Integral and series representations of the digamma and polygamma\n functions

2010/07/30 by Mark W. Coffey, Coffey, Mark W.
Mathematics · #Advanced Mathematical Identities #Mathematical functions and polynomials #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.1008.0040

Abstract

We obtain a variety of series and integral representations of the digamma\nfunction \ψ(a). These in turn provide representations of the evaluations\n\ψ(p/q) at rational argument and for the polygamma function \ψ^(j).\nThe approach is through a limit definition of the zeroth Stieltjes constant\n\γ₀(a)=-\ψ(a). Several other results are obtained, including product\nrepresentations for \exp[\γ₀(a)] and for the Gamma function \Γ(a).\nIn addition, we present series representations in terms of trigonometric\nintegrals Ci and Si for \ψ(a) and the Euler constant \γ=-\ψ(1).\n

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