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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