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 sumtohyper
 express an indefinite sum as a Hypergeometric function
 Sumtohyper
 express an indefinite sum as a Hypergeometric function

 Calling Sequence sumtohyper(f, k) Sumtohyper(f, k)

Parameters

 f - expression k - name, summation variable

Description

 • The function sumtohyper converts the sum

$\sum _{k}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}f\left(k\right)$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{sumtools}\right):$
 > $\mathrm{sumtohyper}\left(\mathrm{binomial}\left(n,k\right),k\right)$
 ${{2}}^{{n}}$ (1)
 > $\mathrm{Sumtohyper}\left(\mathrm{binomial}\left(n,k\right),k\right)$
 ${\mathrm{Hypergeom}}{}\left(\left[{-}{n}\right]{,}\left[{}\right]{,}{-}{1}\right)$ (2)
 > $\mathrm{sumtohyper}\left({\mathrm{binomial}\left(n,k\right)}^{2},k\right)$
 $\frac{{\mathrm{Γ}}{}\left({1}{+}{2}{}{n}\right)}{{{\mathrm{Γ}}{}\left({n}{+}{1}\right)}^{{2}}}$ (3)
 > $\mathrm{sumtohyper}\left({\mathrm{binomial}\left(n,k\right)}^{2}-{\mathrm{binomial}\left(n-1,k\right)}^{2},k\right)$
 $\frac{\left({3}{}{n}{-}{2}\right){}{\mathrm{Γ}}{}\left({2}{}{n}{-}{1}\right)}{{{\mathrm{Γ}}{}\left({n}\right)}^{{2}}{}{n}}$ (4)