On the q-Extension of Apostol-Euler Numbers and Polynomials

Abstract EN

Recently, Choi et al.

(2008) 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 En,q,ξ and q-Euler polynomials En,q,ξ(x).

We obtain the generating functions of En,q,ξ and En,q,ξ(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.