Barnes Integral - Barnes Lemmas

Barnes Lemmas

The first Barnes lemma (Barnes 1908) states

$\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds =\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)}.$

This is an analogue of Gauss's ₂F₁ summation formula, and also an extension of Euler's beta integral. The integral in it is sometimes called Barnes's beta integral.

The second Barnes lemma (Barnes 1910) states

$=\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1-d+a)\Gamma(1-d+b)\Gamma(1-d+c)}{\Gamma(e-a)\Gamma(e-b)\Gamma(e-c)}$

where e = a + b + c − d + 1. This is an analogue of Saalschütz's summation formula.

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