The Multiple Gamma-Functions and the Log-Gamma Integrals

In this paper, which is a companion paper to [W], starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of log . This enables us to locate the genesis of two new functions and considered by Srivastava and Choi. We consider the closely related function A(a) and the Hurwitz zeta function, which render the task easier than working with the functions themselves. We shall also give a direct proof of Theorem 4.1, which is a consequence of [CKK, Corollary 1.1], though. 1. Introduction If is analytic in a domain containing the circle and has no zero on the circle, then the Gauss mean value theorem is true. In [1, page 207] the case is considered where has a zero on the circle, and (1.1) turns out that the Euler integral which is essential in proving a generalization of Jensen's formula [1, pages 207-208]. Let denote the Catalan constant defined by the absolutely convergent series where is the nonprincipal Dirichlet character mod 4. As a next step from (1.2) the relation holds true. In this connection, in [2] we obtained some results on viewing it as an intrinsic value to the Barnes -function. The Barnes -function (which is in the class of multiple gamma functions) is defined as the solution to the difference equation (cf. (2.3)) with the initial condition and the asymptotic formula to be satisfied , where indicates the Euler gamma function (cf., e.g., [3]). Invoking the reciprocity relation for the gamma function it is natural to consider the integrals of or of multiple gamma functions (cf., e.g., [4, 5]). Barnes’ theorem [6, page 283] reads valid for nonintegral values of . In this paper, motivated by the above, we proceed in another direction to developing some generalizations of the above integrals considered by Srivastava and Choi [7]. For -analogues of the results, compare the recent book of the same authors [8]. Our main result is Theorem 2.1 which gives a closed form for and locates its genesis. A slight modification of Theorem 2.1 gives the counterpart of Barnes’ formula (1.9) which reads. Corollary 1.1. Except for integral values of , one has Srivastava and Choi introduced two functions and by (2.9) and (2.9) with formal replacement of by , respectively. They state , which is rather ambiguous as to how we interpret the meaning because (2.9) is defined for [7, page 347, l.11]. They use this function to express the integral , without giving proof. This being the case, it may be of interest to locate the integral of [7, (13), page 349], thereby [7, page 347]. For this purpose