sumtools - Maple Programming Help

Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : SumTools : sumtools/sumrecursion

sumtools

 sumrecursion
 Zeilberger's algorithm

 Calling Sequence sumrecursion(f, k, s(n))

Parameters

 f - expression k - name, summation variable n - name, recurrence variable s - name, recurrence function

Description

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

$\sum _{k}f\left(k\right)$

 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

$\frac{f}{\mathrm{subs}\left(n=n-m,f\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{f}{\mathrm{subs}\left(k=k-l,f\right)}$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{sumtools}\right)$
 $\left[{\mathrm{Hypersum}}{,}{\mathrm{Sumtohyper}}{,}{\mathrm{extended_gosper}}{,}{\mathrm{gosper}}{,}{\mathrm{hyperrecursion}}{,}{\mathrm{hypersum}}{,}{\mathrm{hyperterm}}{,}{\mathrm{simpcomb}}{,}{\mathrm{sumrecursion}}{,}{\mathrm{sumtohyper}}\right]$ (1)
 > $\mathrm{sumrecursion}\left({\mathrm{binomial}\left(n,k\right)}^{3},k,f\left(n\right)\right)$
 ${-}{8}{}{\left({n}{-}{1}\right)}^{{2}}{}{f}{}\left({n}{-}{2}\right){-}\left({7}{}{{n}}^{{2}}{-}{7}{}{n}{+}{2}\right){}{f}{}\left({n}{-}{1}\right){+}{f}{}\left({n}\right){}{{n}}^{{2}}$ (2)
 > $\mathrm{sumrecursion}\left({\mathrm{binomial}\left(n,k\right)}^{2}\mathrm{binomial}\left(2k,n\right),k,s\left(n\right)\right)$
 ${-}{8}{}{\left({n}{-}{1}\right)}^{{2}}{}{s}{}\left({n}{-}{2}\right){-}\left({7}{}{{n}}^{{2}}{-}{7}{}{n}{+}{2}\right){}{s}{}\left({n}{-}{1}\right){+}{s}{}\left({n}\right){}{{n}}^{{2}}$ (3)

Dougall's identity

 > $f≔\frac{\mathrm{hyperterm}\left(\left[a,1+\frac{a}{2},b,c,d,1+2a-b-c-d+n,-n\right],\left[\frac{a}{2},1+a-b,1+a-c,1+a-d,1+a-\left(1+2a-b-c-d+n\right),1+a+n\right],1,k\right)}{\mathrm{hyperterm}\left(\left[1+a,1+a-b-c,1+a-b-d,1+a-c-d,1\right],\left[1+a-b,1+a-c,1+a-d,1+a-b-c-d\right],1,n\right)}$
 ${f}{≔}\frac{{\mathrm{pochhammer}}{}\left({a}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}\frac{{a}}{{2}}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({b}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({c}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({d}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{2}{}{a}{-}{b}{-}{c}{-}{d}{+}{n}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({-}{n}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{b}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{c}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{d}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{b}{-}{c}{-}{d}{,}{n}\right)}{{\mathrm{pochhammer}}{}\left(\frac{{a}}{{2}}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{b}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{c}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{d}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({-}{a}{+}{b}{+}{c}{+}{d}{-}{n}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{+}{n}{,}{k}\right){}{k}{!}{}{\mathrm{pochhammer}}{}\left({1}{+}{a}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{b}{-}{c}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{b}{-}{d}{,}{n}\right){}{\mathrm{pochhammer}}{}\left({1}{+}{a}{-}{c}{-}{d}{,}{n}\right)}$ (4)
 > $\mathrm{sumrecursion}\left(f,k,s\left(n\right)\right)$
 ${s}{}\left({n}\right){-}{s}{}\left({n}{-}{1}\right)$ (5)