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Summary: MULTIPLE GAMMA AND RELATED FUNCTIONS
Junesang Choi, H. M. Srivastava, and V. S. Adamchik
Abstract. The authors give several new (and potentially useful) relationships be-
tween the multiple Gamma functions and other mathematical functions and con-
stants. As by-products of some of these relationships, a classical de nite integral due
to Euler and other de nite integrals are also considered together with closed-form
evaluations of some series involving the Riemann and Hurwitz Zeta functions.
1. Introduction and Preliminaries
The multiple Gamma functions were de ned and studied by Barnes (cf. [7] and
[8]) and others in about 1900. 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 [42, p. 264] and recorded also by
Gradshteyn and Ryzhik [24, p. 661, Entry 6.441(4); p. 937, Entry 8.333]. Recently,
these functions were revived in the study of the determinants of the Laplacians on
the n-dimensional unit sphere S n (see [11], [17], [18], [30], [39], and [41]), and in
evaluations of speci c classes of de nite integrals and in nite series involving, for
example, the Riemann and Hurwitz Zeta functions (see [4], [15], [16], [18], and
[19]). The subject of some of these developments can be traced back to an over
two-century old theorem of Christian Goldbach (1690{1764), as noted in the work
of Srivastava [32, p. 1] who investigated this subject in a systematic and uni ed
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