Theory and Modern Applications

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

## Abstract

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

## 1. Introduction

Let be the set of continuous functions on . Then the classical Bernstein polynomials of degree for are defined by

(11)

where is called the Bernstein operator and

(12)

are called the Bernstein basis polynomials (or the Bernstein polynomials of degree ). Recently, Acikgoz and Araci have studied the generating function for Bernstein polynomials (see [1, 2]). Their generating function for is given by

(13)

where and . Note that

(14)

for (see [1, 2]). In [3], Simsek and Acikgoz defined generating function of the (-)Bernstein-Type Polynomials, as follows:

(15)

where . Observe that

(16)

Hence by the above one can very easily see that

(17)

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 [17]). 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

Let be a fixed prime number. Throughout this paper, , , and will denote the ring of -adic integers, the field of -adic numbers, and the completion of the algebraic closure of , respectively. Let be the normalized exponential valuation of with . For , the bosonic distribution on

(21)

is known as the -adic Haar distribution where (cf. [8]). We will write to remind ourselves that is the variable of integration. Let be the space of uniformly differentiable function on . Then yields the fermionic -adic -integral of a function

(22)

(cf. [8]). Many interesting properties of (2.2) were studied by many authors (cf. [8, 9] and the references given there). For , write . We have

(23)

This identity is to derives interesting relationships involving Bernoulli numbers and polynomials. Indeed, we note that

(24)

where are the Bernoulli polynomials (cf. [8]). From (1.2), we have

(25)

By (2.5), we obtain the following proposition.

Proposition 2.1.

For ,

(26)

From (2.4), we note that

(27)

with the usual convention of replacing by and by . Thus, we have

(28)

for , since . Therefore we obtain the following theorem.

Theorem 2.2.

For ,

(29)

Also we obtain

(210)

Therefore we obtain the following result.

Corollary 2.3.

For ,

(211)

From the property of the Bernstein polynomials of degree , we easily see that

(212)

Continuing this process, we obtain the following theorem.

Theorem 2.4.

The multiplication of the sequence of Bernstein polynomials

(213)

for with different degree under -adic integral on , can be given as

(214)

We put

(215)

Theorem 2.5.

The multiplication of

(216)

Bernstein polynomials with different degrees under -adic integral on can be given as

(217)

Theorem 2.6.

The multiplication of

(218)

Bernstein polynomials with different degrees with different powers under -adic integral on can be given as

(219)

Problem 2.

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

## References

1. 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 press

2. 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;

3. Simsek Y, Acikgoz M:A new generating function of (-) Bernstein-type polynomials and their interpolation function. Abstract and Applied Analysis 2010, 2010:-12.

4. Bernstein S: Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities. Communications of the Kharkov Mathematical Society 1913, 13: 1-2.

5. Jang L-C, Kim W-J, Simsek Y:A study on the p-adic integral representation on associated with Bernstein and Bernoulli polynomials. Advances in Difference Equations 2010, 2010:-6.

6. Kim T, Jang L-C, Yi H:A note on the modified -bernstein polynomials. Discrete Dynamics in Nature and Society 2010, 2010:-12.

7. Phillips GM:Bernstein polynomials based on the -integers. Annals of Numerical Mathematics 1997,4(1–4):511-518.

8. Kim T:On a -analogue of the -adic log gamma functions and related integrals. Journal of Number Theory 1999,76(2):320-329. 10.1006/jnth.1999.2373

9. Kim T, Choi J, Kim Y-H:Some identities on the -Bernstein polynomials, -Stirling numbers and -Bernoulli numbers. Advanced Studies in Contemporary Mathematics 2010,20(3):335-341.

## 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.

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Correspondence to Taekyun Kim.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kim, MS., Kim, T., Lee, B. et al. Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials. Adv Differ Equ 2010, 305018 (2010). https://doi.org/10.1155/2010/305018

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

### Keywords

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