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Multiple Gamma Function and Its Application to Computation
of Series and Products
V. S. Adamchik
Department of Computer Science, Carnegie Mellon University, Pittsburgh, USA
Abstract. The multiple gamma function n, defined by a recurrencefunctional
equation as a generalization of the Euler gamma function, was originally introduced
by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of
Conrey, Katz and Sarnak, interest in the Barnes function has been revived. This
paper discusses some theoretical aspects of the n function and their applications
to summation of series and infinite products.
Keywords: Barnes function, Gamma function, Riemann zeta function, Hurwitz
zeta function, Stirling numbers, Stieltjes constants, Catalan's constant, harmonic
numbers, Glaisher's constant
AMS: 33E, 11M, 1Y
1. Introduction
The Hurwitz zeta function, one of the fundamental transcendental
functions, is traditionally defined (see [11]) by the series
