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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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References:

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