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

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.

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

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

$\frac{\mathrm{pochhammer}\left(1+a\,n\right)\mathrm{pochhammer}\left(a-b-c+1\,n\right)\mathrm{pochhammer}\left(a-b-d+1\,n\right)\mathrm{pochhammer}\left(a-c-d+1\,n\right)}{\mathrm{pochhammer}\left(1+a-b\,n\right)\mathrm{pochhammer}\left(1+a-c\,n\right)\mathrm{pochhammer}\left(1+a-d\,n\right)\mathrm{pochhammer}\left(a-b-c-d+1\,n\right)}$

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

$\mathrm{Hyperterm}\left(\left[1\,1+a\,a-b-c+1\,a-b-d+1\,a-c-d+1\right]\,\left[1+a-b\,1+a-c\,1+a-d\,a-b-c-d+1\right]\,1\,n\right)$

$\mathrm{Hypersum}\left(\left[-n\,n+3a\,a\right]\,\left[\frac{3}{2}a\,\frac{3a+1}{2}\right]\,\frac{3}{4}\,n\right)$

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