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Gamma Integral

$\large \int_{-\infty}^{\infty} \dfrac{ \mathrm{d} x}{\Gamma(\alpha +x) \Gamma (\beta - x)} = \dfrac{2^{\alpha + \beta -2}}{\Gamma(\alpha + \beta -1)} \quad ; \quad \Re (\alpha + \beta) > 1$

Prove the equation above without using the method of residues.

Notation: $$\Gamma(\cdot)$$ denotes the gamma function.

This is a part of the set Formidable Series and Integrals.

Note by Ishan Singh
9 months, 2 weeks ago

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I wonder if it is possible to prove it without contour, well here is its solution, just put $$n=0$$ at the end. @Ishan Singh · 9 months ago