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 Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 305018 7 pages doi 2010 305018 Research Article Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials Min-Soo Kim 1 Taekyun Kim 2 Byungje Lee 3 and Cheon-Seoung Ryoo4 1 Department of Mathematics KAIST 373-1 Guseong-dong Yuseong-gu Daejeon 305-701 Republic of Korea 2 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 3 Department of Wireless Communications Engineering Kwangwoon University Seoul 139-701 Republic of Korea 4 Department of Mathematics Hannam University Daejeon 306-791 Republic of Korea Correspondence should be addressed to Taekyun Kim tkkim@ Received 30 August 2010 Accepted 27 October 2010 Academic Editor Istvan Gyori Copyright 2010 Min-Soo 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. We investigate some interesting properties of the Bernstein polynomials related to the bosonic p-adic integrals on Zp 1. Introduction Let C 0 1 be the set of continuous functions on 0 1 Then the classical Bernstein polynomials of degree n for f e C 0 1 are defined by Bn f Ẻf k Bk n xr 0 x 1 k 0 n where Bn f is called the Bernstein operator and Bk n x n xk x - 1 n-k 1 1 1 2 2 Advances in Difference Equations are called the Bernstein basis polynomials or the Bernstein polynomials of degree n . Recently Acikgoz and Araci have studied the generating function for Bernstein polynomials see 1 2 . Their generating function for Bk n x is given by Fk i x ỉ- Ể Bkn x s n 0 where k 0 1 . and x e 0 1 . Note that Bk n x xk 1 - x n-k. if n k 0 if n k for n 0 1 . see 1 2 . In 3 Simsek and Acikgoz defined generating function of the q- Bernstein-Type Polynomials Yn k x q as follows fM1- fn Fkrq f x ------------ 2 Yn k x q --
