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SumTools[Hypergeometric]

 ExtendedGosper
 perform extended Gosper's algorithm

 Calling Sequence ExtendedGosper(T, n)

Parameters

 T - list or set of hypergeometric terms of n n - variable

Description

 • 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)

$T=\left\{{t}_{1}\left(n\right),...,{t}_{p}\left(n\right)\right\}$

 where the ${t}_{i}\left(n\right)$ are hypergeometric terms of n, the ExtendedGosper(T,n) command returns a set (list)

$S=\left\{{s}_{1}\left(n\right),...,{s}_{q}\left(n\right)\right\}$

 of hypergeometric terms ${s}_{i}\left(n\right)$ such that

$\left(\mathrm{En}-1\right)\sum _{i=1}^{q}{s}_{i}\left(n\right)=\sum _{j=1}^{p}{t}_{j}\left(n\right)$

 if each of the hypergeometric term ${s}_{i}\left(n\right)$ exists. Otherwise, the ExtendedGosper routine returns the error message no solution found''.

Examples

 > $\mathrm{with}\left({\mathrm{SumTools}}_{\mathrm{Hypergeometric}}\right):$
 > $T≔\left\{{\left(-1\right)}^{k}\mathrm{binomial}\left(n,k\right){k}^{j},{n}^{2}{a}^{n},-{n}^{2}{a}^{n}+{\left(n+1\right)}^{2}{a}^{n+1}\right\}$
 ${T}{≔}\left\{{{n}}^{{2}}{}{{a}}^{{n}}{,}{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{{k}}^{{j}}{,}{-}{{n}}^{{2}}{}{{a}}^{{n}}{+}{\left({n}{+}{1}\right)}^{{2}}{}{{a}}^{{n}{+}{1}}\right\}$ (1)
 > $\mathrm{ExtendedGosper}\left(T,n\right)$
 $\left\{\frac{\left({{a}}^{{2}}{}{{n}}^{{2}}{-}{2}{}{a}{}{{n}}^{{2}}{-}{2}{}{a}{}{n}{+}{{n}}^{{2}}{+}{a}{+}{2}{}{n}{+}{1}\right){}{{a}}^{{n}{+}{1}}}{{{a}}^{{3}}{-}{3}{}{{a}}^{{2}}{+}{3}{}{a}{-}{1}}{,}{-}\frac{\left({-}{n}{+}{k}\right){}{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{{k}}^{{j}}}{{k}{+}{1}}\right\}$ (2)
 > $T≔\left[\frac{{n}^{2}{4}^{n}}{\left(n+1\right)\left(n+2\right)},\frac{{2}^{2n-1}}{n\left(2n+1\right)\mathrm{binomial}\left(2n,n\right)},-\frac{{n}^{2}{4}^{n}}{\left(n+1\right)\left(n+2\right)}+\frac{{\left(n+1\right)}^{2}{4}^{n+1}}{\left(n+2\right)\left(n+3\right)}\right]$
 ${T}{≔}\left[\frac{{{n}}^{{2}}{}{{4}}^{{n}}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}{,}\frac{{{2}}^{{2}{}{n}{-}{1}}}{{n}{}\left({2}{}{n}{+}{1}\right){}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}{,}{-}\frac{{{n}}^{{2}}{}{{4}}^{{n}}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}{+}\frac{{\left({n}{+}{1}\right)}^{{2}}{}{{4}}^{{n}{+}{1}}}{\left({n}{+}{2}\right){}\left({n}{+}{3}\right)}\right]$ (3)
 > $\mathrm{ExtendedGosper}\left(T,n\right)$
 $\left[{-}\frac{{{2}}^{{2}{}{n}{-}{1}}}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right){}{n}}{,}\frac{\left({n}{-}{1}\right){}{{4}}^{{n}{+}{1}}}{{3}{}\left({n}{+}{2}\right)}\right]$ (4)

No solution found:

 > $T≔\left\{{n}^{2}{a}^{n},\frac{\left(2n+1\right){n}^{2}{a}^{n}}{{n}^{2}-3}\right\}$
 ${T}{≔}\left\{{{n}}^{{2}}{}{{a}}^{{n}}{,}\frac{\left({2}{}{n}{+}{1}\right){}{{n}}^{{2}}{}{{a}}^{{n}}}{{{n}}^{{2}}{-}{3}}\right\}$ (5)
 > $\mathrm{ExtendedGosper}\left(T,n\right)$

References

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

 See Also