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

TL;DR

This paper derives explicit formulas for higher derivatives of the Barnes zeta function at non-positive integers, expressing them in terms of Hurwitz zeta functions, generalized Stieltjes constants, and Riemann zeta values, with applications to approximation.

Contribution

It provides new explicit formulas for derivatives of Barnes zeta functions at non-positive integers, generalizing classical gamma function formulas and enabling numerical approximations.

Findings

Explicit formulas involving Hurwitz zeta and Stieltjes constants.

Generalizations of Kummer's and Koyama-Kurokawa's formulas.

Method for numerical approximation when parameters are positive reals.

Abstract

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s \zeta_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $\zeta_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. 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 $w_1, \ldots, w_N$ and $x$ are positive real numbers.