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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article New Approach to q-Euler Numbers and Polynomials | Hindawi Publishing Corporation Advances in Difference Equations Volume 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 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 South Korea 2 Department of Mathematics and Computer Science Konkuk University Chungju 380-701 South Korea 3 Department of Mathematics Education Kyungpook National University Taegu 702-701 South Korea Correspondence should be addressed to Young-Hee Kim yhkim@kw.ac.kr 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. 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 X 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. 1. Introduction Let C be the complex number field. We assume that q e C with q 1 and that the q-number is defined by x q 1 - qx 1 - q in this paper. Recently many mathematicians have studied for q-Euler and q-Bernoulli polynomials and numbers see 1-18 . Specially there are papers for the q-extensions of Euler polynomials and numbers approaching with two kinds of viewpoint among remarkable papers see 7 10 . It is known that the Euler polynomials are .
