A note on \(q\)-analogues of Dirichlet series
type: Item
instance of: scholarly article
title: A note on \(q\)-analogues of Dirichlet series
zbMATH Open document ID: 0931.11031
author: Hirofumi Tsumura
published in: Proceedings of the Japan Academy. Series A
publication date: 2000-02-22T00:00:00Z
review text: The paper is devoted to the function \[ Z_q(s)=\sum\limits_{n=1}^\infty \frac{q^n}{[n]^s}, \] where \(q\in \mathbb C\), \(|q|<1\), \([n]=\frac{1-q^n}{1-q}\). The values of \(Z_q(s)\), when \(s\) is a non-positive integer, coincide with the modified \(q\)-Bernoulli numbers. The author studies the case when \(s\) is a natural number. The series representation is obtained, which gives in the limit \(q\to 1\) the Euler formulas for \(\zeta (2k)\) and the representation for \(\zeta (2k+1)\) found by \textit{D. Cvijović} and \textit{J. Klinowski} [Proc. Am. Math. Soc. 125, 1263-1271 (1997; Zbl 0863.11055)]. The connection between \(Z_q(s)\) and Jackson's \(q\)-\(\Gamma\)-function is also found.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 33D05
zbMATH DE Number: 1335027
zbMATH Keywords: Dirichlet series
zbMATH Keywords: \(q\)-zeta function
zbMATH Keywords: \(q\)-Bernoulli numbers
zbMATH Keywords: Jackson's \(q\)-\(\Gamma\)-function
reviewed by: Anatoly N. Kochubei
MaRDI profile type: Publication
full work available at URL: pjaa.75.23
OpenAlex ID: W2077951350
cites work: Theq-Gamma andq-Beta Functions†
cites work: Euler and the Zeta Function
cites work: New rapidly convergent series representations for $\zeta (2n+1)$
cites work: Q3377568
cites work: A note on \(q\)-analogues of the Dirichlet series and \(q\)-Bernoulli numbers
cites work: On evaluation of the Dirichlet series at positive integers by \(q\)- calculation
DOI: 10.3792/PJAA.75.23
description: scientific article; zbMATH DE number 1335027