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

2009/03/25 by Donal F. Connon, Connon, Donal F.
Mathematics · #11M06 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.0903.4323

openalex publication_date 2009/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I
• Tables of Integrals, Series, and Products

Related

• The Fourier series of the log-Barnes function
• Contributions to the Theory of the Barnes Function
• Multiple Gamma Function and Its Application to Computation of Series
• On the definition of higher Gamma functions
• Asymptotic Expansions for the multiple gamma functions of Barnes-Milnor type