Abstract and Applied Analysis
Volume 2008 (2008), Article ID 296159, 10 pages
doi:10.1155/2008/296159
Research Article
On the q-Extension of Apostol-Euler Numbers and Polynomials
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
Received 4 October 2008; Accepted 21 November 2008
Academic Editor: Lance Littlejohn
Copyright © 2008 Young-Hee Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.