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MULTIPLE GAMMA AND RELATED FUNCTIONS Junesang Choi, H. M. Srivastava, and V. S. Adamchik

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


Source: Adamchik, Victor - School of Computer Science, Carnegie Mellon University


Collections: Computer Technologies and Information Sciences