Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions

TL;DR

This paper derives Fourier series representations for the logarithms of the Euler gamma function and Barnes multiple gamma functions, extending classical results and exploring their applications.

Contribution

It introduces a Fourier series for the Barnes double gamma function, extending Kummer's classical series for the log gamma function, with potential for higher order functions.

Findings

Derived Fourier series for Barnes double gamma function

Extended classical Fourier series to multiple gamma functions

Explored applications of these Fourier series

Abstract

Kummer's Fourier series for the log gamma function is well known, having been discovered in 1847. In this paper we develop a corresponding Fourier series for the logarithm of the Barnes double gamma function (and the method may be easily extended to higher order multiple gamma functions). Some applications of these Fourier series are explored.