Integrating gamma products and quotients over a vertical line

2012-01-29T21:21:44Z

The Wolfram functions collection contains a small number of integrals of products and quotients of terms $\Gamma(a_i\pm t)$ over a vertical line, all of which can be evaluated in terms of only gamma functions.

Does somebody know references concerning formulae of this type? If they are obtained by the residue theorem, how?

The second one seems to be false. By putting $2b=a-c+1$, we would get from it, after using the doubling formula, $$\int_{\gamma-i\infty}^{\gamma+i\infty}\Gamma(a+t)\Gamma(c-t)dt=2\pi i\cdot2^{a+c}{\Gamma(a+c)},$$ whereas for $a,c\in\mathbb Z$ such that $a+c\ge1$, we have $$\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\Gamma(a+t)\Gamma(c-t)dt=\pi i\int\limits_{-\infty}^{\infty}\frac{\overbrace{(1-a-t)\cdots(c-1-t)}^{(a+c-1)\ {terms}}}{\cosh \pi t}dt=\pi i\int\limits_{-\infty}^{\infty}\frac{\Gamma(c-t)\; dt}{\Gamma(1-a-t)\cosh \pi t}$$ which may be written in umbral form as $(1-a-\frac E2)\cdots(c-1-\frac E2)$, i.e. as a linear combination of terms $\int\limits_{-\infty}^{\infty}\dfrac{t^{k}dt}{\cosh \pi t}=\dfrac{E_k}{2^k}$, where the $E_k$ are the (absolute) Euler numbers, thus it can definitely not be written in terms of $\Gamma(a+c)$ etc.

Some others of the Wolfram identities look like they could be generalized, e.g. the first and third one. Is it true e.g. that for $n\ge3$, $$\int_{\gamma-i\infty}^{\gamma+i\infty}\prod\limits_{j=1}^n\Bigl(\Gamma(a_j+t)\Gamma(b_j-t)\Bigr)dt=2\pi i\frac{\prod_j\prod_k\Gamma(a_j+b_k)}{\Gamma(\sum a_j+\sum b_j)^{n-1}}$$ where all products and sums run from $1$ to $n$? (Note that again, this does not hold for $n=1$.)

Integrating gamma products and quotients over a vertical line - MathOverflow

Integrating gamma products and quotients over a vertical line