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 for the Generalized Twisted Bernoulli Polynomials | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 164743 8 pages doi 2009 164743 Research Article On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials Taekyun Kim and Young-Hee Kim Division of General Education-Mathematics Kwangwoon University Seoul 139-701 South Korea Correspondence should be addressed to Young-Hee Kim yhkim@ Received 6 July 2009 Accepted 18 October 2009 Recommended by Narendra Kumar Govil We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the p-adic invariant integral. Copyright 2009 T. Kim and . Kim. 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 the symbols Z Zp Qp and Cp denote the ring of rational integers the ring of p-adic integers the field of p-adic rational numbers and the completion of 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 vp pi p . Let UD Zp be the space of uniformly differentiable function on Zp. For f e UD Zp the p-adic invariant integral on Zp is defined as 1 f f f fdx im -N E f x V1 JZp N- pN x 0 see 1 . From we note that CfO If f 0 2 Journal of Inequalities and Applications where 0 df x dx x 0 and f1 x f x 1 . For n e N let fn x f x n . Then we can derive the following equation from n-1 I f I ff f i 0 see 1-7 . Let d be a fixed positive integer. For n e N let X Xd limj- Z dpN Z X1 Zp X u a dp Zp 0 a dp a p 1 a dpNZp Ịx e X x a mod dpN Ị where a e Z lies in 0 a dpN. It is easy to see that ị f x dx f x dx for f e UD
