Barnes integral
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In mathematics, a Barnes integral or Mellin-Barnes integral is a contour integral involving a product of gamma functions. It is named for Ernest William Barnes.
Hypergeometric series
The hypergeometric function is given as a Barnes integral (Barnes 1908) by
Barnes lemmas
The first Barnes lemma (Barnes 1908) states
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
where e = a + b + c − d + 1. This is an analogue of Saalschutz's summation formula.
References
- Barnes, E. W. (1908) A New Development in the Theory of the Hypergeometric Functions Proc. London Math. Soc. 6, 141-177.