle b^2; a b, b c, b d, b e, b f}; q, {\displaystyle {{q^2}\over {a b c d e f}}})}}\\ - \frac {a} {c}{{ { {(\let \over / a^2 q, a b, {b\over a}, {c\over a}, {{c q}\over d}, {{c q}\over e}, {{c q}\over f}, {q\over {c d}}, {q\over {c e}}, {q\over {c f}} ;q) _\infty} \over {(\let \over / {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {q\over {a d}}, {q\over {a e}}, {q\over {a f}}, c^2 q, b c, {a\over c}, {b\over c} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle c^2; a c, b c, c d, c e, c f}; q, {\displaystyle {{q^2}\over {a b c d e f}}})}} \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T8810 \Description Transformation formula (\cite{\GaRaAA}, (5.6.1); Appendix (III.38)) in form of a rule. \NoBlackBoxes $$\multline {} _{8} \psi _{8} \! \left [ \matrix \let \over / {\sqrt{a}} q, - {\sqrt{a}} q , b, c, d, e, f, g\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {{a q}\over g}\endmatrix ;q, {\displaystyle {{{a^3} {q^2}}\over {b c d e f g}}} \right ] \\ \hskip-7pt\longrightarrow { {(\let \over / f, {f\over a}, a q, {q\over a}, q, {{a q}\over {b g}}, {{a q}\over {c g}}, {{a q}\over {d g}}, {{a q}\over {e g}}, {{g q}\over b}, {{g q}\over c}, {{g q}\over d}, {{g q}\over e} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {q\over b}, {q\over c}, {q\over d}, {q\over e}, {q\over g}, {{a q}\over g}, {f\over g}, {{f g}\over a}, {{{g^2} q}\over a} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{{g^2}}\over a}; {{b g}\over a}, {{c g}\over a}, {{d g}\over a}, {{e g}\over a}, {{f g}\over a}}; q, {\displaystyle {{{a^3} {q^2}}\over {b c d e f g}}}) \\ + { {(\let \over / g, {g\over a}, a q, {q\over a}, q, {{a q}\over {b f}}, {{a q}\over {c f}}, {{a q}\over {d f}}, {{a q}\over {e f}}, {{f q}\over b}, {{f q}\over c}, {{f q}\over d}, {{f q}\over e} ;q) _\infty} \over {(\let \over / {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {q\over b}, {q\over c}, {q\over d}, {q\over e}, {q\over f}, {{a q}\over f}, {g\over f}, {{f g}\over a}, {{{f^2} q}\over a} ;q) _\infty} } {} _{8} W _{7} ({\displaystyle {{{f^2}}\over a}; {{b f}\over a}, {{c f}\over a}, {{d f}\over a}, {{e f}\over a}, {{f g}\over a}}; q, {\displaystyle {{{a^3} {q^2}}\over {b c d e f g}}}) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10901 \Description Transformation formula (\cite{\GaRaAA}, (2.9.1); Appendix (III.28)) in form of a rule. \BlackBoxes $$\multline {}_{10}W _9(a;b,c,d,e,f,{{{a^3} {q^{2 + n}}}\over {b c d e f}},{q^{-n}};q,q)\\\longrightarrow { {{(\let\over/ a q,{{a q}\over {e f}},{{{a^2} {q^2}}\over {b c d e}},{{{a^2} {q^2}}\over {b c d f}};q)}_{n}}\over{{(\let\over/ {{a q}\over e},{{a q}\over f},{{{a^2} {q^2}}\over {b c d e f}},{{{a^2} {q^2}}\over {b c d}};q)}_{n}} } {}_{10}W _9({{{a^2} q}\over {b c d}};{{a q}\over {c d}},{{a q}\over {b d}},{{a q}\over {b c}},e,f,{{{a^3} {q^{2 + n}}}\over {b c d e f}},{q^{-n}};q,q) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10902 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.13(i), reversed) in