\input amstex \input amsppt.sty \def\[{\left[} \def\]{\right]} \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 \textonlyfont@\bf\twelvebf \textonlyfont@\smc\twelvesmc \textonlyfont@\tt\twelvett %Erg\"anzung des fetten Small-Capitals-Fonts: % \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}% \let\Big\big \let\bigg\big \let\Bigg\big \else \textfont\z@=\twelverm \scriptfont\z@=\tenrm \scriptscriptfont\z@=\sevenrm \textfont\@ne=\twelvei \scriptfont\@ne=\teni \scriptscriptfont\@ne=\seveni \textfont\tw@=\twelvesy \scriptfont\tw@=\tensy \scriptscriptfont\tw@=\sevensy \textfont\thr@@=\twelveex \scriptfont\thr@@=\tenex \scriptscriptfont\thr@@=\tenex \textfont\itfam=\twelveit \scriptfont\itfam=\tenit \scriptscriptfont\itfam=\tenit \textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf \scriptscriptfont\bffam=\sevenbf \setbox\strutbox\hbox{\vrule height10.2\p@ depth4.2\p@ width\z@}% \setbox\strutbox@\hbox{\lower.6\normallineskiplimit\vbox{% \kern-\normallineskiplimit\copy\strutbox}}% \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.4\ht\z@ \fi \normalbaselines\rm\ex@.2326ex\jot3.6\ex@\the\twelvepoint@} \font@\fourteenrm=cmr10 scaled\magstep2 \font@\fourteenit=cmti10 scaled\magstep2 \font@\fourteensl=cmsl10 scaled\magstep2 \font@\fourteensmc=cmcsc10 scaled\magstep2 \font@\fourteentt=cmtt10 scaled\magstep2 \font@\fourteenbf=cmbx10 scaled\magstep2 \font@\fourteeni=cmmi10 scaled\magstep2 \font@\fourteensy=cmsy10 scaled\magstep2 \font@\fourteenex=cmex10 scaled\magstep2 \font@\fourteenmsa=msam10 scaled\magstep2 \font@\fourteeneufm=eufm10 scaled\magstep2 \font@\fourteenmsb=msbm10 scaled\magstep2 \newtoks\fourteenpoint@ \def\fourteenpoint{\normalbaselineskip15\p@ \abovedisplayskip18\p@ plus4.3\p@ minus12.9\p@ \belowdisplayskip\abovedisplayskip \abovedisplayshortskip\z@ plus4.3\p@ \belowdisplayshortskip10.1\p@ plus4.3\p@ minus5.8\p@ \textonlyfont@\rm\fourteenrm \textonlyfont@\it\fourteenit \textonlyfont@\sl\fourteensl \textonlyfont@\bf\fourteenbf \textonlyfont@\smc\fourteensmc \textonlyfont@\tt\fourteentt %Erg\"anzung des fetten Small-Capitals-Fonts: % \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}% \let\Big\big \let\bigg\big \let\Bigg\big \else \textfont\z@=\fourteenrm \scriptfont\z@=\twelverm \scriptscriptfont\z@=\tenrm \textfont\@ne=\fourteeni \scriptfont\@ne=\twelvei \scriptscriptfont\@ne=\teni \textfont\tw@=\fourteensy \scriptfont\tw@=\twelvesy \scriptscriptfont\tw@=\tensy \textfont\thr@@=\fourteenex \scriptfont\thr@@=\twelveex \scriptscriptfont\thr@@=\twelveex \textfont\itfam=\fourteenit \scriptfont\itfam=\twelveit \scriptscriptfont\itfam=\twelveit \textfont\bffam=\fourteenbf \scriptfont\bffam=\twelvebf \scriptscriptfont\bffam=\tenbf \setbox\strutbox\hbox{\vrule height12.2\p@ depth5\p@ width\z@}% \setbox\strutbox@\hbox{\lower.72\normallineskiplimit\vbox{% \kern-\normallineskiplimit\copy\strutbox}}% \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@=1.7\ht\z@ \fi \normalbaselines\rm\ex@.2326ex\jot4.3\ex@\the\fourteenpoint@} \catcode`\@=13 \def\ZeilAV{9} \def\ZeilAN{8} \def\ZeilAM{7} \def\StreAD{6} \def\SlatAC{5} \def\PaScAA{4} \def\GospAB{3} \def\GaRaAA{2} \def\BailAA{1} \hsize15cm \vsize20cm \hoffset-1.2truecm \def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\smc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} \def\poq#1#2{(#1;q)_#2} \def\po#1#2{(#1)_#2} {\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 height8pt 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}} \head\fourteenpoint\bf Summary of the Main Features of HYP and HYPQ \endhead \subhead 1. The package HYP\endsubhead \subsubhead 1.1. The basic objects\endsubsubhead Of course, the basic objects of HYP are the binomial coefficient $\binom nk$, the Pochhammer symbol $(a)_n=a(a+1)\cdots(a+n-1)$, the Gamma function $\Gamma(x)$, and the (generalized) hypergeometric series $${}_r F_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; z\right]=\sum _{n=0} ^{\infty}\frac {\po{a_1}{n}\cdots\po{a_r}{n}} {n!\,\po{b_1}{n}\cdots\po{b_s}{n}} z^n\ .$$ (All the notation and terminology is adopted from \cite{\GaRaAA, pp.~1--6}.) The example below shows how to enter these basic objects. \MATH \goodbreakpoint% In[1]:= Binomial[n,k]*p[a,n]*GAMMA[x]*F[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,e% \MATHrbrace ,z] \goodbreakpoint% ( ) % \MATHluEck % \MATHruEck ( n ) % \MATHvStrich a, b, c % \MATHvStrich Out[1]= ( ) F % \MATHvStrich ; z % \MATHvStrich \MATHGamma (x) (a) ( k ) 3 2% \MATHvStrich d, e % \MATHvStrich n ( ) % \MATHloEck % \MATHroEck \goodbreakpoint% \endMATH \subsubhead 1.2. Converting binomial sums into hypergeometric notation\endsubsubhead As is well-known, ``almost all" binomial sums can be written in hypergeometric notation, which is sort of a ``normal form" for binomial series. For accomplishing this task quickly there is the rule \hbox{\tt SUMF}. As an example we