tailieunhanh - Báo cáo hóa học: " Research Article A Study on the p-Adic q-Integral Representation on p Associated with the Weighted q-Bernstein and q-Bernoulli Polynomials"

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 A Study on the p-Adic q-Integral Representation on p Associated with the Weighted q-Bernstein and q-Bernoulli Polynomials | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 513821 8 pages doi 2011 513821 Research Article A Study on the p-Adic q-Integral Representation on Zp Associated with the Weighted q-Bernstein and q-Bernoulli Polynomials T. Kim 1 A. Bayad 2 and . Kim1 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 2 Departement de Mathématiques Universite d Evry Val d Essonne Boulevard Francois Mitterrand 91025 Evry Cedex France Correspondence should be addressed to A. Bayad abayad@ Received 6 December 2010 Accepted 15 January 2011 Academic Editor Vijay Gupta Copyright 2011 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. We investigate some interesting properties of the weighted q-Bernstein polynomials related to the weighted q-Bernoulli numbers and polynomials by using p-adic q-integral on zp. 1. Introduction and Preliminaries Let p be a fixed prime number. Throughout this paper zpjQp and cp will denote the ring of p-adic integers the field of p-adic rational numbers and the completion of the algebraic closure of Op respectively. Let N be the set of natural numbers and let z N u 0 . Let vp be the normalized exponential valuation of Cp with p p p V D 1 p. Let q be regarded as either a complex number q e c or a p-adic number q e cp .If q e c then we always assume q 1. If q eCp we assume that 1 - q p 1. In this paper we define the q-number as x q 1 - qx 1 - q see 1-13 . Let C 0 1 be the set of continuous functions on 0 1 . For a e N and n k el the weighted q-Bernstein operator of order n for f e C 0 1 is defined by 10 a f fik n Frlk. ri _ n-k p k B a 11 n qf x - zf fxiqa L1 xiq-a 2-if Bk n x q - O-1 k 0 n w k 0 n Here B jkalt x q is called the weighted q-Bernstein polynomials of degree n