 
Summary: CONTRIBUTIONS TO THE THEORY OF THE
BARNES FUNCTION
V. S. ADAMCHIK
Abstract. This paper presents a family of new integral represen
tations and asymptotic series of the multiple gamma function. The
numerical schemes for highprecision computation of the Barnes
gamma function and Glaisher's constant are also discussed
1. Introduction
In a sequence of papers published between 18991904, Barnes intro
duced and studied (see [9, 10, 11, 12]) a generalization of the classi
cal Euler gamma function, called the multiple gamma function # n (z).
The function # n (z) satisfies the following recurrencefunctional equa
tion [31, 32]:
(1)
# n+1 (z + 1) =
# n+1 (z)
# n (z)
, z # C, n # N,
# 1 (z) = #(z),
# n (1) = 1,
