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

JOURNAL OF NUMBER THEORY

卷:194

Article

Tangedal, Brett A.¹ 

[1] Univ N Carolina, Dept Math & Stat, Greensboro, NC 27412 USA

关键词: Barnes double zeta-function;    Double gamma function;    Double sine function;    Double Fourier series;   

DOI  :  10.1016/j.jnt.2018.07.011

来源: Elsevier

【 摘 要 】

A classical formula due to Kummer and Malmsten 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 S-2(x, (w(1), w(2))), and of the two derivations given for this expansion, the first requires that w(1) and w(2) are incommensurable, and the second removes this restriction and holds in general. This second derivation relies upon an integral representation of the function log S-2(x, (w(1), w(2))) 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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