Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
On values of the higher derivatives of the Barnes zeta function at non-positive integers
Shinpei SakaneMiho Aoki
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2022 Volume 45 Issue 1 Pages 65-95


Let x be a complex number which has a positive real part, and w1, ..., wN be positive rational numbers. We show that ws ζN(s, x|w1, ..., wN) can be expressed as a finite linear combination of the Hurwitz zeta functions over Q(x), where ζN(s,x|w1, ..., wN) is the Barnes zeta function and w is a positive rational number explicitly determined by w1, ..., wN. 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, ..., wN and x are positive real numbers.

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