Let $T\left(m\,k\right)$ be a hypergeometric term of m and k. Let $V\left(n\,k\right)=\sum _{m=-n+b}^{n+d}T\left(m\,k\right)$ where b and d are integers. The ExtendedZeilberger(V, n, k, En) command, an extension to Zeilberger's algorithm, constructs the minimal Z-pair for $V\left(n\,k\right)$ provided that it exists.

It can be shown that a Z-pair for $V\left(n\,k\right)$ exists if and only if a Z-pair for the hypergeometric term $T\left(n\,k\right)$ exists.

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

$T≔\frac{{\left(-1\right)}^k\mathrm{binomial}\left(m\,k\right)\mathrm{binomial}\left(2k\,k\right)\cdot 1}{2^{2k}}$

$T≔\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{m}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{2^{2k}}$

$V≔\mathrm{Sum}\left(T\,m=0..n\right)$

$V≔\sum _{m=0}^n\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{m}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{2^{2k}}$

$\mathrm{ExtendedZeilberger}\left(V\,n\,k\,\mathrm{En}\right)$

$\left[\left(2n+4\right){\mathrm{En}}^2+\left(-4n-7\right)\mathrm{En}+2n+3\,\frac{2k^2{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{n+1}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{\left(-n+k-2\right)2^{2k}}\right]$

$\mathrm{\_EnvDoubleSum}≔\mathrm{true}$

$\mathrm{\_EnvDoubleSum}≔\mathrm{true}$

$\mathrm{Sum}\left(V\,k=0..n\right)=\mathrm{DefiniteSum}\left(V\,n\,k\,0..n\right)$

$\sum _{k=0}^n\sum _{m=0}^n\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{m}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{2^{2k}}=\frac{\left(2n+1\right)\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}{4^n}$

$\mathrm{sum}\left(\mathrm{sum}\left(T\,m=0..n\right)\,k=0..n\right)$

$4^{-n}\left(n+1\right)\left(\genfrac{}{}{0ex}{}{2n+1}{n}\right)+\left(\sum _{k=0}^n\frac{2\sin \left(\mathrm{\pi }k\right){\left(\mathrm{-1}\right)}^k{4}^{-k}\left(\genfrac{}{}{0ex}{}{2k-1}{k-1}\right)}{\mathrm{\pi }\left(k+1\right)}\right)$

Le, H.Q. "Computing the Minimal Telescoper for Sums of Hypergeometric Terms." SIGSAM Bulletin: Communications on Computer Algebra. Vol. 35 No. 3. (2001): 2-10.