<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\input amstex
\input amsppt.sty

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\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}

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\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

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