- 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

## 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

## References

- Acikgoz M, Araci S: A study on the integral of the product of several type Bernstein polynomials. IST Transaction of Applied Mathematics-Modelling and Simulation. In pressGoogle Scholar
- Acikgoz M, Araci S:
**On the generating function of the Bernstein polynomials.**In*Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM '10), March 2010, Rhodes, Greece*. AIP;Google Scholar - Bernstein S:
**Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities.***Communications of the Kharkov Mathematical Society*1913,**13:**1-2.Google Scholar - Kim T, Jang LC, Yi H:
**A note on the modified**q**-Bernstein polynomials.***Discrete Dynamics in Nature and Society*2010,**2010:**-12.Google Scholar - Phillips GM:
**Bernstein polynomials based on the**q**-integers.***Annals of Numerical Mathematics*1997,**4**(1–4):511-518.MATHMathSciNetGoogle Scholar - Kim MS, Kim D, Kim T:
**On the q-Euler numbers related to modified q-Bernstein polynomials.**http://arxiv.org/abs/1007.3317 - Kim T: q
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288-299.MATHMathSciNetGoogle Scholar - Kim T: q
**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51-57.MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**Note on the Euler**q**-zeta functions.***Journal of Number Theory*2009,**129**(7):1798-1804. 10.1016/j.jnt.2008.10.007MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**On a**q**-analogue of the**p**-adic log gamma functions and related integrals.***Journal of Number Theory*1999,**76**(2):320-329. 10.1006/jnth.1999.2373MATHMathSciNetView ArticleGoogle Scholar - Kim T: Barnes-type multiple q -zeta functions and q -Euler polynomials. Journal of Physics A 2010.,43(25):Google Scholar
- Kim T:
**Power series and asymptotic series associated with the**q**-analog of the two-variable**p**-adic**L**-function.***Russian Journal of Mathematical Physics*2005,**12**(2):186-196.MATHMathSciNetGoogle Scholar - Kim T, Choi J, Kim Y-H:
**Some identities on the q-Bernstein polynomials, g-Stirling numbers and q-Bernoulli numbers.***Advanced Studies in Contemporary Mathematics*2010,**20**(3):335-341.MATHMathSciNetGoogle Scholar - Kim T:
**Some identities on the**q**-Euler polynomials of higher order and**q**-Stirling numbers by the fermionic**p**-adic integral on****.***Russian Journal of Mathematical Physics*2009,**16**(4):484-491. 10.1134/S1061920809040037MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**On**p**-adic interpolating function for**q**-Euler numbers and its derivatives.***Journal of Mathematical Analysis and Applications*2008,**339**(1):598-608. 10.1016/j.jmaa.2007.07.027MATHMathSciNetView ArticleGoogle Scholar - Kim T, Park D-W, Rim S-H:
**On multivariate**p**-adic**q**-integrals.***Journal of Physics A*2001,**34**(37):7633-7638. 10.1088/0305-4470/34/37/315MATHMathSciNetView ArticleGoogle Scholar

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