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 Symmetry Properties of Higher-Order Bernoulli Polynomials | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 318639 6 pages doi 2009 318639 Research Article Symmetry Properties of Higher-Order Bernoulli Polynomials Taekyun Kim 1 Kyung-Won Hwang 2 and Young-Hee Kim1 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 South Korea 2 Department of General Education Kookmin University Seoul 136-702 South Korea Correspondence should be addressed to Taekyun Kim tkkim@ and Kyung-Won Hwang khwang7@ Received 11 March 2009 Revised 6 July 2009 Accepted 2 August 2009 Recommended by Patricia J. Y. Wong We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariate p-adic invariant integral on Zp. 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 Let p be a fixed prime number. Throughout this paper Zp Qp and Cp will respectively denote the ring of p-adic rational integers the field of p-adic rational numbers and the completion of algebraic closure of Qp. For x e Cp we use the notation x q 1 - qx 1 - q . Let UD Zp be the space of uniformly differentiable functions on Zp and let vp be the normalized exponential valuation of Cp with p p p v-w 1 p. For q e Cp with 1 - q p 1 the q-Volkenborn integral on Zp is defined as f 1 p -1 Iq f f x dpq x hm r Nr X f xtf JZp N p f e UD Zp see 1 2 . The ordinary p-adic invariant integral on Zp is given by I1 f lim Iq f f x dx q 1 Zp 2 Advances in Difference Equations see 1-15 . Let 0 df x dx x 0. Then we easily see that h 1 I1f f 0 where f1 x f x 1 . From we can derive I Zp ex-dx e- n b n n 0 t 1 see 2 8-10 where Bn are the nth Bernoulli numbers. By and we easily see that n Z extdx J .