On values of the higher derivatives of the Barnes zeta function at non-positive integers

Abstract

Let x be a complex number which has a positive real part, and w₁, ..., w_N be positive rational numbers. We show that w^s ζ_N(s, x|w₁, ..., w_N) can be expressed as a finite linear combination of the Hurwitz zeta functions over Q(x), where ζ_N(s,x|w₁, ..., w_N) is the Barnes zeta function and w is a positive rational number explicitly determined by w₁, ..., w_N. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function in the case that x is a positive rational number, involving the generalized Stieltjes constants and the values at positive integers of the Riemann zeta function. Our formulae also makes it possible to calculate an approximation in the case that w₁, ..., w_N and x are positive real numbers.