Discrete Dynamics in Nature and Society

Volume 2010 (2010), Article ID 706483, 12 pages

http://dx.doi.org/10.1155/2010/706483

## A Note on the Modified -Bernstein Polynomials

^{1}Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea^{2}Department of Mathematics and Computer Science, Konkuk University, Chungju 138-701, Republic of Korea^{3}Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 20 May 2010; Revised 11 July 2010; Accepted 14 July 2010

Academic Editor: Leonid Shaikhet

Copyright © 2010 Taekyun Kim et al. 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.

#### Abstract

We propose the modified -Bernstein polynomials of degree which are different -Bernstein polynomials of Phillips (1997). From these modified -Bernstein polynomials of degree , we derive some recurrence formulae for the modified -Bernstein polynomials.

#### 1. Introduction

Let denote the set of continuous function on . For , Bernstein introduced the following well-known linear positive operators in [1]:
where . Here is called the * Bernstein operator of order ** for *. For , the * Bernstein polynomial of degree * is defined by
where . For example,
Also, for , because .

Some people have studied the Bernstein polynomials in the area of approximation theory (see [2] through [3]). Note that for and , Because , we obtain the generating function for as follows: (see [4, 5]), where and . Notice that for (see [2]).

Let . Define * the **-number of * by
See [2] through [3] for details and related facts. Note that . In [6], Phillips proposed a generalization of the classical Bernstein polynomials based on -integers. In the last decade some new generalizations of well-known positive linear operators, based on -integers were introduced and studied by several authors (see [1–13]). Recently, Simsek and Acikgoz have also studied the -extension of Bernstein-type polynomials [5]. Their -Bernstein-type polynomials are given by
where are the second-kind stirling number. In [5], we can find some interesting formulae related to -extension of Bernstein polynomials which are different -Bernstein polynomials of Phillips. In the conference of Jangjeon Mathematical Society which was held in IRAN (on Feb.2010), Acikgoz and Arci has introduced several-type Bernstein polynomials (see [2]). The Acikgoz paper [2] announced in the conference is actually what motivated us to write this paper. In this paper, we considered the -extension of Bernstein polynomials which were introduced by Acikgoz at the conference of Jangjeon Mathematical Society on Feb. 2010. First, we consider the -extension of the generating function of Bernstein polynomials in (1.5). Indeed, this generating function is also treated by Simsek and Acikgoz in a previous paper (see [5]). From this -extension of the generating function for the Bernstein polynomials, we propose the modified -Bernstein polynomials of degree which are different -Bernstein polynomials of Phillips. By using the properties of the modified -Bernstein polynomials, we obtain some recurrence formulae for the modified -Bernstein polynomials of degree .

#### 2. The Modified -Bernstein Polynomials

For , consider the -extension of (1.5) as follows:
where and . Note that . We define the * modified **-Bernstein polynomials* as follows:
where and .

*Remark 2. *This generating function is also introduced by Simsek and Acikgoz in a previous paper (see [5]).

By comparing the coefficients of (2.1) and (2.2), we obtain the following theorem.

Theorem 2.1. *For and ,
*

For , we have and the derivatives of the modified -Bernstein polynomials of degree are also polynomials of degree , that is, Therefore, we obtain the following recurrence formulae.

Theorem 2.2 (recurrence formulae for ()). *For and for ,
*

Let be a continuous function on . Then the * modified **-Bernstein operator of order ** for * is defined by
where , . We get from Theorem 2.1 and (2.7) that for ,
We also see from Theorem 2.1 that

The modified -Bernstein polynomials are symmetric polynomials in the following sense: Therefore, we get the following theorem.

Theorem 2.3. *For and ,
*

For , and for , consider where is a circle around the origin and integration is in the positive direction. We see from the definition of the modified -Bernstein polynomials and the basic theory of complex analysis including Laurent series that We get from (2.12) and (2.13) that We also get from (2.12) and (2.15) that Therefore, we see from (2.14) and (2.16) that Note that Therefore, we can write the modified -Bernstein polynomials as a linear combination of polynomials of higher order as follows.

Theorem 2.4. *For and ,
*

We easily see from (2.17) that for , Thus, the following corollary holds.

Corollary 2.5. *For and ,
*

Note from the definition of the modified -Bernstein polynomials and the binomial theorem that for , Therefore, we showed that the following theorem holds.

Theorem 2.6. *For and ,
*

It is possible to write as a linear combination of the modified -Bernstein polynomials by using the degree evaluation formulae and mathematical induction. We easily see from the property of the modified -Bernstein polynomials that and that Continuing this process, we obtain for . Therefore, we obtain the following theorem.

Theorem 2.7. *For and ,
*

For , the * Bernoulli polynomial of order * is defined by
and are called the th* Bernoulli numbers of order *. It is well known that the * second kind stirling number* is defined by
for . We note from (2.2) that
We have from (2.2) and (2.30) that
and .

*Remark 2. *The Equations (2.30) and (2.31) are already known by Simsek and Acikgoz in a previous paper [5, page 7].

Let be the * shift difference operator* defined by . We see from the iterative method that
for . We get from (2.29) and (2.32) that
By comparing the coefficients on both sides above, we have
for . Thus, we get from (2.31) and (2.34) that
Let be the * shift operator*. Then the *-difference operator* is defined by
where is an identity operator(see [7] through [11]). For and , we have
where is the * Gaussian binomial coefficient* defined by

Let be the generating function of the *-extension of the second kind stirling number* as follows:
We have from (2.39) that
where . It is not difficult to see that
See also [7] through [11] for details and related facts for above. Then, we get from (2.41) and Theorem 2.7 that
Therefore, this completes the proof of the following theorem.

Theorem 2.8. *For and ,
*

#### Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2010.

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