Some analytic continuations of the Barnes zeta function in two and higher dimensions
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 \[ \Delta_1 =\sum_{k=1}^{\infty} \bigg(\frac{2 \pi k}{| r_1| }\bigg)^{s-1} e^{-2 \pi k a /| r_1| }(1-e^{-2 \pi k r_2/| r_1| })^{-1}. \]
scientific article; zbMATH DE number 5163915
scientific article