This function is an implementation of Koepf's extension of Zeilberger's algorithm, calculating a (downward) recurrence equation for the sum

the sum to be taken over all integers k, with respect to n. 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 resulting expression equals zero.

The output is a recurrence which equals zero. The recurrence is output as a function of n, the recurrence variable, and $s\left(n\right),s\left(n-1\right),\mathrm{...}$.

The command with(sumtools,hyperrecursion) allows the use of the abbreviated form of this command.

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

$\mathrm{hyperrecursion}\left(\left[-n\,a\right]\,\left[b\right]\,1\,f\left(n\right)\right)$

$\left(-n+a-b+1\right)f\left(n-1\right)+\left(b+n-1\right)f\left(n\right)$

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

$-\left(a+n\right)\left(a-c-d+n\right)\left(a-b-d+n\right)\left(a-b-c+n\right)s\left(n-1\right)+\left(a-d+n\right)\left(a-c+n\right)\left(a-b+n\right)\left(a-b-c-d+n\right)s\left(n\right)$

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

$\left(b-2+3n\right)\left(b+1-3n\right)\left(2a-1+3n\right)\left(2a+3n\right)s\left(n-1\right)+\left(3n-1\right)\left(3n-2\right)\left(3n-1+b+2a\right)\left(3n-b+2a\right)s\left(n\right)$