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 $\mathbf Q(x)$, 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.
Funding Statement
This work was supported by JSPS KAKENHI Grant Number JP21K03181.
Acknowledgment
The authors would like to thank Professor Shin-ya Koyama and Professor Nobushige Kurokawa for useful advice. They also express their gratitude to the referee for suggesting the addition of the generalizetion of Kummer's formula in §3.
Citation
Shinpei Sakane. Miho Aoki. "On values of the higher derivatives of the Barnes zeta function at non-positive integers." Kodai Math. J. 45 (1) 65 - 95, March 2022. https://doi.org/10.2996/kmj/kmj45105
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