- Research Article
- Open Access

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

**2010**:163217

https://doi.org/10.1155/2010/163217

© Lee-Chae Jang et al. 2010

**Received:**13 August 2010**Accepted:**15 September 2010**Published:**19 September 2010

## Abstract

## Keywords

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

## 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.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.