The function sumtohyper converts the sum

to be taken over all integers k, involving exponentials, factorials, binomial coefficients, and Pochhammer symbols into hypergeometric notation. The procedure Sumtohyper is the corresponding inert form which gives the result in terms of an unevaluated %hypergeom expression.

The command with(sumtools,sumtohyper) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{sumtools}\right)\:$

$\mathrm{sumtohyper}\left(\mathrm{binomial}\left(n\,k\right)\,k\right)$

$2^n$

$\mathrm{Sumtohyper}\left(\mathrm{binomial}\left(n\,k\right)\,k\right)$

$\mathrm{hypergeom}\left(\left[-n\right]\,\left[\right]\,\mathrm{-1}\right)$

$\mathrm{sumtohyper}\left({\mathrm{binomial}\left(n\,k\right)}^2\,k\right)$

$\mathrm{hypergeom}\left(\left[-n\,-n\right]\,\left[1\right]\,1\right)$

$\mathrm{sumtohyper}\left({\mathrm{binomial}\left(n\,k\right)}^2-{\mathrm{binomial}\left(n-1\,k\right)}^2\,k\right)$

$\left(2n-1\right)\mathrm{hypergeom}\left(\left[-n+1\,-n+1\,-2n+2\right]\,\left[2\,-2n+1\right]\,1\right)$