The Fourier series of the log-Barnes function

TL;DR

This paper derives the Fourier series expansion of the log-Barnes function, extending classical results, and uses it to evaluate related integrals and sums involving the log-Gamma and log-harmonic functions.

Contribution

It provides the first Fourier series expansion of the log-Barnes function and applies it to evaluate new integrals and sums involving log-Gamma functions.

Findings

Fourier series expansion of the log-Barnes function derived

New evaluations of log-Gamma integrals with respect to log-G

Explicit evaluation of a log-harmonic sum.

Abstract

In this paper we determine the Fourier series expansion of the log-Barnes function. This is the analogue of the classical result of Kummer and Malmsten. Applying this expansion we get some integrals similar to the Espinosa-Moll log-Gamma integrals with respect to $\log G$. During the course of the paper some interesting log-Gamma integrals and a log-harmonic sum are evaluated.