form of a rule. $$ {} _{10} W _{9} ({\displaystyle a; b, -b, b {\sqrt{q}}, - b {\sqrt{q}} , c, d, {{a^2 q}\over {b^2 c d}}}; q, {\displaystyle {{a q}\over {b^2}}}) \longrightarrow { {(\let \over / a q, {{a^2 q}\over {{b^4}}} ;q) _\infty} \over {(\let \over / {{a q}\over {b^2}}, {{a^2 q}\over {b^2}} ;q) _\infty} } {} _{4} \phi _{3} \! \left [ \matrix \let \over / b^2, {{b^2 c}\over a}, {{b^2 d}\over a}, {{a q}\over {c d}}\\ \let \over / {{a q}\over c}, {{a q}\over d}, {{b^2 c d}\over a}\endmatrix ;q, {\displaystyle {{a^2 q}\over {{b^4}}}} \right ] $$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10903 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.19) in form of a rule. $$\multline {} _{10} W _{9} ({\displaystyle a; b, c, d, e, f, g, {q^{-n}}}; q, {\displaystyle q}) \\ \longrightarrow {e^n} {{({\let \over / a q, {{a q}\over {c e}}, {{a q}\over {d e}}, {{a q}\over {e f}}, {{a q}\over {e g}}, b}; q) _{n}}\over {({\let \over / {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {{a q}\over g}, {b\over e}}; q) _{n}}} {} _{10} W _{9} ({\displaystyle {e\over {b {q^n}}}; e, {{a q}\over {b c}}, {{a q}\over {b d}}, {{a q}\over {b f}}, {{a q}\over {b g}}, {e\over {a {q^n}}}, {q^{-n}}}; q, {\displaystyle q}) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10904 \Description Transformation formula (\cite{\GaRaAA}, (3.10.4)) in form of a rule. $$\multline {} _{10} W _{9} ({\displaystyle a; b, x, -x, y, -y, -{q^{-n}}, {q^{-n}}}; q, {\displaystyle -{{{a^3} {q^{3 + 2 n}}}\over {b x^2 y^2}}})\\ \longrightarrow {{({\let \over / a^2 q^2, {{a^2 q^2}\over {x^2 y^2}}}; q^2) _{n}}\over {({\let \over / {{a^2 q^2}\over {x^2}}, {{a^2 q^2}\over {y^2}}}; q^2) _{n}}} {} _{5} \phi _{4} \! \left [ \matrix \let \over / {q^{-2 n}}, x^2, y^2, -{{a q}\over b}, -{{a q^2}\over b}\\ \let \over / {{x^2 y^2}\over {a^2 {q^{2 n}}}}, {{a^2 q^2}\over {b^2}}, - a q , - a q^2 \endmatrix ;q^2, {\displaystyle q^2} \right ] \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10905 \Description Transformation formula (\cite{\GaRaAA}, Ex.~3.21(iii)) in form of a rule.\NoBlackBoxes $$\multline {} _{10} \phi _{9} \! \left [ \matrix \let \over / a, {\sqrt{a}} q, - {\sqrt{a}} q , b, c, {a\over {b c}}, {C\over {A {q^n}}}, {1\over {B C {q^n}}}, {B\over {A {q^n}}}, {q^{-n}}\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, b c q, {1\over {C {q^n}}}, {{B C}\over {A {q^n}}}, {1\over {B {q^n}}}, {1\over {A {q^n}}}\endmatrix ;q, {\displaystyle q} \right ] \\\longrightarrow {{({\let \over / a q, b q, c q, {{a q}\over {b c}}, {{A q}\over B}, {{A q}\over C}, B C q}; q) _{n}}\over {({\let \over / A q, B q, C q, {{A q}\over {B C}}, {{a q}\over b}, {{a q}\over c}, b c q}; q) _{n}}} {} _{10} \phi _{9} \! \left [ \matrix \let \over / A, {\sqrt{A}} q, - {\sqrt{A}} q , B, C, {A\over {B C}}, {c\over {a {q^n}}}, {1\over {b c {q^n}}}, {b\over {a {q^n}}}, {q^{-n}}\\ \let \over / {\sqrt{A}}, -{\sqrt{A}}, {{A q}\over B}, {{A q}\over C}, B C q, {1\over {c {q^n}}}, {{b c}\over {a {q^n}}}, {1\over {b {q^n}}}, {1\over {a {q^n}}}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10906 \Description Transformation formula (\cite{\RaVeAA}, (7.7), reversed) in form of a rule.