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

- Lee-Chae Jang
^{1}Email author, - Won-Joo Kim
^{2}and - Yilmaz Simsek
^{3}

**2010**:163217

**DOI: **10.1155/2010/163217

© Lee-Chae Jang et al. 2010

**Received: **13 August 2010

**Accepted: **15 September 2010

**Published: **19 September 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

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

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

By (2.11), we obtain the following theorem.

Theorem 2.1.

where are the th Bernoulli numbers.

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

Theorem 2.2.

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

Theorem 2.3.

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

Corollary 2.4.

where are the th Bernoulli numbers.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

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