SumTools[Hypergeometric] - Maple Programming Help

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SumTools[Hypergeometric]

 IsHolonomic
 test if a given bivariate hypergeometric term is holonomic
 IsProperHypergeometricTerm
 test if a given bivariate hypergeometric term is proper

 Calling Sequence IsHolonomic(T, n, k) IsProperHypergeometricTerm(T, n, k)

Parameters

 T - hypergeometric term of n and k n - variable k - variable

Description

 • The IsProperHypergeometricTerm(T,n,k) command returns true if $T\left(n,k\right)$ is a proper hypergeometric term. Otherwise, it returns false.
 • The IsHolonomic(T,n,k) command returns true if the bivariate hypergeometric term $T\left(n,k\right)$ is holonomic. Otherwise, it returns false.
 • A bivariate hypergeometric term $T\left(n,k\right)$ is proper if it can be written as $T\left(n,k\right)=P\left(n,k\right)\mathrm{Tp}\left(n,k\right)$ where $P\left(n,k\right)$ is a polynomial of n and k, and $\mathrm{Tp}\left(n,k\right)=\frac{{u}^{n}{v}^{k}\left({\prod }_{i=1}^{l}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(k{b}_{i}+n{a}_{i}+{g}_{i}\right)!\right)}{{\prod }_{i=1}^{m}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\mathrm{ap}}_{i}+{\mathrm{bp}}_{i}+{\mathrm{gp}}_{i}\right)!}$, ${a}_{i},{b}_{i},{\mathrm{ap}}_{i},{\mathrm{bp}}_{i}$ are integers, and $l,m$ are non-negative integers, ${g}_{i},{\mathrm{gp}}_{i},u,v$ are complex numbers.
 • It can be shown that $T\left(n,k\right)$ is proper if and only if it is holonomic.
 Note: If $T\left(n,k\right)$ is a proper hypergeometric term, the termination of Zeilberger's algorithm is guaranteed.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}[\mathrm{Hypergeometric}]\right):$
 > $T≔\left(-\frac{4-k-4n}{-1-k-4n}+\frac{3-k-4n}{-2-k-4n}\right){\left(-1\right)}^{k}\mathrm{binomial}\left(n+1,k\right)\mathrm{binomial}\left(2n-2k+1,n\right)$
 ${T}{≔}\left({-}\frac{{4}{-}{k}{-}{4}{}{n}}{{-}{1}{-}{k}{-}{4}{}{n}}{+}\frac{{3}{-}{k}{-}{4}{}{n}}{{-}{2}{-}{k}{-}{4}{}{n}}\right){}{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({n}{+}{1}{,}{k}\right){}{\mathrm{binomial}}{}\left({2}{}{n}{-}{2}{}{k}{+}{1}{,}{n}\right)$ (1)
 > $\mathrm{ConjugateRTerm}[1]\left(T,n,k,'\mathrm{listform}'\right)$
 $\left[{-}\frac{{2}}{\sqrt{{\mathrm{π}}}}{,}\frac{\left({n}{-}{k}\right){!}{}\left({k}{+}{4}{}{n}\right){!}{}{\left({-}\frac{{1}}{{4}}\right)}^{{k}}{}{{4}}^{{n}}{}\left({n}{-}{k}{+}\frac{{1}}{{2}}\right){!}{}\left({n}{+}{1}\right){!}}{\left({n}{+}{1}{-}{k}\right){!}{}\left({2}{+}{k}{+}{4}{}{n}\right){!}{}{k}{!}{}\left({n}{-}{2}{}{k}{+}{1}\right){!}{}{n}{!}}\right]$ (2)
 > $\mathrm{IsProperHypergeometricTerm}\left(T,n,k\right)$
 ${\mathrm{true}}$ (3)
 > $T≔\frac{\left(-48-94n+10k\right){\left(-1\right)}^{k}\mathrm{binomial}\left(n+1,k\right)\mathrm{binomial}\left(2n-2k+1,n\right)}{{\left(5-5n+k{9}^{\frac{1}{3}}\right)}^{3}}$
 ${T}{≔}\frac{\left({-}{48}{-}{94}{}{n}{+}{10}{}{k}\right){}{\left({-}{1}\right)}^{{k}}{}{\mathrm{binomial}}{}\left({n}{+}{1}{,}{k}\right){}{\mathrm{binomial}}{}\left({2}{}{n}{-}{2}{}{k}{+}{1}{,}{n}\right)}{{\left({5}{-}{5}{}{n}{+}{k}{}{{9}}^{{1}{/}{3}}\right)}^{{3}}}$ (4)
 > $\mathrm{ConjugateRTerm}[1]\left(T,n,k,'\mathrm{listform}'\right)$
 $\left[{-}\frac{{54}}{{5}{}\sqrt{{\mathrm{π}}}{}{\left({5}{}{{3}}^{{1}{/}{3}}{}{n}{-}{5}{}{{3}}^{{1}{/}{3}}{-}{3}{}{k}\right)}^{{3}}}{,}\frac{\left({24}{+}{47}{}{n}{-}{5}{}{k}\right){!}{}\left({n}{-}{k}\right){!}{}{\left({-}\frac{{1}}{{4}}\right)}^{{k}}{}{{4}}^{{n}}{}\left({n}{-}{k}{+}\frac{{1}}{{2}}\right){!}{}\left({n}{+}{1}\right){!}}{\left({23}{+}{47}{}{n}{-}{5}{}{k}\right){!}{}\left({n}{+}{1}{-}{k}\right){!}{}{k}{!}{}\left({n}{-}{2}{}{k}{+}{1}\right){!}{}{n}{!}}\right]$ (5)
 > $\mathrm{IsProperHypergeometricTerm}\left(T,n,k\right)$
 ${\mathrm{false}}$ (6)

References

 Abramov, S.A., and Petkovsek, M. "Proof of a Conjecture of Wilf and Zeilberger." Preprint series. Vol. 39. (2001): 748. University of Ljubljana, ISSN 1318--4865.