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arXiv · 1008.0040

Integral and series representations of the digamma and polygamma functions

Mark W. Coffey
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Abstract

We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $\gamma_0(a)=-\psi(a)$. Several other results are obtained, including product representations for $\exp[\gamma_0(a)]$ and for the Gamma function $\Gamma(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $\psi(a)$ and the Euler constant $\gamma=-\psi(1)$.

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Mark W. Coffey. 2010-07-31. Integral and series representations of the digamma and polygamma functions. https://arxiv.org/abs/1008.0040

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