consider the Vandermonde sum. \MATH \goodbreakpoint% In[2]:= SUM[Binomial[N,l]*Binomial[M,K-l],% \MATHlbrace l,0,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck ( ) ( ) \MATHbackslash ( M ) ( N ) Out[2]= \MATHgroesser ( ) ( ) / ( K - l ) ( l ) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck ( ) ( ) l=0 \endMATH and convert it into hypergeometric notation \MATH \goodbreakpoint% In[3]:= \%/.SUMF \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -K, -N % \MATHvStrich F % \MATHvStrich ; 1 % \MATHvStrich (1 - K + M) 2 1% \MATHvStrich 1 - K + M % \MATHvStrich K % \MATHloEck % \MATHroEck Out[3]= -------------------------------- (1) K \endMATH \subsubhead 1.3. Summations for hypergeometric series\endsubsubhead The package HYP includes 29 summation formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. Besides, there is the rule \hbox{\tt SListe} which for a hypergeometric series gives a list of applicable summations. First we ask which summations might be (directly) applicable to our sum. \MATH \goodbreakpoint% In[4]:= \%3/.SListe Is N a nonnegative integer? [y|n]: y \goodbreakpoint% Be sure to apply "FOrdne" before using the following information! \goodbreakpoint% S2101 (1 - K + M) S2103 (1 - K + M) K K Out[4]= % \MATHlbrace % \MATHlbrace ------------------% \MATHrbrace , % \MATHlbrace ------------------% \MATHrbrace % \MATHrbrace (1) (1) K K \endMATH There are two of them. If we want to know how they look like and what they are we could consult the manual, or display the information on the screen as shown below. \MATH \goodbreakpoint% In[5]:= Sgl2101 Do you want to set values for the equation? [y|n]: n \goodbreakpoint% % \MATHluEck % \MATHruEck (-a + c) % \MATHvStrich a, -n % \MATHvStrich n Out[5]= F % \MATHvStrich ; 1 % \MATHvStrich == --------- 2 1% \MATHvStrich c % \MATHvStrich (c) % \MATHloEck % \MATHroEck n \goodbreakpoint% In[6]:= ?S2101 Summation formula (Slater, Appendix (III.4)) in form of a rule. See also: SListe, SUMListe, Ers, PosListe. \endMATH Let us apply {\tt S2101} (the Vandermonde summation \cite{\SlatAC, (1.7.7), Appendix (III.4)}; {\tt S2103} is Gauss' $_2F_1[1]$-evaluation \cite{\SlatAC, (1.7.6)}). \MATH \goodbreakpoint% In[6]:= \%3/.FOrdne/.S2101 Is N a nonnegative integer? [y|n]: y \goodbreakpoint% (1 + M) (1 - K + M) N K Out[6]= --------------------- (1) (1 - K + M) K N \endMATH \subsubhead 1.4. Manipulations of hypergeometric expressions\endsubsubhead The result in {\tt Out[6]} is not completely convincing since everybody knows that the result for the Vandermonde sum {\tt Out[2]} should read $\binom {M+N}K$. To do simplifications of hypergeometric expressions there are 15 rules which allow to do all the manipulations which are the contents of Appendix~I in \cite{\SlatAC}. Besides, there are two functions, \hbox{\tt PosListe} and \hbox{\tt Ers}, for {\it controlled} application of rules: \hbox{\tt PosListe} gives a list of all subexpressions of an expression, together with their respective positions. {\tt Ers} allows the application of a rule to a specified subexpression. Now starting with {\tt Out[6]}, we first want to replace $(1+M)_N\,(1-K+M)_K$ by $(1-K+M)_{K+N}$. This is done with the help of the rule {\tt erw1}, which replaces $(a)_n$ by $(a)_{n+m}/(a+n)_m$ where $m$ has to be entered on request. \MATH \goodbreakpoint% In[7]:= PosListe[\%6] \goodbreakpoint% 1 Out[7]= % \MATHlbrace % \MATHlbrace ----, % \MATHlbrace % \MATHlbrace 1% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace (1 + M) , % \MATHlbrace % \MATHlbrace 2% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace (1 - K + M) , % \MATHlbrace % \MATHlbrace 3% \MATHrbrace % \MATHrbrace % \MATHrbrace , (1) N K K 1 \MATHgroesser % \MATHlbrace ------------, % \MATHlbrace % \MATHlbrace 4% \MATHrbrace % \MATHrbrace % \MATHrbrace % \MATHrbrace (1 - K + M) N \goodbreakpoint% In[8]:= Ers[\%6,erw1,% \MATHlbrace 3% \MATHrbrace ] top-extend by: N \goodbreakpoint% (1 - K + M) K + N Out[8]= ----------------- (1) (1 - K + M) K N \endMATH Next we want to replace $(1-K+M)_{K+N}/(1-K+M)_{N}$ by $(1-K+M+N)_K$. This is done with the help of {\tt zerl} which splits $(a)_n$ into $(a)_m\,(a+m)_{n-m}$ where $m$ has to be entered on request. \MATH \goodbreakpoint% In[9]:= PosListe[\%] \goodbreakpoint% 1 1 Out[9]= % \MATHlbrace % \MATHlbrace ----, % \MATHlbrace % \MATHlbrace 1% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace ------------, % \MATHlbrace % \MATHlbrace 2% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace (1 - K + M) , % \MATHlbrace % \MATHlbrace 3% \MATHrbrace % \MATHrbrace % \MATHrbrace % \MATHrbrace (1) (1 - K + M) K + N K N \goodbreakpoint% In[10]:= Ers[\%\%,zerl,% \MATHlbrace 3% \MATHrbrace ] bottom-split by: N \goodbreakpoint% (1 - K + M + N) K Out[10]= ---------------- (1) K \endMATH The last expression clearly is identical with $\binom{M+N}K$. Additional tools are