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

Abstract

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 $E_{n,q,\xi }$ and $q$-Euler polynomials $E_{n,q,\xi }\left(x\right)$. We obtain the generating functions of $E_{n,q,\xi }$ and $E_{n,q,\xi }\left(x\right)$, 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.