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**On the \(q\)-extension of Apostol-Euler numbers and polynomials.**
*(English)*
Zbl 1247.11028

Summary: Recently, J. Choi et al. [Appl. Math. Comput. 199, No. 2, 723–737 (2008; Zbl 1146.33001)] have studied the \(q\)-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order \(n\) and multiple Hurwitz zeta function. In this paper, we define Apostol’s type \(q\)-Euler numbers \(E_{n,q,\xi }\) and \(q\)-Euler polynomials \(E_{n,q,\xi }(x)\). We obtain the generating functions of \(E_{n,q,\xi }\) and \(E_{n,q,\xi }(x)\), respectively. We also have the distribution relation for Apostol’s type \(q\)-Euler polynomials. Finally, we obtain \(q\)-zeta function associated with Apostol’s type \(q\)-Euler numbers and Hurwitz’s type \(q\)-zeta function associated with Apostol’s type \(q\)-Euler polynomials for negative integers.

### MSC:

11B68 | Bernoulli and Euler numbers and polynomials |

11B65 | Binomial coefficients; factorials; \(q\)-identities |

11M35 | Hurwitz and Lerch zeta functions |

11S40 | Zeta functions and \(L\)-functions |

### Citations:

Zbl 1146.33001
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\textit{Y.-H. Kim} et al., Abstr. Appl. Anal. 2008, Article ID 296159, 10 p. (2008; Zbl 1247.11028)

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### References:

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