Advances in Difference Equations
Volume 2010 (2010), Article ID 431436, 9 pages
doi:10.1155/2010/431436
Research Article

New Approach to q-Euler Numbers and Polynomials

Taekyun Kim,1 Lee-Chae Jang,2 Young-Hee Kim,1 and Seog-Hoon Rim3

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
2Department of Mathematics and Computer Science, Konkuk University, Chungju 380-701, South Korea
3Department of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea

Received 11 January 2010; Accepted 14 March 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Taekyun 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

We give a new construction of the q-extensions of Euler numbers and polynomials. We present new generating functions which are related to the q-Euler numbers and polynomials. We also consider the generalized q-Euler polynomials attached to Dirichlet's character χ and have the generating functions of them. We obtain distribution relations for the q-Euler polynomials and have some identities involving q-Euler numbers and polynomials. Finally, we derive the q-extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the q-Euler polynomials at negative integers.