This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following:
$$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi [I_0(1)+L_0(1)] $$ Then i came up with a solution of my own that is:
$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = \sum_{n=0}^{\infty}( \frac{1}{n!} \sum_{a=0}^{\infty} [\frac{\Gamma(\frac{1-n}{2}+a)}{ a!(2a+1) \Gamma(\frac{1-n}{2})}]) $
Therefore my questions isn't really to prove that the statements are equal, but just to prove that:
$$ \sum_{n=0}^{\infty}( \frac{1}{n!} \sum_{a=0}^{\infty} \frac{\Gamma(\frac{1-n}{2}+a)}{ a!(2a+1) \Gamma(\frac{1-n}{2})}) = 2\pi [ I_0(1) + L_0(1)] $$
Without converting the series back into a hyper-geometric function if possible, or perhaps shed some light on the matter because i couldn't simplify the series anymore with the identities i know about gamma functions to explicitly connect the two (in my eyes). Also i have used Mathmatica to check that the expression is true.