 
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 byproducts of some of these relationships, a classical denite integral due
to Euler and other denite integrals are also considered together with closedform
evaluations of some series involving the Riemann and Hurwitz Zeta functions.
1. Introduction and Preliminaries
The multiple Gamma functions were dened 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 wellknown 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 ndimensional unit sphere S n (see [11], [17], [18], [30], [39], and [41]), and in
evaluations of specic classes of denite integrals and innite 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
twocentury 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 unied
