Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article On the Twisted q-Analogs of the Generalized Euler Numbers and Polynomials of Higher Order | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 875098 11 pages doi 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 Kim3 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 Received 12 April 2010 Accepted 28 June 2010 Academic Editor Istvan Gyori Copyright 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 X. 1. Introduction and Preliminaries Let p be an odd prime number. For n e Z N u 0 let Cpn Z Ự 1 be the cyclic group of order pn and let Tp limn. cp Un 0 Cpn Cp be the space of locally constant functions in the p-adic number field Cp. When one talks of q-extension q is variously considered as an indeterminate a complex number q e C or p-adic number q e Cp. If q e C one normally assumes that q 1. If q e Cp one normally assumes that 1 - q p 1. In this paper we use the notation x q 1 - qx Ĩ-Ĩ x -q 1 - -qf 1 q Let d be a fixed positive odd integer. For N e N we set limvrZ X Xd N- X1 Zp dpNZ 2 Advances in Difference Equations X u a dpZp 0 a dp a p 1 a dpnZp x e X x a mod dpn ị where a e Z lies in 0 a dpn compared to 1-16 . Let X be the Dirichlet s character with an odd conductor d e N. Then the generalized Z-Euler polynomials attached to X Enxz x are defined as Wx t 9 vd 1 1 1ơ1t 22j 1 0 I-1 XV Z e xt .
