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sumtools

 hypersum
 Zeilberger-Koepf's hypersum algorithm
 Hypersum
 Zeilberger-Koepf's algorithm

 Calling Sequence hypersum(U, L, z, n) Hypersum(U, L, z, n)

Parameters

 U, L - lists of the upper and lower parameters z - evaluation point n - name, recurrence variable

Description

 • This function is an implementation of Zeilberger-Koepf's algorithm, and calculates a closed form for the sum

$\sum _{k}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\mathrm{hyperterm}\left(U,L,k\right)$

 the sum to be taken over all integers k, with respect to n, whenever an extension of Zeilberger's algorithm gives a suitable recurrence equation. Here, U and L denote the lists of upper and lower parameters, and z is the evaluation point. The arguments of U and L are assumed to be rational-linear with respect to n.  The procedure Hypersum is the corresponding inert form which remains unevaluated.
 • The command with(sumtools,hypersum) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{sumtools}\right):$

Dougall's identity

 > $\mathrm{hypersum}\left(\left[a,1+\frac{a}{2},b,c,d,1+2a-b-c-d+n,-n\right],\left[\frac{a}{2},1+a-b,1+a-c,1+a-d,1+a-\left(1+2a-b-c-d+n\right),1+a+n\right],1,n\right)$
 $\frac{{\mathrm{pochhammer}}{}\left({1}{+}{a}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({a}{-}{b}{-}{c}{+}{1}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({a}{-}{b}{-}{d}{+}{1}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({a}{-}{c}{-}{d}{+}{1}{,}{n}\right)}{{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{b}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{c}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{d}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({a}{-}{b}{-}{c}{-}{d}{+}{1}{,}{n}\right)}$ (1)
 > $\mathrm{Hypersum}\left(\left[a,1+\frac{a}{2},b,c,d,1+2a-b-c-d+n,-n\right],\left[\frac{a}{2},1+a-b,1+a-c,1+a-d,1+a-\left(1+2a-b-c-d+n\right),1+a+n\right],1,n\right)$
 ${\mathrm{Hyperterm}}{}\left(\left[{1}{,}{1}{+}{a}{,}{a}{-}{b}{-}{c}{+}{1}{,}{a}{-}{b}{-}{d}{+}{1}{,}{a}{-}{c}{-}{d}{+}{1}\right]{,}\left[{1}{+}{a}{-}{b}{,}{1}{+}{a}{-}{c}{,}{1}{+}{a}{-}{d}{,}{a}{-}{b}{-}{c}{-}{d}{+}{1}\right]{,}{1}{,}{n}\right)$ (2)

Andrews

 > $\mathrm{Hypersum}\left(\left[-n,n+3a,a\right],\left[\frac{3a}{2},\frac{3a+1}{2}\right],\frac{3}{4},n\right)$
 ${{}\begin{array}{cc}{\mathrm{Hyperterm}}{}\left(\left[{1}{,}\frac{{2}}{{3}}{,}\frac{{1}}{{3}}\right]{,}\left[\frac{{2}}{{3}}{+}{a}{,}{a}{+}\frac{{1}}{{3}}\right]{,}{1}{,}\frac{{1}}{{3}}{}{n}\right)& {\mathrm{irem}}{}\left({n}{,}{3}\right){=}{0}\\ {0}& {\mathrm{irem}}{}\left({n}{,}{3}\right){=}{1}\\ {0}& {\mathrm{irem}}{}\left({n}{,}{3}\right){=}{2}\end{array}$ (3)