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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 864247, 12 pages doi:10.1155/2010/864247 Research Article On the Fermionic p-adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials T. Kim,1 J. Choi,1 Y. H. Kim,1 and C. S. Ryoo2 1 2 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr Received 30 August 2010; Accepted 3 December 2010 Academic Editor: Paolo E. Ricci Copyright q 2010 T. Kim et al. This is an open access article distributed under the Creative. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 864247 12 pages doi 10.1155 2010 864247 Research Article On the Fermionic p-adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials T. Kim 1 J. Choi 1 Y. H. Kim 1 and C. S. Ryoo2 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 2 Department of Mathematics Hannam University Daejeon 306-791 Republic of Korea Correspondence should be addressed to T. Kim tkkim@kw.ac.kr Received 30 August 2010 Accepted 3 December 2010 Academic Editor Paolo E. Ricci Copyright 2010 T. 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. The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic p-adic integral on zp. From these properties we derive some interesting identities on the Euler numbers and polynomials. 1. Introduction Throughout this paper let p be an odd prime number. The symbol zp Op and cp denote the ring of p-adic integers the field of p-adic rational numbers the complex number field 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-VpW 1 p. Note that zp x x p 1 lim. z pNZp. When one talks of q-extension q is variously considered as an indeterminate a complex number q e c or p-adic number q e cp .If q e c we normally assume q 1 and if q eCp we always assume 1 - q p 1. We say that f is uniformly differentiable function at a point a e Zp and write f e UD Z p if the difference quotient Ff x y f x - f yf x - y has a limit f a as x y a a . For f e UD Zp the fermionic p-adic q-integral on zp is defined as f 1 q pN-1 . . I-q f f x dy-q x firn JN s f W
