KoepfGosper - Maple Help

SumTools[Hypergeometric]

 KoepfGosper
 indefinite summation of j-fold hypergeometric terms

 Calling Sequence KoepfGosper(T, n)

Parameters

 T - hypergeometric term in n n - name; specifies summation index

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $T≔n\left(\frac{n}{2}\right)!$
 ${T}{≔}{n}{}\left(\frac{{n}}{{2}}\right){!}$ (1)
 > $\mathrm{KoepfGosper}\left(T,n\right)$
 ${2}{}\left(\frac{{n}}{{2}}\right){!}{+}{2}{}\left(\frac{{n}}{{2}}{+}\frac{{1}}{{2}}\right){!}$ (2)
 > $\mathrm{IsHypergeometricTerm}\left(T,n\right)$
 ${\mathrm{false}}$ (3)

Note that T is not a hypergeometric term in n. Hence, Gosper'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.