A two-variable generalization of the Kummer–Malmstén formula for the logarithm of the double gamma and double sine functions

Abstract

A classical formula due to Kummer and Malmstén expresses the logarithm of the gamma function as a Fourier series. We prove a generalization of this formula which expresses the logarithm of the double gamma function as a double Fourier series. We also obtain a double Fourier series expansion for the logarithm of the double sine function S2(x,(ω1,ω2)), and of the two derivations given for this expansion, the first requires that ω1 and ω2 are incommensurable, and the second removes this restriction and holds in general. This second derivation relies upon an integral representation of the function log⁡S2(x,(ω1,ω2)) which appears to be new. A generalization of Raabe's formula for the gamma function to a two-variable version for the double gamma function is obtained, and double Fourier series expansions of the double Bernoulli polynomials are derived as well.