## 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

Zbl 1146.33001
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