On Barnes' Multiple Zeta and Gamma Functions

Abstract

We show how various known results concerning the Barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions. The pertinent functions involve Laplace transforms, and their asymptotics is obtained by exploiting this. We also demonstrate how Barnes' multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations. Both of these new approaches to the Barnes functions give rise to novel integral representations.

Supersymmetric Casimir energy and the anomaly polynomial

“…The appearance of the Bernoulli polynomials in the exponential is of fundamental importance here. The multiple sine function is also defined as a regularized infinite product [86,87]: , where x k = e 2πiz/ω k , q k = (e 2πiω 1 /ω k , · · · , e 2πiω k−1 /ω k , e 2πiω k+1 /ω k , · · · , e 2πiωr/ω k ).…”

Essays on the theory of elliptic hypergeometric functions

“…As a starting point in his work serves a function generalizing the Hurwitz zeta-function [1] ζ(s, u) = ∞ n=0 1 (u + n) s , Re(s) > 1, which reduces to the Riemann zeta-function for u = 1. Because of the grown interest the Barnes theory was sufficiently widely discussed in the recent literature, for instance, in [19,26,27,28,37].…”

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

“…The following theorem due to Chiang and Ruijsenaars [21] contains as special cases some of the Ruijsenaars' earlier results on minimal solutions [116,117]. branch of the logarithm) satisfies the conditions of theorem 6.4 can be constructed by using Hadamard products [21].…”

“…If the q-difference equation (117) There are a number of effective methods from different areas of complex analysis which can be applied to study the value distribution of meromorphic solutions of the Schröder equation (117), and q-difference equations in general. For instance, Eremenko and Sodin [28] used methods from complex dynamics to show that the Valiron and Nevanlinna deficient values of meromorphic solutions of the autonomous Schröder equation (117) always coincide with the exceptional values of R(z). On the other hand, Ishizaki and Yanagihara [66] showed that this is not true in general for the non-autonomous Schröder equation.…”