Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials

Min-Soo Kim,Byungje Lee,Cheon-Seoung Ryoo,Taekyun Kim

doi:10.1186/1687-1847-2010-305018

Abstract

We investigate some interesting properties of Bernstein polynomials associated with boson p-adic integrals on Zp.

Highlights

Let C[0, 1] be the set of continuous functions on [0, 1]
We investigate some interesting properties of the Bernstein polynomials related to the bosonic p-adic integrals on Zp
We introduce the Bernstein polynomials on the ring of padic integers Zp

Summary

Introduction

Let C[0, 1] be the set of continuous functions on [0, 1]. the classical Bernstein polynomials of degree n for f ∈ C[0, 1] are defined by (1.1) n Bn(f ) = f k nBk,n(x), k=0 where Bn(f ) is called the Bernstein operator and 0≤x≤1 (1.2) Bk,n(x) =n xk(x − 1)n−k k are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n) (see [10]). We investigate some interesting properties of the Bernstein polynomials related to the bosonic p-adic integrals on Zp. Let C[0, 1] be the set of continuous functions on [0, 1]. The classical Bernstein polynomials of degree n for f ∈ C[0, 1] are defined by N xk(x − 1)n−k k are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n) (see [10]).

Results

Conclusion