On theq-Extension of Apostol-Euler Numbers and Polynomials

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,E-q,E-xi and q-Euler polynomials E-n,E-q,E-xi(x). We obtain the generating functions of E-n,E-q,E-xi and E-n,E-q,E-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. Copyright (C) 2008 Young-Hee Kim et al.