A note on \(q\)-analogues of Dirichlet series (Q1302092)

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.