I am trying to compute the following double summation over the indices, $m$ and $n$, which involves the hypergeometric function, ${}_2 F_1$, an exponential function and, factorials as a part of a bigger calculation.
$G(r,\alpha,\beta,\gamma,n',p)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{r! c(n)^2 \beta ^{2 m} e^{-\alpha m-\gamma n-\beta ^2} \, _2F_1(-n,-m-n+r;-n+r+1;-1){}^2}{n! ((r-n)!)^2 (m+n-r)!}$
where, $c(n)=\binom{n'}{n}p^n (1-p)^{n'-n}$ is the binomial distribution.
Any guidance on how to go proceed with this summation (either numerically or analytically) would be really appreciated.