\BlackBoxes $$\multline {} _{10} W _{9} ({\displaystyle a^2 {q^n}; c, d, e, a {q^{{1\over 2} + n}}, - a {q^{{1\over 2} + n}} , {{{a^4} {q^{1 + n}}}\over {c d e}}, {q^{-n}}}; q, {\displaystyle -{q^{1 + n}}}) \\\longrightarrow {{({\let \over / {{a^2 q}\over c}, {{a^2 q}\over d}, {{a^2 q}\over e}, {{a^2 q}\over {c d e}}}; q) _{n}}\over {({\let \over / a^2 q, {{a^2 q}\over {c d}}, {{a^2 q}\over {c e}}, {{a^2 q}\over {d e}}}; q) _{n}}} {} _{12} W _{11} ({\displaystyle a^2; {q^{-2 n}}, c, c q, d, d q, e, e q, {{{a^4} {q^{1 + n}}}\over {c d e}}, {{{a^4} {q^{2 + n}}}\over {c d e}}}; q^2, {\displaystyle q^2}) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10907 \Description Transformation formula (\cite{\RaVeAA}, (7.8), reversed) in form of a rule. $$\multline {} _{10} W _{9} ({\displaystyle {{a^2}\over e}; {{a {\sqrt{q}}}\over e}, -{{a {\sqrt{q}}}\over e}, c, d, e, {{{a^4} {q^{1 + n}}}\over {c d e}}, {q^{-n}}}; q, {\displaystyle -{q\over e}}) \\\longrightarrow {{({\let \over / {{a^2 q}\over c}, {{a^2 q}\over d}, {{a^2 q}\over e}, {{a^2 q}\over {c d e}}}; q) _{n}}\over {({\let \over / a^2 q, {{a^2 q}\over {c d}}, {{a^2 q}\over {c e}}, {{a^2 q}\over {d e}}}; q) _{n}}} {} _{12} W _{11} ({\displaystyle a^2; e^2, c, c q, d, d q, {{{a^4} {q^{1 + n}}}\over {c d e}}, {{{a^4} {q^{2 + n}}}\over {c d e}}, {q^{1 - n}}, {q^{-n}}}; q^2, {\displaystyle q^2}) \endmultline$$ where $n$ is a nonnegative integer. \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10961 \Description Transformation formula (\cite{\GaRaAA}, (2.12.9); Appendix (III.39)) in form of a rule. $$\multline {}_{10}W_9(a; b,c,d,e,f,g,{{{a^3} {q^2}}\over {b c d e f g}}; q,q)\\ \longrightarrow \frac {{( \let\over/ a q,{b\over a},f,g,{{{a^3} {q^2}}\over {b c d e f g}},{{b q}\over f},{{b q}\over g},{{{b^2} c d e f g}\over {{a^3} q}},{{a q}\over {d e}},{{a q}\over {c e}},{{a q}\over {c d}} ;q)_\infty}} {{( \let\over/ {{{b^2} c d e}\over {{a^2}}},{{{a^2} q}\over {b c d e}},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over g},{{b c d e f g}\over {{a^2} q}},{{b c}\over a},{{b d}\over a},{{b e}\over a};q)_\infty}} \\ \frac {{( \let\over/ {{b d e}\over a},{{b c e}\over a},{{b c d}\over a} ;q)_\infty}} {{(\let\over/ {{b f}\over a},{{b g}\over a},{{{a^2} {q^2}}\over {c d e f g}};q)_\infty}} {}_{10}W_9({{{b^2} c d e}\over {{a^2} q}}; b,{{b c}\over a},{{b d}\over a},{{b e}\over a}, {{b c d e f}\over {{a^2} q}},{{b c d e g}\over {{a^2} q}}, {{a q}\over {f g}}; q,q) \\+ \frac {{( \let\over/ a q,{b\over a},{{{a^2} {q^2}}\over {c d e f}},{{{a^2} {q^2}}\over {c d e g}},{{b f g}\over a},{{b c d e f}\over {{a^2} q}},{{b c d e g}\over {{a^2} q}},{{a q}\over {f g}};q)_\infty}} {{( \let\over/ {{{a^2} {q^2}}\over {c d e}},{{b c d