 ExtendedZeilberger - Maple Help

SumTools[Hypergeometric]

 ExtendedZeilberger
 construct a minimal Z-pair Calling Sequence ExtendedZeilberger(V, n, k, En) Parameters

 V - definite sum of hypergeometric term n - name k - name En - name denoting the shift operator with respect to n Description

 • Let $T\left(m,k\right)$ be a hypergeometric term of m and k. Let $V\left(n,k\right)={\sum }_{m=-n+b}^{n+d}T\left(m,k\right)$ where b and d are integers. The ExtendedZeilberger(V, n, k, En) command, an extension to Zeilberger's algorithm, constructs the minimal Z-pair for $V\left(n,k\right)$ provided that it exists.
 • It can be shown that a Z-pair for $V\left(n,k\right)$ exists if and only if a Z-pair for the hypergeometric term $T\left(n,k\right)$ exists. Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $T≔\frac{{\left(-1\right)}^{k}\mathrm{binomial}\left(m,k\right)\mathrm{binomial}\left(2k,k\right)\cdot 1}{{2}^{2k}}$
 ${T}{≔}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{m}}{{k}}\right){}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right)}{{{2}}^{{2}{}{k}}}$ (1)
 > $V≔\mathrm{Sum}\left(T,m=0..n\right)$
 ${V}{≔}{\sum }_{{m}{=}{0}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{m}}{{k}}\right){}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right)}{{{2}}^{{2}{}{k}}}$ (2)
 > $\mathrm{ExtendedZeilberger}\left(V,n,k,\mathrm{En}\right)$
 $\left[\left({2}{}{n}{+}{4}\right){}{{\mathrm{En}}}^{{2}}{+}\left({-}{4}{}{n}{-}{7}\right){}{\mathrm{En}}{+}{2}{}{n}{+}{3}{,}\frac{{2}{}{{k}}^{{2}}{}{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{n}{+}{1}}{{k}}\right){}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right)}{\left({-}{n}{+}{k}{-}{2}\right){}{{2}}^{{2}{}{k}}}\right]$ (3)
 > $\mathrm{_EnvDoubleSum}≔\mathrm{true}$
 ${\mathrm{_EnvDoubleSum}}{≔}{\mathrm{true}}$ (4)
 > $\mathrm{Sum}\left(V,k=0..n\right)=\mathrm{DefiniteSum}\left(V,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{m}{=}{0}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{m}}{{k}}\right){}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right)}{{{2}}^{{2}{}{k}}}{=}\frac{\left({2}{}{n}{+}{1}\right){}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}{{{4}}^{{n}}}$ (5)

Try the Maple command sum:

 > $\mathrm{sum}\left(\mathrm{sum}\left(T,m=0..n\right),k=0..n\right)$
 $\left({\sum }_{{k}{=}{0}}^{{n}}{}\frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{k}\right){}{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{2}{}{k}}{{k}}\right){}{{4}}^{{-}{k}}}{{\mathrm{\pi }}{}\left({k}{+}{1}\right)}\right){+}\left({2}{}{n}{+}{1}\right){}{{4}}^{{-}{n}}{}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)$ (6) References

 Le, H.Q. "Computing the Minimal Telescoper for Sums of Hypergeometric Terms." SIGSAM Bulletin: Communications on Computer Algebra. Vol. 35 No. 3. (2001): 2-10.