## A note on $$q$$-analogues of Dirichlet series.(English)Zbl 0931.11031

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 D. Cvijović and 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.

### MSC:

 11M41 Other Dirichlet series and zeta functions 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals

Zbl 0863.11055
Full Text:

### References:

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