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 On the Symmetric Properties of Higher-Order Twisted q-Euler Numbers and Polynomials | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 765259 8 pages doi 2010 765259 Research Article On the Symmetric Properties of Higher-Order Twisted q-Euler Numbers and Polynomials Eun-Jung Moon 1 Seog-Hoon Rim 2 Jeong-Hee Jin 1 and Sun-Jung Lee1 1 Department of Mathematics Kyungpook National University Daegu 702-701 South Korea 2 Department of Mathematics Education Kyungpook National University Daegu 702-701 South Korea Correspondence should be addressed to Seog-Hoon Rim shrim@ Received 14 December 2009 Accepted 19 March 2010 Academic Editor Panayiotis Siafarikas Copyright 2010 Eun-Jung Moon 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. In 2009 Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher-order recently. In this paper we extend our result to the higher-order twisted q-Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher-order twisted q-Euler numbers and polynomials by the properties of p-adic invariant integral on Zp. Especially if q 1 we derive the result of Kim et al. 2009 . 1. Introduction Let p be a fixed odd prime number. Throughout this paper the symbols Z Zp Qp C and Cp will denote the ring of rational integers the ring of p-adic integers the field of p-adic rational numbers the complex number field and the completion of the algebraic closure of Qp respectively Let N be the set of natural numbers and Z NU 0 . Let vp be the normalized exponential valuation of Cp with p p p-vfp 1 p. When one talks of q-extension q is variously considered as an indeterminate a complex q e C or a p-adic number q e Cp. If q e C one normally assumes that q 1. If q e Cp then we assume that q - 1 p p-1 P-1 so that qx exp x log q for each x e Zp. We use the .