provided for reversing finite summations, for splitting summations, for shifting summation indices, for exchanging sums, etc. \subsubhead 1.5. Transformations of hypergeometric series\endsubsubhead The package HYP includes 86 transformation formulas in form of rules. All the available transformations (with references) are listed and displayed in the manual. Besides, there is the rule \hbox{\tt TListe} which for a hypergeometric series gives a list of applicable transformations. As an example, we find a proof for \MATH \goodbreakpoint% In[11]:= SUM[Binomial[n,j]\MATHhoch 3,% \MATHlbrace j,0,Infinity% \MATHrbrace ]== SUM[Binomial[n,k]\MATHhoch 2*Binomial[2*(n-k),n],% \MATHlbrace k,0,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty \MATHinfty % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck ( ) % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck ( ) ( ) \MATHbackslash ( n )3 \MATHbackslash ( n )2 ( 2 (-k + n) ) Out[11]= \MATHgroesser ( ) == \MATHgroesser ( ) ( ) / ( j ) / ( k ) ( n ) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck ( ) % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck ( ) ( ) j=0 k=0 \endMATH an identity that occured in the work of V.~Strehl \cite{\StreAD}. Of course, the first step is to transform {\tt Out[11]} into hypergeometric notation. \MATH \goodbreakpoint% In[12]:= \%/.SUMF \goodbreakpoint% 1 n -n % \MATHluEck -n, - - -, -- % \MATHruEck % \MATHvStrich 2 2 2 % \MATHvStrich F % \MATHvStrich ; 1 % \MATHvStrich (1 + n) 3 2% \MATHvStrich 1 % \MATHvStrich n % \MATHluEck % \MATHruEck % \MATHloEck 1, - - n % \MATHroEck % \MATHvStrich -n, -n, -n % \MATHvStrich 2 Out[12]= F % \MATHvStrich ; -1 % \MATHvStrich == -------------------------------- 3 2% \MATHvStrich 1, 1 % \MATHvStrich (1) % \MATHloEck % \MATHroEck n \endMATH Now, let us continue with the left-hand side, \MATH \goodbreakpoint% In[13]:= \%[[1]] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -n, -n, -n % \MATHvStrich Out[13]= F % \MATHvStrich ; -1 % \MATHvStrich 3 2% \MATHvStrich 1, 1 % \MATHvStrich % \MATHloEck % \MATHroEck \endMATH We apply \hbox{\tt TListe} to find out which transformation might be applicable. \MATH \goodbreakpoint% In[14]:= \%/.TListe \goodbreakpoint% Be sure to apply "FOrdne" before using the following information! \goodbreakpoint% Out[14]= % \MATHlbrace % \MATHlbrace T3239% \MATHrbrace % \MATHrbrace \endMATH There is only a single transformation provided by HYP (namely \cite{\BailAA, Ex.~4.(iv), p.~97}) which can be applied. Again we may display the identity and a reference on the screen. \MATH \goodbreakpoint% In[15]:= Tgl3239 Do you want to set values for the equation? [y|n]: n \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, c % \MATHvStrich Out[15]= F % \MATHvStrich ; z % \MATHvStrich == 3 2% \MATHvStrich 1 + a - b, 1 + a - c % \MATHvStrich % \MATHloEck % \MATHroEck % \MATHluEck a 1 a % \MATHruEck % \MATHvStrich -, - + -, 1 + a - b - c -4 z % \MATHvStrich F % \MATHvStrich 2 2 2 ; --------- % \MATHvStrich 3 2% \MATHvStrich 2 % \MATHvStrich % \MATHloEck 1 + a - b, 1 + a - c (-1 + z) % \MATHroEck \MATHgroesser ----------------------------------------- a (1 - z) \goodbreakpoint% In[16]:= ?T3239 Transformation formula (Bailey, Ex. 4.(iv), p. 97) in form of a rule. See also: TListe, TransListe, Ers, PosListe. \endMATH Now, let us apply this transformation. \MATH \goodbreakpoint% In[16]:= \%13/.T3239 \goodbreakpoint% % \MATHluEck -n 1 - n % \MATHruEck n % \MATHvStrich --, -----, 1 + n % \MATHvStrich Out[16]= 2 F % \MATHvStrich 2 2 ; 1 % \MATHvStrich 3 2% \MATHvStrich % \MATHvStrich % \MATHloEck 1, 1 % \MATHroEck \endMATH Now there are several $_3F_2[1]$-transformations which can be applied. \MATH \goodbreakpoint% In[17]:= \%/.TListe Is -1 - n a nonnegative integer? [y|n]: y \goodbreakpoint% Be sure to apply "FOrdne" before using the following information! \goodbreakpoint% n n n n n Out[17]= % \MATHlbrace % \MATHlbrace 2 T3204% \MATHrbrace , % \MATHlbrace 2 T3205% \MATHrbrace , % \MATHlbrace 2 T3206% \MATHrbrace , % \MATHlbrace 2 T3207% \MATHrbrace , % \MATHlbrace 2 T3217% \MATHrbrace , n n n n \MATHgroesser % \MATHlbrace 2 FPerm[2, 1, 3, u], 2 T3232% \MATHrbrace , % \MATHlbrace 2 FPerm[2, 1, 3, u], 2 T3233% \MATHrbrace , n n n n n n \MATHgroesser % \MATHlbrace 2 T3237% \MATHrbrace , % \MATHlbrace 2 T3261% \MATHrbrace , % \MATHlbrace 2 T3262% \MATHrbrace , % \MATHlbrace 2 T3263% \MATHrbrace , % \MATHlbrace 2 T3264% \MATHrbrace , % \MATHlbrace 2 T3267% \MATHrbrace , n \MATHgroesser % \MATHlbrace 2 T3268% \MATHrbrace % \MATHrbrace \endMATH This seems to be too much. However a short inspection reveals that only \hbox{\tt T3204}--\hbox{\tt T3207} make sense in this context. (\hbox{\tt T3261}, \hbox{\tt T3262}, \hbox{\tt T3263}, \hbox{\tt T3264}, \hbox{\tt T3267} are three-term transformations, \hbox{\tt T3232} and \hbox{\tt T3233} would lead back, and \hbox{\tt T3237} transforms our $_3F_2$ into a $_6F_5$.) The task is to apply {\tt T23}--{\tt T26} and parameter permutations (being performed by the rules \hbox{\tt FPerm} and \hbox{\tt FTausche}) in some order until we arrive at the right-hand side of {\tt Out[12]}. Even this seems to be a time-consuming task. However, there is another feature of HYP which can be used here. For each transformation included in the package there is also a rule that finds all possible outcomes when combining this transformation with parameter permutations. So let us try \hbox{\tt T3204} first. \MATH \goodbreakpoint% In[18]:= \%16/.Tli3204 \goodbreakpoint% % \MATHluEck -n 1 - n % \MATHruEck n % \MATHvStrich --, -----, 1 + n % \MATHvStrich Out[18]= % \MATHlbrace 1, 2 F % \MATHvStrich 2 2 ; 1 % \MATHvStrich , 3 2% \MATHvStrich % \MATHvStrich % \MATHloEck 1, 1 % \MATHroEck \goodbreakpoint% 1 - n n 1 - n n % \MATHluEck -----, -n, 1 + - % \MATHruEck % \MATHluEck 1, 1 - ----- - - % \MATHruEck n % \MATHvStrich 2 2 % \MATHvStrich % \MATHvStrich 2 2 % \MATHvStrich \MATHgroesser % \MATHlbrace % \MATHlbrace 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich % \MATHrbrace , 3 2% \MATHvStrich n % \MATHvStrich % \MATHvStrich 1 - n n % \MATHvStrich % \MATHloEck 1, 1 - - % \MATHroEck % \MATHloEck 1 - -----, 1 - - % \MATHroEck 2 2 2 \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 3, 1, u], T3204% \MATHrbrace % \MATHrbrace , \goodbreakpoint% -n 1 - n 1 - n n % \MATHluEck --, 1 - -----, -n % \MATHruEck % \MATHluEck 1, 1 - ----- - - % \MATHruEck n % \MATHvStrich 2 2 % \MATHvStrich % \MATHvStrich 2 2 % \MATHvStrich \MATHgroesser % \MATHlbrace % \MATHlbrace 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich % \MATHrbrace , T3204% \MATHrbrace , 3 2% \MATHvStrich 1 - n % \MATHvStrich % \MATHvStrich n 1 - n % \MATHvStrich % \MATHloEck 1, 1 - ----- - n % \MATHroEck % \MATHloEck 1 + -, 1 - ----- - n % \MATHroEck 2 2 2 \goodbreakpoint% n 1 - n 1 - n n % \MATHluEck 1 + n, 1 + -, 1 - ----- % \MATHruEck % \MATHluEck 1, 1 - ----- - - % \MATHruEck n % \MATHvStrich 2 2 % \MATHvStrich % \MATHvStrich 2 2 % \MATHvStrich \MATHgroesser % \MATHlbrace % \MATHlbrace 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich % \MATHrbrace , 3 2% \MATHvStrich 1 - n n % \MATHvStrich % \MATHvStrich 1 - n n % \MATHvStrich % \MATHloEck 1, 2 - ----- + - % \MATHroEck % \MATHloEck -n, 2 - ----- + - % \MATHroEck 2 2 2 2 \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[3, 1, 2, u], T3204% \MATHrbrace % \MATHrbrace % \MATHrbrace \endMATH And now \hbox{\tt T3207}. \MATH \goodbreakpoint% In[19]:= \%/.Tli3207 Is -1 - n/2 a nonnegative integer? [y|n]: n Is n a nonnegative integer? [y|n]: y Is -1/2 - n/2 a nonnegative integer? [y|n]: n Is -1 - n a nonnegative integer? [y|n]: n Is -1/2 + n/2 a nonnegative integer? [y|n]: n Is n/2 a nonnegative integer? [y|n]: n \endMATH Finally the result {\tt Out[19]} is displayed, which is simplified in the next step. \MATH \goodbreakpoint% In[20]:= ExpandAll[\%19] \goodbreakpoint% % \MATHluEck -n 1 n % \MATHruEck n % \MATHvStrich --, - - -, 1 + n % \MATHvStrich Out[20]= % \MATHlbrace 2, 2 F % \MATHvStrich 2 2 2 ; 1 % \MATHvStrich , 3 2% \MATHvStrich % \MATHvStrich % \MATHloEck 1, 1 % \MATHroEck \goodbreakpoint% n 1 n 1 % \MATHluEck 1 + n, 1 + -, - + - % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace % \MATHlbrace 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich % \MATHrbrace , 3 2% \MATHvStrich 3 % \MATHvStrich % \MATHvStrich 3 % \MATHvStrich % \MATHloEck 1, - + n % \MATHroEck % \MATHloEck -n, - + n % \MATHroEck 2 2 \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[3, 1, 2, u], T3204% \MATHrbrace % \MATHrbrace % \MATHrbrace , T3207% \MATHrbrace , \goodbreakpoint% n 1 n 1 % \MATHluEck 1 + n, 1 + -, - + - % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace % \MATHlbrace 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich % \MATHrbrace , 3 2% \MATHvStrich 3 % \MATHvStrich % \MATHvStrich 3 % \MATHvStrich % \MATHloEck 1, - + n % \MATHroEck % \MATHloEck -n, - + n % \MATHroEck 2 2 \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[3, 1, 2, u], T3204% \MATHrbrace % \MATHrbrace % \MATHrbrace , T3207% \MATHrbrace , \goodbreakpoint% 1 % \MATHluEck 1 n n % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich -n, - + -, 1 + - % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 2 F % \MATHvStrich 2 2 2; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (-n) 3 2% \MATHvStrich % \MATHvStrich % \MATHvStrich n 1 n % \MATHvStrich n % \MATHloEck 1, 1 % \MATHroEck % \MATHloEck 1 + -, - - - % \MATHroEck 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace % \MATHlbrace -----------------------------------------------------% \MATHrbrace , T3204% \MATHrbrace % \MATHrbrace , 1 n (- - -) 2 2 n \goodbreakpoint% 1 % \MATHluEck n 1 n % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich -n, 1 + -, - + - % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 2 F % \MATHvStrich 2 2 2; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (-n) 3 2% \MATHvStrich % \MATHvStrich % \MATHvStrich 1 n n % \MATHvStrich n % \MATHloEck 1, 1 % \MATHroEck % \MATHloEck - + -, 1 - - % \MATHroEck 2 2 2 \MATHgroesser T3207% \MATHrbrace , % \MATHlbrace % \MATHlbrace % \MATHlbrace % \MATHlbrace -----------------------------------------------------% \MATHrbrace , n (1 - -) 2 n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 3, 1, u], T3204% \MATHrbrace % \MATHrbrace % \MATHrbrace , T3207% \MATHrbrace , \goodbreakpoint% 1 n 1 % \MATHluEck -n, - - -, -n % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 1 n 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (- + -) 3 2% \MATHvStrich n 1 3 n % \MATHvStrich % \MATHvStrich 1 n n % \MATHvStrich 2 2 n % \MATHloEck 1 - -, - - --- % \MATHroEck % \MATHloEck - + -, 1 - - % \MATHroEck 2 2 2 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace ------------------------------------------------------% \MATHrbrace , (1) n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 3, 1, u], T3204% \MATHrbrace % \MATHrbrace , \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 1, 3, u], FPerm[2, 1, l], T3207% \MATHrbrace % \MATHrbrace , \goodbreakpoint% 1 n 1 1 % \MATHluEck -n, - + -, - % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 1 n 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (- - -) 3 2% \MATHvStrich 1 n 1 n % \MATHvStrich % \MATHvStrich n 1 n % \MATHvStrich 2 2 n % \MATHloEck - - -, - - - % \MATHroEck % \MATHloEck 1 + -, - - - % \MATHroEck 2 2 2 2 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace ----------------------------------------------------% \MATHrbrace , T3204% \MATHrbrace , (1) n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 1, l], T3207% \MATHrbrace % \MATHrbrace , \goodbreakpoint% -n 1 n 1 % \MATHluEck -n, --, - - - % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 1 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (-) 3 2% \MATHvStrich 1 % \MATHvStrich % \MATHvStrich n 1 n % \MATHvStrich 2 n % \MATHloEck 1, - - n % \MATHroEck % \MATHloEck 1 + -, - - - % \MATHroEck 2 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace -------------------------------------------------% \MATHrbrace , T3204% \MATHrbrace , 1 n (- - -) 2 2 n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 1, 3, u], T3207% \MATHrbrace % \MATHrbrace , \goodbreakpoint% 1 n -n 1 % \MATHluEck -n, - - -, -- % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 1 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (-) 3 2% \MATHvStrich 1 % \MATHvStrich % \MATHvStrich 1 n n % \MATHvStrich 2 n % \MATHloEck 1, - - n % \MATHroEck % \MATHloEck - + -, 1 - - % \MATHroEck 2 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace -------------------------------------------------% \MATHrbrace , n (1 - -) 2 n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 3, 1, u], T3204% \MATHrbrace % \MATHrbrace , % \MATHlbrace FPerm[2, 1, 3, u], T3207% \MATHrbrace % \MATHrbrace , \goodbreakpoint% -n 1 % \MATHluEck -n, --, -n % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich n 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (1 + -) 3 2% \MATHvStrich 1 n -3 n % \MATHvStrich % \MATHvStrich n 1 n % \MATHvStrich 2 n % \MATHloEck - - -, ---- % \MATHroEck % \MATHloEck 1 + -, - - - % \MATHroEck 2 2 2 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace ---------------------------------------------------% \MATHrbrace , T3204% \MATHrbrace , (1) n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 1, 3, u], FPerm[2, 1, l], T3207% \MATHrbrace % \MATHrbrace , \goodbreakpoint% 1 % \MATHluEck 1 n n % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich -n, - + -, 1 + - % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 2 F % \MATHvStrich 2 2 2; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (-n) 3 2% \MATHvStrich % \MATHvStrich % \MATHvStrich n 1 n % \MATHvStrich n % \MATHloEck 1, 1 % \MATHroEck % \MATHloEck 1 + -, - - - % \MATHroEck 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace % \MATHlbrace -----------------------------------------------------% \MATHrbrace , T3204% \MATHrbrace % \MATHrbrace , 1 n (- - -) 2 2 n \goodbreakpoint% 1 % \MATHluEck n 1 n % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich -n, 1 + -, - + - % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich 2 F % \MATHvStrich 2 2 2; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (-n) 3 2% \MATHvStrich % \MATHvStrich % \MATHvStrich 1 n n % \MATHvStrich n % \MATHloEck 1, 1 % \MATHroEck % \MATHloEck - + -, 1 - - % \MATHroEck 2 2 2 \MATHgroesser T3207% \MATHrbrace , % \MATHlbrace % \MATHlbrace % \MATHlbrace % \MATHlbrace -----------------------------------------------------% \MATHrbrace , n (1 - -) 2 n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 