Multiple Gamma Functions and Their Applications

Abstract

The double Gamma function Γ 2 and the multiple Gamma functions Γ n were defined and studied systematically by Barnes in about 1900. Before their investigation by Barnes, these functions had been introduced in a different form by, for example, Holder, Alexeiewsky, and Kinkelin. Although these functions did not appear in the tables of the most well-known special functions, yet the double Gamma function was cited in the exercises by Whittaker and Watson’s book and recorded also by Gradshteyn and Ryzhik’s book. In about the middle of the 1980s, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere S n . Here, in this expository paper, from the middle of the 1980s until today, we aim at giving an eclectic review for recent developments and applications of the simple and multiple Gamma functions.