The KoepfGosper(T, n) command solves the problem of indefinite summation of j-fold hypergeometric terms, that is, for the input j-fold hypergeometric term T of n, it constructs a function $G$ which is a sum of hypergeometric terms of n such that $T\left(n\right)=G\left(n+1\right)-G\left(n\right)$, provided that such a $G$ exists. Otherwise, the function returns the error message ``no solution found''.

The parameter T is a j-fold hypergeometric term in n if $\frac{T\left(n+j\right)}{T\left(n\right)}$ is a rational function in n.

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

$T≔n\left(\frac{n}{2}\right)!$

$T≔n\left(\frac{n}{2}\right)!$

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

$2\left(\frac{n}{2}\right)!+2\left(\frac{n}{2}+\frac{1}{2}\right)!$

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

$\mathrm{false}$

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.