KoepfZeilberger - Maple Help

SumTools[Hypergeometric]

 KoepfZeilberger
 perform Koepf-Zeilberger's algorithm

 Calling Sequence KoepfZeilberger(T, n, k, En)

Parameters

 T - (m, l)-fold hypergeometric term in n and k n - name k - name En - name; denote the shift operator with respect to n

Description

 • For a specified (m, l)-fold hypergeometric term $T\left(n,k\right)$ in n and k, the KoepfZeilberger(T, n, k, En) command constructs for $T\left(n,k\right)$ a Z-pair $L,G$ that consists of a linear difference operator with coefficients that are polynomials of n over the complex number field

$L={a}_{v}\left(n\right){\mathrm{En}}^{v}+\mathrm{...}+{a}_{1}\left(n\right)\mathrm{En}+{a}_{0}\left(n\right)$

 and a function $G\left(n,k\right)$ such that

$LT\left(n,k\right)=G\left(n,k+1\right)-G\left(n,k\right).$

 • A function $T\left(n,k\right)$ is an (m, l)-fold hypergeometric term if $\frac{T\left(n+m,k\right)}{T\left(n,k\right)}$ and $\frac{T\left(n,k+l\right)}{T\left(n,k\right)}$ are rational functions of n and k.
 • The output from the KoepfZeilberger command is a list of two elements $\left[L,G\right]$ representing the computed Z-pair $L,G$.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $T≔\mathrm{binomial}\left(\frac{2}{3}n,2k\right)$
 ${T}{≔}\left(\genfrac{}{}{0}{}{\frac{{2}{}{n}}{{3}}}{{2}{}{k}}\right)$ (1)
 > $\mathrm{Zpair}≔\mathrm{KoepfZeilberger}\left(T,n,k,\mathrm{En}\right)$
 ${\mathrm{Zpair}}{≔}\left[{{\mathrm{En}}}^{{3}}{-}{4}{,}\frac{{6}{}\left({k}{-}\frac{{n}}{{2}}{-}{1}\right){}\left({2}{}{k}{-}{1}\right){}{k}{}\left(\genfrac{}{}{0}{}{\frac{{2}{}{n}}{{3}}}{{2}{}{k}}\right)}{\left({-}\frac{{n}}{{3}}{+}{k}{-}{1}\right){}\left({-}\frac{{2}{}{n}}{{3}}{+}{2}{}{k}{-}{1}\right){}{n}}\right]$ (2)
 > $\mathrm{Verify}\left(T,\mathrm{Zpair},n,k,\mathrm{En}\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsHypergeometricTerm}\left(T,n\right)$
 ${\mathrm{false}}$ (4)

Note that since T is not a hypergeometric term in n, Zeilberger's algorithm is not applicable to T.

References

 Koepf, W. "Algorithms for m-fold Hypergeometric Summation." Journal of Symbolic Computation. Vol. 20 No. 4. (1995): 399-417.
 Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.