On the Laurent series expansions of the Barnes double zeta function

TL;DR

This paper explores the Laurent series expansions of the Barnes double zeta-function at singular points, deriving explicit formulas and analyzing their asymptotic behavior to deepen understanding of its analytic structure.

Contribution

It provides explicit limit expressions for Laurent coefficients of the Barnes double zeta-function and compares their structure to classical zeta-functions.

Findings

Explicit formulas for Laurent coefficients at singularities

Representation of coefficients via finite double sums with logs

Simpler asymptotic structure compared to Hurwitz zeta-function

Abstract

The Laurent series expansions of zeta-functions play an important role in understanding their behavior near singularities, and their coefficients often encode significant arithmetic information. In the case of the Riemann and Hurwitz zeta-functions, these coefficients are given by the Euler-Stieltjes constants and their generalizations. In this paper, we investigate the Laurent series expansions of the Barnes double zeta-function $\zeta_2(s,\alpha;v,w)$ at the singular points $s=1$ and $s=2$. We derive explicit limit expressions for the Laurent coefficients, providing analogues of the Euler-Stieltjes constants in this setting. In particular, we obtain representations of the coefficients in terms of finite double sums together with logarithmic correction terms. Furthermore, we study the asymptotic behavior of the Laurent coefficients and show that, in contrast to the Hurwitz zeta-function, they exhibit a much simpler structure. These results provide a clearer understanding of the analytic structure of the Barnes double zeta-function and highlight notable differences from the classical theory.