Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 875098, 11 pages doi:10.1155/2010/875098 Research Article On the Twisted q-Analogs of the Generalized Euler Numbers and Polynomials of Higher Order Lee Chae Jang, 1 Byungje Lee, 2 and Taekyun Kim 3 1 Department of Mathematics and Computer Science, KonKuk University, Chungju 138-701, Republic of Korea 2 Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea 3 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Correspondence should be addressed to Lee Chae Jang, leechae.jang@kku.ac.kr Received 12 April 2010; Accepted 28 June 2010 Academic Editor: Istvan Gyori Copyright q 2010 Lee Chae Jang 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. We consider the twisted q-extensions of the generalized Euler numbers and polynomials attached to χ. 1. Introduction and Preliminaries Let p be an odd prime number. For n ∈ Z N ∪{0}, let C p n {ζ | ζ p n 1} be the cyclic group of order p n , and let T p lim n →∞ C p n n≥0 C p n C p ∞ be the space of locally constant functions in the p-adic number field C p . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ C p . If q ∈ C, one normally assumes that |q| < 1. If q ∈ C p , one normally assumes that |1 - q| p < 1. In this paper, we use the notation x q 1 - q x 1 - q , x -q 1 - ( -q ) x 1 q . 1.1 Let d be a fixed positive odd integer. For N ∈ N, we set X X d lim ←- N Z dp N Z , X 1 Z p ,