e}\over {{a^2} q}},{{a q}\over f},{{a q}\over g},{{b c d e f g}\over {{a^2} q}},{{b f}\over a},{{b g}\over a},{{{a^2} {q^2}}\over {c d e f g}};q)_\infty}} \\ {}_{10}W_9({{{a^2} q}\over {c d e}}; b,{{a q}\over {d e}},{{a q}\over {c e}}, {{a q}\over {c d}},f,g,{{{a^3} {q^2}}\over {b c d e f g}}; q,q)\\ - \frac {{( \let\over/ a q,{b\over a},c,d,e,f,g,{{{a^3} {q^2}}\over {b c d e f g}},{{b q}\over c},{{b q}\over d},{{b q}\over e},{{b q}\over f} ;q)_\infty}} {{( \let\over/ {{{b^2} q}\over a},{a\over b},{{a q}\over c},{{a q}\over d},{{a q}\over e},{{a q}\over f},{{a q}\over g},{{b c d e f g}\over {{a^2} q}},{{b c}\over a},{{b d}\over a},{{b e}\over a},{{b f}\over a};q)_\infty}} \\ \frac {{( \let\over/ {{b q}\over g},{{{b^2} c d e f g}\over {{a^3} q}} ;q)_\infty}} {{( \let\over/ {{b g}\over a},{{{a^2} {q^2}}\over {c d e f g}};q)_\infty}} {}_{10}W_9({{{b^2}}\over a}; b,{{b c}\over a},{{b d}\over a},{{b e}\over a}, {{b f}\over a},{{b g}\over a},{{{a^2} {q^2}}\over {c d e f g}}; q,q) \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10962 \Description Transformation formula (\cite{\GaRaAA}, (3.5.7)) in form of a rule. $$\multline {} _{10} W _{9} ({\displaystyle a; b, c, c q, d, d q, e, e q}; q^2, {\displaystyle {{{a^3} {q^3}}\over {b c^2 d^2 e^2}}}) \longrightarrow { {(\let \over / a q, {{a q}\over {c d}}, {{a q}\over {c e}}, {{a q}\over {d e}} ;q) _\infty} \over {(\let \over / {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{a q}\over {c d e}} ;q) _\infty} } {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{\sqrt{a q}}}\over {{\sqrt{b}}}}, -{{{\sqrt{a q}}}\over {{\sqrt{b}}}}, c, d, e\\ \let \over / {\sqrt{a q}}, -{\sqrt{a q}}, {{a q}\over b}, {{c d e}\over a}\endmatrix ;q, {\displaystyle q} \right ]\\+ { {(\let \over / a q^2, {{{a^3} {q^3}}\over {c^2 d^2 e^2}} ;q^2) _\infty} \over {(\let \over / {{a q^2}\over b}, {{{a^3} {q^3}}\over {b c^2 d^2 e^2}} ;q^2) _\infty} } { {(\let \over / c, d, e, {{a^2 q^2}\over {b c d e}} ;q) _\infty} \over {(\let \over / {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{c d e}\over {a q}} ;q) _\infty} } \\ {} _{5} \phi _{4} \! \left [ \matrix \let \over / {{{{\left( a q \right) }^{{3\over 2}}}}\over {{\sqrt{b}} c d e}}, -{{{{\left( a q \right) }^{{3\over 2}}}}\over {{\sqrt{b}} c d e}}, {{a q}\over {c d}}, {{a q}\over {c e}}, {{a q}\over {d e}}\\ \let \over / {{{{\left( a q \right) }^{{3\over 2}}}}\over {c d e}}, -{{{{\left( a q \right) }^{{3\over 2}}}}\over {c d e}}, {{a^2 q^2}\over {b c d e}}, {{a q^2}\over {c d e}}\endmatrix ;q, {\displaystyle q} \right ] \endmultline$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T10963 \Description Transformation formula (\cite{\GaRaAA}, Ex.