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

*Advances in Difference Equations*
**volume 2010**, Article number: 163217 (2010)

## 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

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

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

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:

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

For ,

Then, we have

From (2.4), we note that

where

The Stirling number of the first kind is defined by

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

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

where . Note that, for and ,

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

By (2.11), we obtain the following theorem.

Theorem 2.1.

For with , one has

where are the th Bernoulli numbers.

In [12], it is known that

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

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

Theorem 2.2.

For , and , one has

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

In [16], it is known that

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

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

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

Theorem 2.3.

For and , one has

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

Corollary 2.4.

For , one has

where are the th Bernoulli numbers.

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## Acknowledgment

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

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### Keywords

- Ordinary Differential Equation
- Functional Equation
- Prime Number
- Rational Number
- Difference Operator