# calculus and analysis – Evaluation of a double summation invovlving hypergeometric and exponential functions

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.