tailieunhanh - Báo cáo hóa học: " Research Article On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to χ of Higher Order"

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 Generalized Bernoulli Polynomials Attached to χ of Higher Order | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 640152 7 pages doi 2009 640152 Research Article On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to X of Higher Order Taekyun Kim 1 Lee-Chae Jang 2 Young-Hee Kim 1 and Kyung-Won Hwang3 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 South Korea 2 Department of Mathematics and Computer Science Konkook University Chungju 139-701 South Korea 3 Department of General Education Kookmin University Seoul 136-702 South Korea Correspondence should be addressed to Taekyun Kim tkkim@ Received 5 June 2009 Accepted 5 August 2009 Recommended by Vijay Gupta We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached to X of higher order using 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 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 N u 0 . Let Vp be the normalized exponential valuation of Cp with p p p V p p-1 see 1-24 . Let UD Zp be the space of uniformly differentiable function on Zp. Let d be a fixed positive integer. For n e N let Z X Xd --lim dpNZ X1 Zp X c a dpZp 0 a dp a p 1 a dpNZp Ịx e X x a mod dpN Ị 2 Journal of Inequalities and Applications where a e Z lies in 0 a dpN. For f e UD X the p-adic invariant integral on X is defined as 1 dpN-1 1 ư J Xf Mdx - Nm dpN 10f x see 11-19 . From we note that f If f 0 where f 0 df x dx x 0 and fi x f x 1 . .