# 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.

• Please specify: n1, p, r, beta. – Daniel Huber Feb 20 at 14:44
• Please take n1=40, p=0.5, and beta=5. r is a variable here. – Jayanth Jayakumar Feb 20 at 21:58
• Take the sum for 0..1, 0..2, 0..3 ... You will see that the expression gets longer and longer. The main reason is that Hypergeometric2F1 with the variable r `can not be expanded. – Daniel Huber Feb 21 at 8:39