Some analytic continuations of the Barnes zeta function in two and higher dimensions
type: Item
instance of: scholarly article
title: Some analytic continuations of the Barnes zeta function in two and higher dimensions
zbMATH Open document ID: 1148.11047
author: Emilio Elizalde
published in: Applied Mathematics and Computation
publication date: 2007-06-13T00:00:00Z
review text: The Barnes zeta-function in two dimensions \(\zeta_B(s; a| \overrightarrow{r})\) is defined by \[ \zeta_B(s;a| \overrightarrow{r})=\sum_{n_1,n_2=0}^{\infty}(a+r_1n_1+r_2n_2)^{-s}, \quad \Re s>2, \quad r_1,r_2>0, \] It is assumed that, for any value \(n_1, n_2\), the expression in brackets never vanish. The function \(\zeta_B(s;a| \overrightarrow{r})\) can be written as an infinite sum of Hurwitz zeta-functions \(\zeta_H\) or by asymptotic expansion via Hurwitz zeta-function as follows \[ \zeta_B(s;a| \overrightarrow{r}) =r_2^{-s} \sum_{n=0}^\infty \zeta_H \bigg(s, \frac{a}{r_2} +\frac{r_1}{r_2}n\bigg) \] \[ \sim \frac{r_1^{1-s}r_2^{-1}}{s-1} \zeta_H\bigg(s-1, \frac{a}{r_1}\bigg) +\frac{r_1^{-s}}{2} \zeta_H\bigg(s,\frac{a}{r_1}\bigg) +\frac{sB_2}{2} r_1^{-1-s}r_2 \zeta_H\bigg(s+1,\frac{a}{r_1}\bigg)+O(s), \] where \(B_\nu\) is a \(\nu\)th Bernoulli number. In the present paper, the author constructs the formulas for the analytic continuation of the Barnes zeta-function, and some affine extensions thereof in two and higher dimensions. The expressions are used to deal with determinants of multidimensional harmonic oscillators. For example, the one of obtained results is the analytic continuation in the parameter \(r_1\) of the function \(\zeta_B(s;a| \overrightarrow{r})\). It has the form \[ \sum_{n_1,n_2=0}^{\infty} (a+r_1n_1+r_2n_2)^{-s} =\frac{1}{\Gamma(s)} \int_{\mathcal R}d \tau \tau^{s-1} \frac{e^{-a\tau}}{(1-e^{-r_1 \tau})(1-e^{-r_2\tau})} +2 \pi \Delta_1, \] \(\Re a\), \( \Re r_2>0\), \( \Re r_1<0\), where \(\mathcal R\) is a straight-line in the second quadrant of the \(r_1\)-complex plane, starting at the origin, and \[ \Delta_1 =\sum_{k=1}^{\infty} \bigg(\frac{2 \pi k}{| r_1| }\bigg)^{s-1} e^{-2 \pi k a /| r_1| }(1-e^{-2 \pi k r_2/| r_1| })^{-1}. \]
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5163915
zbMATH Keywords: Barnes zeta function
zbMATH Keywords: Hurwitz (or generalized) zeta
zbMATH Keywords: determinant
zbMATH Keywords: multiplicative anomaly (or defect)
zbMATH Keywords: quantum harmonic oscillators
reviewed by: Roma Kačinskaitė
MaRDI profile type: Publication
full work available at URL: j.amc.2006.08.110
OpenAlex ID: W2155634278
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DOI: 10.1016/J.AMC.2006.08.110
description: scientific article; zbMATH DE number 5163915