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 Identities of Symmetry for the ζ-Euler Polynomials of Higher Order | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 273545 9 pages doi 2009 273545 Research Article On the Identities of Symmetry for the Z-Euler Polynomials of Higher Order Taekyun Kim 1 Kyoung Ho Park 2 and Kyung-won Hwang3 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 South Korea 2 Department of Mathematics Sogang University Seoul 121-742 South Korea 3 Department of General Education Kookmin University Seoul 139-702 South Korea Correspondence should be addressed to Taekyun Kim tkkim@ Received 19 February 2009 Revised 31 May 2009 Accepted 18 June 2009 Recommended by Agacik Zafer The main purpose of this paper is to investigate several further interesting properties of symmetry for the multivariate p-adic fermionic integral on Zp. From these symmetries we can derive some recurrence identities for the Z-Euler polynomials of higher order which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the Z-Euler polynomials of higher order we can obtain many identities related to the Frobenius-Euler polynomials of higher order. Copyright 2009 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 Definition Let p be a fixed odd prime number. Throughout this paper Zp Qp C and Cp will respectively denote the ring of p-adic rational integer the field of p-adic rational numbers the complex number field and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with p p p V p p-1. Let UD Zp be the space of uniformly differentiable functions on Zp. For f e UDfZp q e Cp with 1 - qlp 1 the fermionic p-adic q-integral on Zp is defined as . f . 1 q pN-1 . . I-q f f x dp-q x lim N z f x -q JZp N - 1 qp x 0 see 1 .