Does anyone know how to evaluate the following integral $$ \int_{0}^{\infty} \frac{e^{-qs}\alpha^{s}}{\Gamma(s)\Gamma(s)}\text{d}s $$ where $q,\,\alpha > 0$? I've done some digging in usual resources on tables of integrals, but nothing has jumped out at me.
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Does anyone know how to evaluate the following integral ?
No. No one knows how to evaluate that integral. The proof is by reduction to the absurd: If anyone would have known how to express that integral in closed form, then the much simpler case with $q=0,~\alpha=1,$ and only one $\Gamma(s)$ in the denominator would also have been known to possess a closed form. Unfortunately, no such form is known for the famous Fransen-Robinson constant. QED.
Lucian
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Thanks for your answer. How about a representation in terms of other special functions, a series, or an infinite product? – Billy Pilgrim May 05 '15 at 08:37
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@BillyPilgrim: As far as special functions are concerned, the same observation holds. As for the latter two, I have no idea. I'd start by google-ing series representation Fransen-Robinson or product representation Fransen-Robinson, to see if by chance any might exist, though I wouldn't hold my breath. Then, if I'd find any, I'd try and draw some inspiration from them, and adapt them to the current problem. – Lucian May 05 '15 at 11:47