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

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

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 $L,G$.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\:$

$T≔\mathrm{binomial}\left(\frac{2}{3}n\,2k\right)$

$T≔\left(\genfrac{}{}{0ex}{}{\frac{2n}{3}}{2k}\right)$

$\mathrm{Zpair}≔\mathrm{KoepfZeilberger}\left(T\,n\,k\,\mathrm{En}\right)$

$\mathrm{Zpair}≔\left[{\mathrm{En}}^3-4\,\frac{6\left(k-\frac{n}{2}-1\right)\left(2k-1\right)k\left(\genfrac{}{}{0ex}{}{\frac{2n}{3}}{2k}\right)}{\left(-\frac{n}{3}+k-1\right)\left(-\frac{2n}{3}+2k-1\right)n}\right]$

$\mathrm{Verify}\left(T\,\mathrm{Zpair}\,n\,k\,\mathrm{En}\right)$

$\mathrm{true}$

$\mathrm{IsHypergeometricTerm}\left(T\,n\right)$

$\mathrm{false}$

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.