~2.30) in form of a rule. $$\align {} _{10} W& _{9} ({\displaystyle a; b, c, d, e, f, g, {{a^3 q^2}\over {b c d e f g}}}; q, {\displaystyle q}) \\&\longrightarrow - { {(\let \over / {{b q}\over f}, {{b q}\over g}, {{b^2 c d e f g}\over {a^3 q}}, a q, c, d, e, f, g, {{a^3 q^2}\over {b c d e f g}}, {b\over a}, {{b q}\over c}, {{b q}\over d}, {{b q}\over e} ;q) _\infty} \over {(\let \over / {{a q}\over f}, {{a q}\over g}, {{b c d e f g}\over {a^2 q}}, {{b^2 q}\over a}, {{b c}\over a}, {{b d}\over a}, {{b e}\over a}, {{b f}\over a}, {{b g}\over a}, {{a^2 q^2}\over {c d e f g}}, {a\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e} ;q) _\infty} }\\&\hskip3cm {} _{10} W _{9} ({\displaystyle {b^2\over a}; b, {{b c}\over a}, {{b d}\over a}, {{b e}\over a}, {{b f}\over a}, {{b g}\over a}, {{a^2 q^2}\over {c d e f g}}}; q, {\displaystyle q})\\&\hskip1cm + { {(\let \over / {{b e g}\over a}, {{b f g}\over a}, a q, {b\over a}, {{a^3 q^2}\over {b c d e f g}}, {{b^2 c d e f g}\over {a^3 q}}, {{a q}\over {c g}}, {{a q}\over {d g}}, {{a q}\over {e g}}, {{a q}\over {f g}}, {{b c g}\over a}, {{b d g}\over a} ;q) _\infty} \over {(\let \over / {{b e}\over a}, {{b f}\over a}, {{b^2 c d e f g^2}\over {a^3 q}}, {{b g}\over a}, {{a^3 q^2}\over {b c d e f g^2}}, {{a q}\over g}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {{b c}\over a}, {{b d}\over a} ;q) _\infty} }\\&\hskip3cm {} _{10} W _{9} ({\displaystyle {{b^2 c d e f g^2}\over {a^3 q^2}}; b, {{b d e f g}\over {a^2 q}}, {{b c e f g}\over {a^2 q}}, {{b c d f g}\over {a^2 q}}, {{b c d e g}\over {a^2 q}}, {{b g}\over a}, g}; q, {\displaystyle q}) \\&\hskip1cm+ { {(\let \over / {{a^2 q^2}\over {c d f g}}, {{a^2 q^2}\over {c d e g}}, a q, {b\over a}, g, {{b q}\over g}, {{b d e f g}\over {a^2 q}}, {{b c e f g}\over {a^2 q}}, {{b c d f g}\over {a^2 q}}, {{b c d e g}\over {a^2 q}} ;q) _\infty} \over {(\let \over / {{b e}\over a}, {{b f}\over a}, {{a^3 q^3}\over {c d e f g^2}}, {{a^2 q^2}\over {c d e f g}}, {{b c d e f g^2}\over {a^3 q^2}}, {{b c d e f g}\over {a^2 q}}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{a q}\over f} ;q) _\infty} }\\&\hskip3cm {{(\let \over /{{a^2 q^2}\over {d e f g}}, {{a^2 q^2}\over {c e f g}};q)_{\infty}}\over {(\let \over / {{b c}\over a}, {{b d}\over a}; q)_{\infty}}} {} _{10} W _{9} ({\displaystyle {{a^3 q^2}\over {c d e f g^2}}; b, {{a q}\over {c g}}, {{a q}\over {d g}}, {{a q}\over {e g}}, {{a q}\over {f g}}, {{a^2 q^2}\over {c d e f g}}, {{a^3 q^2}\over {b c d e f g}}}; q, {\displaystyle q}) \endalign$$ \Seealso S2103, S3201, TListe, TransListe, Ers, PosListe. \Name T101010 \Description Transformation formula (\cite{\GaRaAA}, (5.6.3); Appendix (III.40)) in form of a rule. $$\multline {} _{10} \psi _{10} \! \left [ \matrix \let \over / {\sqrt{a}} q, - {\sqrt{a}} q , b, c, d, e, f, g, h, k\\ \let \over / {\sqrt{a}}, -{\sqrt{a}}, {{a q}\over b}, {{a q}\over c}, {{a q}\over d}, {{a q}\over e}, {{a q}\over f}, {{a q}\over g}, {{a q}\over h}, {{a q}\over k}\endmatrix ;q, {\displaystyle {{{a^4} {q^3}}\over {b c d e f g h k}}} \right ] \\ \longrightarrow { {(\let \over / g, h, {g\over a}, {h\over a}, a q, {q\over a}, q, {{a q}\over {b k}}, {{a q}\over {c k}}, {{a q}\over {d k}}, {{a q}\over {e k}}, {{a q}\over {f k}},