Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 179430, 9 pages
doi:10.1155/2010/179430
Research Article

On 𝑝 -Adic Analogue of 𝑞 -Bernstein Polynomials and Related Integrals

T. Kim,1 J. Choi,1 Y. H. Kim,1 and L. C. Jang2

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, Republic of Korea

Received 17 September 2010; Accepted 22 December 2010

Academic Editor: Binggen Zhang

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.

Abstract

Recently, Kim's work (in press) introduced 𝑞 -Bernstein polynomials which are different Phillips' 𝑞 -Bernstein polynomials introduced in the work by (Phillips, 1996; 1997). The purpose of this paper is to study some properties of several type Kim's 𝑞 -Bernstein polynomials to express the 𝑝 -adic 𝑞 -integral of these polynomials on ℤ 𝑝 associated with Carlitz's 𝑞 -Bernoulli numbers and polynomials. Finally, we also derive some relations on the 𝑝 -adic 𝑞 -integral of the products of several type Kim's 𝑞 -Bernstein polynomials and the powers of them on ℤ 𝑝 .