`Discrete Dynamics in Nature and SocietyVolumeΒ 2010, Article IDΒ 179430, 9 pageshttp://dx.doi.org/10.1155/2010/179430`
Research Article

## On -Adic Analogue of -Bernstein Polynomials and Related Integrals

T. Kim,1Β J. Choi,1Β Y. H. Kim,1Β and L. C. Jang2

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, Republic of Korea

Received 17 September 2010; Accepted 22 December 2010

Copyright Β© 2010 T. 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

Recently, Kim's work (in press) introduced -Bernstein polynomials which are different Phillips' -Bernstein polynomials introduced in the work by (Phillips, 1996; 1997). The purpose of this paper is to study some properties of several type Kim's -Bernstein polynomials to express the -adic -integral of these polynomials on associated with Carlitz's -Bernoulli numbers and polynomials. Finally, we also derive some relations on the -adic -integral of the products of several type Kim's -Bernstein polynomials and the powers of them on .

#### 1. Introduction

Let denote the set of continuous functions on . For and , Kim introduced the -extension of Bernstein linear operator of order for as follows: where (see [1]). Here is called Kim's -Bernstein operator of order for . For , are called the Kim's -Bernstein polynomials of degree (see [2β6]).

In [7], Carlitz defined a set of numbers inductively by with the usual convention of replacing by . These numbers are -analogues of ordinary Bernoulli numbers , but they do not remain finite for . So he modified the definition as follows: with the usual convention of replacing by (see [7]). These numbers are called the th Carlitz -Bernoulli numbers. And Carlitz's -Bernoulli polynomials are defined by As , we have and , where and are the ordinary Bernoulli numbers and polynomials, respectively.

Let be a fixed prime number. Throughout this paper, , , , , and will denote the ring of rational integers, the field of rational numbers, the ring of -adic integers, the field of -adic rational numbers and the completion of algebraic closure of , respectively. Let be the normalized exponential valuation of such that .

Let be regarded as either a complex number or a -adic number . If , we assume , and if , we normally assume .

We say that is a uniformly differentiable function at a point and denote this property by if the difference quotient has a limit as (see [1, 3, 8β13]).

For , let us begin with the expression representing a -analogue of the Riemann sums for (see [11]). The integral of on is defined as the limit as of the sums (if exists). The -adic -integral on a function is defined by (see [11]).

As was shown in [3], Carlitz's -Bernoulli numbers can be represented by -adic -integral on as follows: Also, Carlitz's -Bernoulli polynomials can be represented (see [3]).

In this paper, we consider the -adic analogue of Kim's -Bernstein polynomials on and give some properties of the several type Kim's -Bernstein polynomials to represent the -adic -integral on of these polynomials. Finally, we derive some relations on the -adic -integral of the products of several type Kim's -Bernstein polynomials and the powers of them on .

#### 2. -Bernstein Polynomials Associated with -Adic -Integral on

In this section, we assume that with .

From (1.5), (1.7) and (1.8), we note that By (2.1), we get Therefore, we obtain the following theorem.

Theorem 2.1. For , one has

By the definition of Carlitz's -Bernoulli numbers and polynomials, we get Thus, we have the following proposition.

Proposition 2.2. For with , one has

It is easy to show that Hence, we have By (1.8), we get By Theorem 2.1 and (2.8), we see that From (2.9) and Proposition 2.2, we have By (1.7) and (2.10), we obtain the following theorem.

Theorem 2.3. For with , one has

Taking the -adic -integral on for one Kim's -Bernstein polynomials, we get and, by the -symmetric property of , we see that

For , by Theorem 2.3 and (2.13), one has

Let with . Then the -adic -integral for the multiplication of two Kim's -Bernstein polynomials on can be given by the following relation:

By Theorem 2.3 and (2.15), we get

By the simple calculation, we easily get Continuing this process, we obtain

Let and , with . By Theorem 2.3 and (2.18), we get From the definition of binomial coefficient, we note that where and , .

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

Theorem 2.4. (I) For and , with , one has
β(II) For and , , one has

By Theorem 2.4, we obtain the following corollary.

Corollary 2.5. For and , with , one has

Let and , , with . Then one has

From the definition of binomial coefficient, one has

By (2.24) and (2.25), we obtain the following theorem.

Theorem 2.6. For and , , with , one has

#### Acknowledgment

This paper was supported by the research grant of Kwangwoon University in 2010.

#### References

1. T. Kim, βA note on $q$-Bernstein polynomials,β Russian Journal of Mathematical Physics. In press.
2. M. Acikgoz and S. Araci, βA study on the integral of the product of several type Bernstein polynomials,β IST Transaction of Applied Mathematics-Modelling and Simulation, vol. 1, no. 1, pp. 10β14, 2010.
3. T. Kim, βOn a $q$-analogue of the $p$-adic log gamma functions and related integrals,β Journal of Number Theory, vol. 76, no. 2, pp. 320β329, 1999.
4. G. M. Phillips, βOn generalized Bernstein polynomials,β in Numerical Analysis, pp. 263β269, World Scientific, River Edge, NJ, USA, 1996.
5. G. M. Phillips, βBernstein polynomials based on the $q$-integers,β Annals of Numerical Analysis, vol. 4, pp. 511β514, 1997.
6. Y. Simsek and M. Acikgoz, βA new generating function of ($q$-) Bernstein-type polynomials and their interpolation function,β Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.
7. L. Carlitz, β$q$-Bernoulli numbers and polynomials,β Duke Mathematical Journal, vol. 15, pp. 987β1000, 1948.
8. M. Cenkci, V. Kurt, S. H. Rim, and Y. Simsek, βOn $\left(i,q\right)$ Bernoulli and Euler numbers,β Applied Mathematics Letters, vol. 21, no. 7, pp. 706β711, 2008.
9. L.-C. Jang, βA new $q$-analogue of Bernoulli polynomials associated with $p$-adic $q$-integrals,β Abstract and Applied Analysis, vol. 2008, Article ID 295307, 6 pages, 2008.
10. T. Kim, βBarnes-type multiple $q$-zeta functions and $q$-Euler polynomials,β Journal of Physics A, vol. 43, no. 25, Article ID 255201, 11 pages, 2010.
11. T. Kim, β$q$-Volkenborn integration,β Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288β299, 2002.
12. T. Kim, L.-C. Jang, and H. Yi, βA note on the modified $q$-Bernstein polynomials,β Discrete Dynamics in Nature and Society, vol. 2010, Article ID 706483, 12 pages, 2010.
13. B. A. Kupershmidt, βReflection symmetries of $q$-Bernoulli polynomials,β Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, pp. 412β422, 2005.