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 Note on q-Extensions of Euler Numbers and Polynomials of Higher Order | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 371295 9 pages doi 2008 371295 Research Article Note on -Extensions of Euler Numbers and Polynomials of Higher Order Taekyun Kim 1 Lee-Chae Jang 2 and Cheon-Seoung Ryoo3 1 The School of Electrical Engineering and Computer Science EECS Kyungpook National University Taegu 702-701 South Korea 2 Department of Mathematics and Computer Science KonKuk University Chungju 143-701 South Korea 3 Department of Mathematics Hannam University Daejeon 306-791 South Korea Correspondence should be addressed to Cheon-Seoung Ryoo ryoocs@ Received 1 November 2007 Accepted 22 December 2007 Recommended by Paolo Emilio Ricci In 2007 Ozden et al. constructed generating functions of higher-order twisted h q -extension of Euler polynomials and numbers by using p-adic q-deformed fermionic integral on Zp. By applying their generating functions they derived the complete sums of products of the twisted h q -extension of Euler polynomials and numbers. In this paper we consider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore we study Barnes-type q-Euler zeta functions. Finally we will derive the new formula for sums of products of q-Euler numbers and polynomials by using fermionic p-adic q-integral on Zp. Copyright 2008 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. 1. Introduction and notations Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers Q denotes the field of rational numbers Qp denotes
