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 if f is an (m,l)-fold hypergeometric term with respect to (n,k) for some m and l. The minimal values for m, and l are determined automatically.

The output is a recurrence which equals zero. The recurrence is a function of n the recurrence variable and $s\left(n\right),...$.

An expression f is called (m,l)-fold hypergeometric term with respect to (n,k) if

are rational with respect to n and k. This is typically the case for ratios of products of rational functions, exponentials, factorials, binomial coefficients, and Pochhammer symbols that are rational-linear in their arguments. The implementation supports this type of input.

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

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

$\left[\mathrm{Hypersum}\,\mathrm{Sumtohyper}\,\mathrm{extended\_gosper}\,\mathrm{gosper}\,\mathrm{hyperrecursion}\,\mathrm{hypersum}\,\mathrm{hyperterm}\,\mathrm{simpcomb}\,\mathrm{sumrecursion}\,\mathrm{sumtohyper}\right]$

$\mathrm{sumrecursion}\left({\mathrm{binomial}\left(n\,k\right)}^3\,k\,f\left(n\right)\right)$

$-8{\left(n-1\right)}^{2}f\left(n-2\right)-\left(7n^2-7n+2\right)f\left(n-1\right)+f\left(n\right)n^2$

$\mathrm{sumrecursion}\left({\mathrm{binomial}\left(n\,k\right)}^2\mathrm{binomial}\left(2k\,n\right)\,k\,s\left(n\right)\right)$

$-8{\left(n-1\right)}^{2}s\left(n-2\right)-\left(7n^2-7n+2\right)s\left(n-1\right)+s\left(n\right)n^2$

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

$f≔\frac{\mathrm{pochhammer}\left(a\,k\right)\mathrm{pochhammer}\left(1+\frac{a}{2}\,k\right)\mathrm{pochhammer}\left(b\,k\right)\mathrm{pochhammer}\left(c\,k\right)\mathrm{pochhammer}\left(d\,k\right)\mathrm{pochhammer}\left(1+2a-b-c-d+n\,k\right)\mathrm{pochhammer}\left(-n\,k\right)\mathrm{pochhammer}\left(1+a-b\,n\right)\mathrm{pochhammer}\left(1+a-c\,n\right)\mathrm{pochhammer}\left(1+a-d\,n\right)\mathrm{pochhammer}\left(1+a-b-c-d\,n\right)}{\mathrm{pochhammer}\left(\frac{a}{2}\,k\right)\mathrm{pochhammer}\left(1+a-b\,k\right)\mathrm{pochhammer}\left(1+a-c\,k\right)\mathrm{pochhammer}\left(1+a-d\,k\right)\mathrm{pochhammer}\left(-a+b+c+d-n\,k\right)\mathrm{pochhammer}\left(1+a+n\,k\right)k!\mathrm{pochhammer}\left(1+a\,n\right)\mathrm{pochhammer}\left(1+a-b-c\,n\right)\mathrm{pochhammer}\left(1+a-b-d\,n\right)\mathrm{pochhammer}\left(1+a-c-d\,n\right)}$

$\mathrm{sumrecursion}\left(f\,k\,s\left(n\right)\right)$

$s\left(n\right)-s\left(n-1\right)$