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The Hypergeometric(f, k) command computes a closed form of the indefinite sum of $f$ with respect to $k$.

The following algorithms are used to handle indefinite sums of hypergeometric terms (see the References section):

Gosper's algorithm,

Koepf's extension to Gosper's algorithm, and

the algorithm to compute additive decompositions of hypergeometric terms developed by Abramov and Petkovsek.

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair $s,\left[p\,q\right]$, where $s$ is the closed form of the indefinite sum of $f$ w.r.t. $k$, as above, and $p,q$ are lists of points where $f$ does not exist or the computed sum $s$ is undefined or improper, respectively (see SumTools[IndefiniteSum][Indefinite] for more detailed help).

The command returns $\mathrm{FAIL}$ if it is not able to compute a closed form.

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

$f≔\frac{\left(4n-1\right){\mathrm{binomial}\left(2n\,n\right)}^2}{{\left(2n-1\right)}^2{4}^{2n}}$

$f≔\frac{\left(4n-1\right){\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^2}{{\left(2n-1\right)}^2{4}^{2n}}$

$\mathrm{Hypergeometric}\left(f\,n\right)$

$-\frac{4{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^2{n}^2}{{\left(2n-1\right)}^2{4}^{2n}}$

$f≔\frac{\mathrm{binomial}\left(2n-3\,n\right)}{4^n}$

$f≔\frac{\left(\genfrac{}{}{0ex}{}{2n-3}{n}\right)}{4^n}$

$s,\mathrm{fp}≔\mathrm{Hypergeometric}\left(f\,n\,'\mathrm{failpoints}'\right)$

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

$\mathrm{eval}\left(s\,n=2\right)$

Error, numeric exception: division by zero

$f≔{\mathrm{binomial}\left(m\,j\right)}^2{\mathrm{binomial}\left(m\,k\right)}^2\mathrm{binomial}\left(2m+\frac{n}{3}-j-k\,2m\right)$

$f≔{\left(\genfrac{}{}{0ex}{}{m}{j}\right)}^2{\left(\genfrac{}{}{0ex}{}{m}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{2m+\frac{n}{3}-j-k}{2m}\right)$

$\mathrm{Hypergeometric}\left(f\,n\right)$

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

$f≔\frac{\left(n^2-2n-1\right)2^n}{\left(n+1\right)n^2\left(n+3\right)!}$

$f≔\frac{\left(n^2-2n-1\right)2^n}{\left(n+1\right)n^2\left(n+3\right)!}$

$\mathrm{Hypergeometric}\left(f\,n\right)$

$\frac{\left(n+3\right)\left(\prod _{\mathrm{\_i}=1}^{n-1}\frac{2}{\mathrm{\_i}+4}\right)}{12n}+\left(\sum _n\frac{\left(n^2+2n-1\right)\left(\prod _{\mathrm{\_i}=1}^{n-1}\frac{2}{\mathrm{\_i}+4}\right)}{12n^2}\right)$

$\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\left[\mathrm{Gosper}\right]\left(f\,n\right)$

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

Abramov, S.A., and Petkovsek, M. "Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms." Journal of Symbolic Computing. Vol. 33. (2002): 521-543.

Gosper, R.W., Jr. "Decision Procedure for Indefinite Hypergeometric Summation." Proceedings of the National Academy of Sciences USA. Vol. 75. (1978): 40-42.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

Abramov, S.A. and Petkovsek, M. "Gosper's Algorithm, Accurate Summation, and the discrete Newton-Leibniz formula." Proceedings ISSAC'05. (2005): 5-12.