A Study on the -Adic Integral Representation on Associated with Bernstein and Bernoulli Polynomials

Abstract

We consider the Bernstein polynomials on and investigate some interesting properties of Bernstein polynomials related to Stirling numbers and Bernoulli numbers.

1. Introduction

Let denote the set of continuous function on . Then, Bernstein operator for is defined as

(1.1)

for , where is called Bernstein polynomial of degree . Some researchers have studied the Bernstein polynomials in the area of approximation theory (see [1–6]).

Let be a fixed prime number. Throughout this paper , , , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of uniformly differentiable function on . For , the -adic -integral on is defined by

(1.2)

(see [4, 7–15]).

In the special case, if we set in (1.2), we have

(1.3)

In this paper, we consider Bernstein polynomials on and we investigate some interesting properties of Bernstein polynomials related to Stirling numbers and Bernoulli numbers.

2. Bernstein Polynomials Related to Stirling Numbers and Bernoulli Numbers

In this section, for , we consider Bernstein type operator on as follows:

(2.1)

for , where is called Bernstein polynomial of degree . We consider Newton's forward difference operator as follows:

(2.2)

For ,

(2.3)

Then, we have

(2.4)

From (2.4), we note that

(2.5)

where

(2.6)

The Stirling number of the first kind is defined by

(2.7)

and the Stirling number of the second kind is also defined by

(2.8)

By (2.5), (2.6), (2.7), and (2.8), we see that

(2.9)

where . Note that, for and ,

(2.10)

Thus, we note that is the generating function of Bernstein polynomial. It is easy to show that

(2.11)

By (2.11), we obtain the following theorem.

Theorem 2.1.

For with , one has

(2.12)

where are the th Bernoulli numbers.

In [12], it is known that

(2.13)

(2.14)

for . By (1.1) and (2.14), we see that

(2.15)

for . By (2.15), we obtain the following theorem.

Theorem 2.2.

For , and , one has

(2.16)

From (2.13) and (2.14), we note that

(2.17)

In [16], it is known that

(2.18)

By (2.17), (2.18), and Theorem 2.2, we have

(2.19)

From the definition of the Stirling numbers of the first kind, we drive that

(2.20)

By (2.17), (2.19), and (2.20), we obtain the following theorem.

Theorem 2.3.

For and , one has

(2.21)

By Theorems 2.2 and 2.3, we obtain the following corollary.

Corollary 2.4.

For , one has

(2.22)

where are the th Bernoulli numbers.