Let En be the shift operator with respect to n, defined by $\mathrm{En}\left(f\left(n\right)\right)=f\left(n+1\right)$. For the given set (list)

where the $t_i\left(n\right)$ are hypergeometric terms of n, the ExtendedGosper(T,n) command returns a set (list)

of hypergeometric terms $s_i\left(n\right)$ such that

if each of the hypergeometric term $s_i\left(n\right)$ exists. Otherwise, the ExtendedGosper routine returns the error message ``no solution found''.

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

$T≔\left\{{\left(-1\right)}^k\mathrm{binomial}\left(n\,k\right)k^j\,n^2{a}^n\,-n^2{a}^n+{\left(n+1\right)}^2{a}^{n+1}\right\}$

$T≔\left\{n^2{a}^n\,{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{n}{k}\right)k^j\,-n^2{a}^n+{\left(n+1\right)}^2{a}^{n+1}\right\}$

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

$\left\{\frac{\left(a^2{n}^2-2a{n}^2-2an+n^2+a+2n+1\right)a^{n+1}}{a^3-3a^2+3a-1}\,-\frac{\left(-n+k\right){\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{n}{k}\right)k^j}{k+1}\right\}$

$T≔\left[\frac{n^2{4}^n}{\left(n+1\right)\left(n+2\right)}\,\frac{2^{2n-1}}{n\left(2n+1\right)\mathrm{binomial}\left(2n\,n\right)}\,-\frac{n^2{4}^n}{\left(n+1\right)\left(n+2\right)}+\frac{{\left(n+1\right)}^2{4}^{n+1}}{\left(n+2\right)\left(n+3\right)}\right]$

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

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

$\left[-\frac{2^{2n-1}}{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)n}\,\frac{\left(n-1\right)4^{n+1}}{3\left(n+2\right)}\right]$

$T≔\left\{\frac{2n+1}{n^2-3}{n}^2{a}^n\,n^2{a}^n\right\}$

$T≔\left\{n^2{a}^n\,\frac{\left(2n+1\right)n^2{a}^n}{n^2-3}\right\}$

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

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

Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B. Wellesley, Massachusetts: A. K. Peters Ltd., 1996.