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.

Here’s the code.

`Sum[((E^(-0.6931471805599453` m - 0.6931471805599453` n - 1.0000000000000002` \[Beta]^2) \[Beta]^(2 m) c[n,n1,p]^2 r! Hypergeometric2F1[-n, -m - n + r, 1 - n + r, -1]^2)/(n! (m + n - r)! ((-n + r)!)^2)), {m, 0, \[Infinity]}, {n, 0, \[Infinity]}] `

where `c[n_, n1_, p_] := n1!/(n! (n1 - n)!) p^n (1 - p)^(n1 - n)`

is the binomial distribution.

Any guidance on how to go proceed with this summation (either numerically or analytically) would be really appreciated.