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.
2026-04-06 09:48:35.1775468915
Laplace transform involving the gamma function.
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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.