Integral and series representations of the digamma and polygamma functions

TL;DR

This paper derives various integral and series representations for the digamma and polygamma functions, including evaluations at rational points, using limit definitions and special constants, enriching the analytical tools for these special functions.

Contribution

It introduces new series and integral representations of the digamma and polygamma functions, including evaluations at rational arguments and product formulas, through a novel approach involving Stieltjes constants.

Findings

New integral and series representations for $ ext{psi}(a)$ and $ ext{psi}^{(j)}(a)$.

Product formulas for $ ext{exp}[ ext{gamma}_0(a)]$ and $ ext{Gamma}(a)$.

Series involving trigonometric integrals for $ ext{psi}(a)$ and $ ext{gamma}$.

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 $\gamma=-\psi(1)$.