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 w_1,...,w_N be positive rational numbers. We write w^s \zeta_N (s, x | w_1,...,w_N) as a finite linear combination of the Hurwitz zeta function over Q(x), where \zeta_N (s,x |w_1,...,w_N) is the Barnes zeta function and w is a positive rational number explicitly determined by w_1,...,w_N. Furthermore, in the case that x is a positive rational number, we give an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function involving the generalized Stieltjes constants and the values at positive integers of the Riemann zeta function. At the end of the paper, we give some tables of numerical examples.