Abstract and Applied Analysis
Volume 2008 (2008), Article ID 296159, 10 pages
doi:10.1155/2008/296159

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

Young-Hee Kim1 , Wonjoo Kim2 and Lee-Chae Jang3

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
 2Natural Science Institute, KonKuk University, Chungju 380-701, South Korea
 3Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea

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 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.