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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

Citations:

Zbl 0863.11055
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References:

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[3] D. Cvijović and J. Klinowski: New rapidly convergent series representations for \(\zeta(2n+1)\). Proc. Amer. Math. Soc., 125 , 1263-1271 (1997). · Zbl 0863.11055
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