The Gosper(T,n,r) command solves the problem of indefinite hypergeometric summation, that is, for the given hypergeometric term $T$ of $n$, it constructs another hypergeometric term $G$ of $n$ such that $T\left(n\right)=G\left(n+1\right)-G\left(n\right)$, provided that such a term exists. Otherwise, the function returns the error message "no solution found".

If the third optional argument $r$ is specified, it is assigned the rational function $r\left(n\right)$ such that $G\left(n\right)=r\left(n\right)T\left(n\right)$ if $G$ was computed successfully, and FAIL otherwise.

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

$T≔\frac{4^n{n}^4}{\mathrm{binomial}\left(2n\,n\right)}$

$T≔\frac{4^n{n}^4}{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}$

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

$\frac{\left(2n-1\right)\left(63n^4-140n^3+60n^2+26n-6\right)4^n}{693\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}$

$T≔\frac{\mathrm{Product}\left(j^3\,j=1..n-1\right)}{\mathrm{Product}\left(j^3+1\,j=1..n\right)}$

$T≔\frac{\prod _{j=1}^{n-1}{j}^3}{\prod _{j=1}^n\left(j^3+1\right)}$

$\mathrm{Gosper}\left(T\,n\,'r'\right)$

$\frac{\left(n+1\right)\left(\mathrm{I}\sqrt{3}-2n+1\right)\left(\mathrm{I}\sqrt{3}+2n-1\right)\left(\prod _{j=1}^{n-1}{j}^3\right)}{4\left(\prod _{j=1}^n\left(j^3+1\right)\right)}$

$r$

$\frac{\left(n+1\right)\left(\mathrm{I}\sqrt{3}-2n+1\right)\left(\mathrm{I}\sqrt{3}+2n-1\right)}{4}$

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

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

$\mathrm{Gosper}\left(T\,n\,'r'\right)$

Error, (in SumTools:-Hypergeometric:-Gosper) no solution found

$r$

$\mathrm{FAIL}$

Gosper, R.W., Jr. "Decision procedure for indefinite hypergeometric summation." Proc. Natl. Acad. Sci. USA. Vol. 75. (1977): 40-42.