- Research Article
- Open Access

# Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials

- Min-Soo Kim
^{1}, - Taekyun Kim
^{2}Email author, - Byungje Lee
^{3}and - Cheon-Seoung Ryoo
^{4}

**2010**:305018

https://doi.org/10.1155/2010/305018

© Min-Soo Kim et al. 2010

**Received:**30 August 2010**Accepted:**27 October 2010**Published:**31 October 2010

## Abstract

We investigate some interesting properties of the Bernstein polynomials related to the bosonic -adic integrals on .

## Keywords

- Prime Number
- Algebraic Closure
- Basis Polynomial
- Bernstein Polynomial
- Bernoulli Number

## 1. Introduction

Thus, we have arrived at the generating function in [1, 2] and also in (1.3) as well.

The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. Some researchers have studied the Bernstein polynomials in the area of approximation theory (see [1–7]). In recent years, Acikgoz and Araci [1, 2] have introduced several type Bernstein polynomials.

In the present paper, we introduce the Bernstein polynomials on the ring of -adic integers . We also investigate some interesting properties of the Bernstein polynomials related to the bosonic -adic integrals on the ring of -adic integers .

## 2. Bernstein Polynomials Related to the Bosonic -Adic Integrals on

By (2.5), we obtain the following proposition.

Proposition 2.1.

for , since . Therefore we obtain the following theorem.

Theorem 2.2.

Therefore we obtain the following result.

Corollary 2.3.

Continuing this process, we obtain the following theorem.

Theorem 2.4.

Theorem 2.5.

Theorem 2.6.

Problem 2.

Find the Witt's formula for the Bernstein polynomials in -adic number field.

## Declarations

### Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0001654). The second author was supported by the research grant of Kwangwoon University in 2010.

## Authors’ Affiliations

## References

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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.