3, 1, u], T3204% \MATHrbrace % \MATHrbrace % \MATHrbrace , T3207% \MATHrbrace , \goodbreakpoint% n 1 1 % \MATHluEck -n, 1 + -, - % \MATHruEck % \MATHluEck 1, - % \MATHruEck n % \MATHvStrich 2 2 % \MATHvStrich % \MATHvStrich 2 % \MATHvStrich -n 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (--) 3 2% \MATHvStrich n n % \MATHvStrich % \MATHvStrich 1 n n % \MATHvStrich 2 n % \MATHloEck 1 - -, 1 - - % \MATHroEck % \MATHloEck - + -, 1 - - % \MATHroEck 2 2 2 2 2 \MATHgroesser % \MATHlbrace % \MATHlbrace % \MATHlbrace -------------------------------------------------% \MATHrbrace , (1) n \goodbreakpoint% \MATHgroesser % \MATHlbrace FPerm[2, 3, 1, u], T3204% \MATHrbrace % \MATHrbrace , % \MATHlbrace FPerm[2, 1, l], T3207% \MATHrbrace % \MATHrbrace % \MATHrbrace \endMATH Indeed, our ${}_3F_2\!\[\smallmatrix -n,1/2-n/2,-n/2\\1,1/2-n\endsmallmatrix; 1\]$ which is at the right-hand side of {\tt Out[12]} appears in this list (even several times), for instance on page 11. And we are also given information how this expression was actually obtained. So we may use this information. \MATH \goodbreakpoint% In[21]:= \%16/.T3204/.FPerm[2,1,3,u]/.T3207 Is n a nonnegative integer? [y|n]: y \goodbreakpoint% -n 1 - n 1 - n n % \MATHluEck -n, --, ----- % \MATHruEck % \MATHluEck 1, 1 - ----- - - % \MATHruEck n % \MATHvStrich 2 2 % \MATHvStrich % \MATHvStrich 2 2 % \MATHvStrich 1 - n n 2 F % \MATHvStrich ; 1 % \MATHvStrich \MATHGamma % \MATHvStrich % \MATHvStrich (1 - ----- - -) 3 2% \MATHvStrich 1 - n n % \MATHvStrich % \MATHvStrich n 1 - n % \MATHvStrich 2 2 n % \MATHloEck 1, ----- - - % \MATHroEck % \MATHloEck 1 + -, 1 - ----- - n % \MATHroEck 2 2 2 2 Out[21]= --------------------------------------------------------------------- 1 - n (1 - ----- - n) 2 n \endMATH The $_3F_2$ agrees with the $_3F_2$ at the right-hand side of {\tt Out[12]}. It is easy to show that the other terms simplify to $(1+n)_n/n!$. Hence {\tt Out[11]} is proved. \subsubhead 1.6. Explicit evaluation\endsubsubhead Very often one wants to compute special values of hypergeometric expressions, in particular when checking if some identity might be true or not. For this purpose one should use the {\it evaluation mode} of HYP. So far, all examples were done in the {\it symbolic mode} of HYP. In symbolic mode even expressions like $(a)_3$ are kept unchanged (though it could be evaluated to $a(a+1)(a+2)$). In evaluation mode every expression (that can be evaluated) is evaluated explicitely. There is the switch {\tt P} that toggles between the two modes of HYP. For example, let us evaluate the Vandermonde sum {\tt Out[3]} for $K=2$ (which must result into $\binom {M+N}2$). \MATH \goodbreakpoint% In[22]:= \%3/.K-\MATHgroesser 2 \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich -2, -N % \MATHvStrich F % \MATHvStrich ; 1 % \MATHvStrich (-1 + M) 2 1% \MATHvStrich -1 + M % \MATHvStrich 2 % \MATHloEck % \MATHroEck Out[22]= -------------------------- (1) 2 \goodbreakpoint% In[23]:= P \goodbreakpoint% In[24]:= \%22 Is N a nonnegative integer? [y|n]: n \goodbreakpoint% 2 N (1 - N) N (-1 + M) M (1 + ------ - ----------) -1 + M (-1 + M) M Out[24]= ------------------------------------ 2 \goodbreakpoint% In[25]:= Factor[\%] \goodbreakpoint% (-1 + M + N) (M + N) Out[25]= -------------------- 2 \goodbreakpoint% In[26]:= P \endMATH The last {\tt P} was entered to switch back to symbolic mode. \subsubhead 1.7. Contiguous relations\endsubsubhead A collection of about 50 contiguous relations in form of rules is provided. The application of these rules is very similar to the application of summation and transformation rules as discussed before in subsections~1.3 and 1.5. \subsubhead 1.8. Formal limits of hypergeometric expressions\endsubsubhead The function \hbox{\tt Limes} enables the user to do {\it formal\/} limits of hypergeometric expressions fast. However, it is left to the user to check in each particular situation if taking the limit in a formal way is actually allowed. As an example we derive Bailey's \cite{\SlatAC, Appendix, (III.7)} $_2F_1[\frac {1} {2}]$-sum from Whipple's \cite{\SlatAC, Appendix, (III.24)} $_3F_2$-sum. \MATH \goodbreakpoint% In[27]:= Sgl3234 Do you want to set values for the equation? [y|n]: n \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, 1 - a, c % \MATHvStrich Out[27]= F % \MATHvStrich ; 1 % \MATHvStrich == 3 2% \MATHvStrich d, 1 + 2 c - d % \MATHvStrich % \MATHloEck % \MATHroEck \goodbreakpoint% % \MATHluEck d, 1 + 2 c - d % \MATHruEck 1 - 2 c % \MATHvStrich % \MATHvStrich \MATHgroesser 2 \MATHpi \MATHGamma % \MATHvStrich a d a 1 + 2 c - d 1 a d 1 a 1 + 2 c - d % \MATHvStrich % \MATHvStrich - + -, - + -----------, - - - + -, - - - + ----------- % \MATHvStrich % \MATHloEck 2 2 2 2 2 2 2 2 2 2 % \MATHroEck \goodbreakpoint% In[28]:= Limes[\%,c-\MATHgroesser Infinity] \goodbreakpoint% % \MATHluEck % \MATHruEck 1 - d % \MATHvStrich a, 1 - a 1 % \MATHvStrich 2 Sqrt[\MATHpi ] \MATHGamma (d) Out[28]= F % \MATHvStrich ; - % \MATHvStrich == --------------------- 2 1% \MATHvStrich d 2 % \MATHvStrich 1 a d a d % \MATHloEck % \MATHroEck \MATHGamma (- - - + -) \MATHGamma (- + -) 2 2 2 2 2 \endMATH Incidentally, this example shows another feature of the packages: The objects \hbox{\tt Sgl*} and \hbox{\tt Tgl*} do not only display a formula on the screen, it is also possible to work with it (or with subexpressions). This saves a lot of typing in many situations. \subsubhead 1.9. Gosper's and Zeilberger's algorithms\endsubsubhead The Gosper and Zeilberger algorithms can be used within HYP, provided one gets Peter Paule and Markus Schorn's {\sl Mathematica} implementation of these algorithms. This implementation can be received via e-mail request to peter.paule\@risc.uni-linz.ac.at. (See also \cite{\PaScAA}.) The Gosper algorithm \cite{\GospAB} does definite summation. The Zeilberger algorithm \cite{\ZeilAM, \ZeilAV} finds a polynomial recurrence for a binomial or hypergeometric series (which necessarily exists for a sufficiently large order \cite{\ZeilAN}). For instance, let us find a second order recurrence for the following expression. \MATH \goodbreakpoint% In[29]:= p[a,n]*F[% \MATHlbrace a,b,-n% \MATHrbrace ,% \MATHlbrace c,d% \MATHrbrace ,1] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHvStrich a, b, -n % \MATHvStrich Out[29]= F % \MATHvStrich ; 1 % \MATHvStrich (a) 3 2% \MATHvStrich c, d % \MATHvStrich n % \MATHloEck % \MATHroEck \goodbreakpoint% In[30]:= \%/.ZB[n,2] Peter Paule and Markus Schorn's implementation of the Zeilberger algorithm. (Version 1.1) Is -a a nonnegative integer? [y|n]: n Is -b a nonnegative integer? [y|n]: n \goodbreakpoint% Out[30]= % \MATHlbrace (a + b - c - d - n) (1 + n) (a + n) (1 + a + n) SUM[n] + \MATHgroesser (1 + a + n) (1 - a - b - a b + 2 c + 2 d + c d + 3 n - a n - b n + 2 \MATHgroesser 2 c n + 2 d n + 2 n ) SUM[1 + n] - \MATHgroesser (1 + c + n) (1 + d + n) SUM[2 + n] == 0% \MATHrbrace \endMATH where {\tt SUM[n]} is the expression in {\tt Out[29]}. \subsubhead 1.10. Transforming hypergeometric and basic hypergeometric \hbox{\sl Mathematica} expressions into \TeX-code\endsubsubhead Of course, the package HYP gives full support for writing binomial or hypergeometric expressions in \TeX-code. Besides, the user may choose between Plain\TeX-, \LaTeX-, or \AmSTeX-compatibility. \subhead 2. The package HYPQ\endsubhead All the features and the organization of the package HYPQ are completely analogous to those of HYP. Of course, the basic objects of HYPQ are the $q$-binomial coefficient $\left[n \atop k\right]_q$, the upper $q$-factorial $(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, the infinite $q$-factorial $(a;q)_\infty=\prod _{i=0} ^{\infty}(1-aq^{i})$, and the basic hypergeometric series $${}_r\phi_s\!\left[\matrix a_1,\dots,a_r\\ b_1,\dots,b_s\endmatrix; q, z\right]=\sum _{n=0} ^{\infty}\frac {\poq{a_1}{n}\cdots\poq{a_r}{n}} {\poq{q}{n}\poq{b_1}{n}\cdots\poq{b_s}{n}}\left((-1)^nq^{\binom n2}\right)^{s-r+1}z^n\ .$$ (Cf\. \cite{\GaRaAA, pp.~1--6}.) These objects are entered as follows, \MATH \goodbreakpoint% In[1]:= Binomialq[n,k]*pq[a,n,q\MATHhoch 2]*pqinf[x,1/q]*ph[% \MATHlbrace a,b,c% \MATHrbrace ,% \MATHlbrace d,e% \MATHrbrace ,q,z] \goodbreakpoint% % \MATHluEck % \MATHruEck % \MATHluEck % \MATHruEck % \MATHvStrich n % \MATHvStrich 1 % \MATHvStrich a, b, c % \MATHvStrich 2 Out[1]= % \MATHvStrich % \MATHvStrich (x;-) \MATHphi % \MATHvStrich ; q, z % \MATHvStrich (a; q ) % \MATHvStrich k % \MATHvStrich q \MATHinfty 3 2% \MATHvStrich d, e % \MATHvStrich n % \MATHloEck % \MATHroEck % \MATHloEck % \MATHroEck q \goodbreakpoint% \endMATH The conversion of $q$-binomal sums into basic hypergeometric notation is accomplished by the rule \hbox{\tt SUMph} (see the example below). The package HYPQ includes 18 rules for the simplification of $q$-factorial expressions, 37 summations in form of rules, 106 transformations in form of rules, about 100 contiguous relations in form of rules, and of course the same rules for reversing, splitting, exchanging, etc., sums, and the functions \hbox{\tt PosListe} and {\tt Ers} for the controlled application of rules. Also the writing of $q$-binomial or basic hypergeometric expressions in \TeX-code is fully supported, again leaving the user the choice between Plain\TeX-, \LaTeX-, or \AmSTeX-compatibility. Once being introduced to HYP, a single example should suffice for a demonstration of the abilities of HYPQ. It concerns the $q$-Vandermonde sum (\cite{\GaRaAA, (1.2.3)}; compare with subsections~1.2--1.3). \MATH \goodbreakpoint% In[2]:= SUM[q\MATHhoch ((N-k)*(L-k))*Binomialq[N,k]*Binomialq[M,L-k],% \MATHlbrace k,0,Infinity% \MATHrbrace ] \goodbreakpoint% \MATHinfty % \MATHluEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHruEck % \MATHluEck % \MATHruEck % \MATHluEck % \MATHruEck \MATHbackslash (L - k) (N - k) % \MATHvStrich M % \MATHvStrich % \MATHvStrich N % \MATHvStrich Out[2]= \MATHgroesser q % \MATHvStrich % \MATHvStrich % \MATHvStrich % \MATHvStrich / % \MATHvStrich L - k % \MATHvStrich % \MATHvStrich k % \MATHvStrich % \MATHloEck % \MATHhStrich % \MATHhStrich % \MATHhStrich % \MATHroEck % \MATHloEck % \MATHroEck % \MATHloEck % \MATHroEck k=0 q q \goodbreakpoint% In[3]:= \%/.SUMph \goodbreakpoint% % \MATHluEck -N -L % \MATHruEck L N % \MATHvStrich q , q % \MATHvStrich 1 - L + M q \MATHphi % \MATHvStrich ; q, q % \MATHvStrich (q ; q) 2 1% \MATHvStrich 1 - L + M % \MATHvStrich L % \MATHloEck q % \MATHroEck Out[3]= --------------------------------------------- (q; q) L \goodbreakpoint% In[4]:= \%/.SListe Is N a nonnegative integer? [y|n]: y Is L a nonnegative integer? [y|n]: y \goodbreakpoint% Be sure to apply "phOrdne" before using the following information! \goodbreakpoint% L N 1 - L + M L N 1 - L + M S2101 q (q ; q) S2161 q (q ; q) L L Out[4]= % \MATHlbrace % \MATHlbrace ---------------------------% \MATHrbrace , % \MATHlbrace ---------------------------% \MATHrbrace % \MATHrbrace (q; q) (q; q) L L \goodbreakpoint% In[5]:= 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[5]= \MATHphi % \MATHvStrich a, q ; q, q % \MATHvStrich == ---------- 2 1% \MATHvStrich % \MATHvStrich (c; q) % \MATHloEck c % \MATHroEck n \goodbreakpoint% In[6]:= ?S2101 Summation formula (Gasper/Rahman, Appendix (II.6)) in form of a rule. See also: SListe, SUMListe, Ers, PosListe. \goodbreakpoint% In[6]:= \%3/.phOrdne/.S2101 Is N a nonnegative integer? [y|n]: y \goodbreakpoint% 1 + M 1 - L + M (q ; q) (q ; q) N L Out[6]= ----------------------------- 1 - L + M (q; q) (q ; q) L N \goodbreakpoint% In[7]:= PosListe[\%] \goodbreakpoint% 1 1 + M 1 - L + M Out[7]= % \MATHlbrace % \MATHlbrace -------, % \MATHlbrace % \MATHlbrace 1% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace (q ; q) , % \MATHlbrace % \MATHlbrace 2% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace (q ; q) , % \MATHlbrace % \MATHlbrace 3% \MATHrbrace % \MATHrbrace % \MATHrbrace , (q; q) N L L \goodbreakpoint% 1 \MATHgroesser % \MATHlbrace ----------------, % \MATHlbrace % \MATHlbrace 4% \MATHrbrace % \MATHrbrace % \MATHrbrace % \MATHrbrace 1 - L + M (q ; q) N \goodbreakpoint% In[8]:= Ers[\%\%,erw1,% \MATHlbrace 3% \MATHrbrace ] top-extend by: N \goodbreakpoint% 1 - L + M (q ; q) L + N Out[8]= ------------------------ 1 - L + M (q; q) (q ; q) L N \goodbreakpoint% In[9]:= PosListe[\%] \goodbreakpoint% 1 1 Out[9]= % \MATHlbrace % \MATHlbrace -------, % \MATHlbrace % \MATHlbrace 1% \MATHrbrace % \MATHrbrace % \MATHrbrace , % \MATHlbrace ----------------, % \MATHlbrace % \MATHlbrace 2% \MATHrbrace % \MATHrbrace % \MATHrbrace , (q; q) 1 - L + M L (q ; q) N \goodbreakpoint% 1 - L + M \MATHgroesser % \MATHlbrace (q ; q) , % \MATHlbrace % \MATHlbrace 3% \MATHrbrace % \MATHrbrace % \MATHrbrace % \MATHrbrace L + N \goodbreakpoint% In[10]:= Ers[\%\%,zerl,% \MATHlbrace 3% \MATHrbrace ] bottom-split by: N \goodbreakpoint% 1 - L + M + N (q ; q) L Out[10]= -------------------- (q; q) L \endMATH Clearly this equals the $q$-binomial coefficient $\left[{M+N}\atop R\right]_q$, as could also be shown by using tools of HYPQ (analogous to those being discussed before in subsection~1.4). \Refs \ref\no \BailAA\by W. N. Bailey \yr 1935 \book Generalized hypergeometric series \publ Cambridge University Press \publaddr Cambridge\endref \ref\no \GaRaAA\by G. Gasper and M. Rahman \yr 1990 \book Basic hypergeometric series \publ Encyclopedia of Mathematics And Its Applications~35, Cambridge University Press \publaddr Cambridge\endref \ref\no \GospAB\by R. W. Gosper \yr 1978 \paper Decision procedure for indefinite hypergeometric summation\jour Proc\. Nat\. Acad\. Sci\. USA\vol 75\pages 40--42\endref \ref\no \PaScAA\by P. Paule and M. Schorn\paper A {\sl Mathematica\/} version of Zeilberger's algorithm for proving binomial coefficient identities\jour J. Symbolic Comput\. \toappear \endref \ref\no \SlatAC\by L. J. Slater \yr 1966 \book Generalized hypergeometric functions \publ Cambridge University Press \publaddr Cambridge\endref \ref\no \StreAD\by V. Strehl \yr 19?? \paper Binomial identities --- Combinatorial and algorithmic aspects\jour \vol \pages \endref \ref\no \ZeilAM\by D. Zeilberger \yr 1990 \paper A fast algorithm for proving terminating hypergeometric identities\jour Discrete Math\.\vol 80\pages 207--211\endref \ref\no \ZeilAN\by D. Zeilberger \yr 1990 \paper A holonomic systems approach to special functiions identities\jour J. Comput\. Appl\. Math\.\vol 32\pages 321--368\endref \ref\no \ZeilAV\by D. Zeilberger \yr 1991 \paper The method of creative telescoping\jour J. Symbolic Comput\.\vol 11\pages 195--204\endref \endRefs \enddocument