SumTools[Hypergeometric] - Maple Programming Help

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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 $V\left(n,k\right)={\sum }_{m=-n+b}^{n+d}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T\left(m,k\right)$ 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

 > $\mathrm{with}\left(\mathrm{SumTools}[\mathrm{Hypergeometric}]\right):$
 > $T≔\frac{{\left(-1\right)}^{k}\mathrm{binomial}\left(m,k\right)\mathrm{binomial}\left(2k,k\right)\cdot 1}{{2}^{2k}}$
 ${T}{≔}\frac{{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({m}{,}{k}\right){}{\mathrm{binomial}}{}\left({2}{}{k}{,}{k}\right)}{{{2}}^{{2}{}{k}}}$ (1)
 > $V≔{\sum }_{m=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T$
 ${V}{≔}{\sum }_{{m}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({m}{,}{k}\right){}{\mathrm{binomial}}{}\left({2}{}{k}{,}{k}\right)}{{{2}}^{{2}{}{k}}}$ (2)
 > $\mathrm{ExtendedZeilberger}\left(V,n,k,\mathrm{En}\right)$
 $\left[\left({2}{}{n}{+}{4}\right){}{{\mathrm{En}}}^{{2}}{+}\left({-}{4}{}{n}{-}{7}\right){}{\mathrm{En}}{+}{2}{}{n}{+}{3}{,}\frac{{2}{}{{k}}^{{2}}{}{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({n}{+}{1}{,}{k}\right){}{\mathrm{binomial}}{}\left({2}{}{k}{,}{k}\right)}{\left({-}{n}{-}{2}{+}{k}\right){}{{2}}^{{2}{}{k}}}\right]$ (3)
 > $\mathrm{_EnvDoubleSum}≔\mathrm{true}$
 ${\mathrm{_EnvDoubleSum}}{≔}{\mathrm{true}}$ (4)
 > ${\sum }_{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}V=\mathrm{DefiniteSum}\left(V,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\sum }_{{m}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({m}{,}{k}\right){}{\mathrm{binomial}}{}\left({2}{}{k}{,}{k}\right)}{{{2}}^{{2}{}{k}}}{=}\frac{{2}{}{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{2}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right)}$ (5)

Try the Maple command sum:

 > $\sum _{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\sum _{m=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T$
 ${-}\frac{{1}}{{2}}{}\frac{\left({-}{4}{}{n}{-}{6}{+}{2}{}{\left({-}{1}\right)}^{{n}{+}{1}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{2}}\right)}{\left({n}{+}{2}\right){}\left({n}{+}{1}\right){}\sqrt{{\mathrm{π}}}{}{\mathrm{Γ}}{}\left({n}\right)}{-}\frac{{1}}{{2}}{}\frac{\left({-}{4}{}{n}{-}{6}{+}{2}{}{\left({-}{1}\right)}^{{n}{+}{1}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{2}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{Γ}}{}\left({3}{+}{n}\right)}{-}\frac{\left({-}{1}{+}{\left({-}{1}\right)}^{{n}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{2}}\right)}{\left({n}{+}{2}\right){}\sqrt{{\mathrm{π}}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right)}{+}{\sum }_{{k}{=}{0}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{k}{}{\left({-}{1}\right)}^{{k}}{}\left({{}\begin{array}{cc}{1}& {k}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{π}}{}{k}\right)}{{\mathrm{π}}{}{k}}& {\mathrm{otherwise}}\end{array}\right){}{\mathrm{binomial}}{}\left({2}{}{k}{,}{k}\right)}{\left({k}{+}{1}\right){}{{2}}^{{2}{}{k}}}$ (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.