Gosper - Maple Help

SumTools[Hypergeometric]

 Gosper
 perform indefinite hypergeometric summation

 Calling Sequence Gosper(T, n, r)

Parameters

 T - hypergeometric term of $n$ n - variable r - (optional) name

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $T≔\frac{{4}^{n}{n}^{4}}{\mathrm{binomial}\left(2n,n\right)}$
 ${T}{≔}\frac{{{4}}^{{n}}{}{{n}}^{{4}}}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}$ (1)
 > $\mathrm{Gosper}\left(T,n\right)$
 $\frac{\left({2}{}{n}{-}{1}\right){}\left({63}{}{{n}}^{{4}}{-}{140}{}{{n}}^{{3}}{+}{60}{}{{n}}^{{2}}{+}{26}{}{n}{-}{6}\right){}{{4}}^{{n}}}{{693}{}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}$ (2)
 > $T≔\frac{\mathrm{Product}\left({j}^{3},j=1..n-1\right)}{\mathrm{Product}\left({j}^{3}+1,j=1..n\right)}$
 ${T}{≔}\frac{{\prod }_{{j}{=}{1}}^{{n}{-}{1}}{}{{j}}^{{3}}}{{\prod }_{{j}{=}{1}}^{{n}}{}\left({{j}}^{{3}}{+}{1}\right)}$ (3)
 > $\mathrm{Gosper}\left(T,n,'r'\right)$
 $\frac{\left({n}{+}{1}\right){}\left({I}{}\sqrt{{3}}{+}{2}{}{n}{-}{1}\right){}\left({I}{}\sqrt{{3}}{-}{2}{}{n}{+}{1}\right){}\left({\prod }_{{j}{=}{1}}^{{n}{-}{1}}{}{{j}}^{{3}}\right)}{{4}{}\left({\prod }_{{j}{=}{1}}^{{n}}{}\left({{j}}^{{3}}{+}{1}\right)\right)}$ (4)
 > $r$
 $\frac{\left({n}{+}{1}\right){}\left({I}{}\sqrt{{3}}{+}{2}{}{n}{-}{1}\right){}\left({I}{}\sqrt{{3}}{-}{2}{}{n}{+}{1}\right)}{{4}}$ (5)

No hypergeometric solution found:

 > $T≔\frac{{n}^{2}}{\mathrm{binomial}\left(2n,n\right)}$
 ${T}{≔}\frac{{{n}}^{{2}}}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}$ (6)
 > $\mathrm{Gosper}\left(T,n,'r'\right)$
 > $r$
 ${\mathrm{FAIL}}$ (7)

References

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