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

*Advances in Difference Equations*
**volume 2010**, Article number: 305018 (2010)

## 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

where is called the Bernstein operator and

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

where and . Note that

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

where . Observe that

Hence by the above one can very easily see that

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

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

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

(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

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

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

By (2.5), we obtain the following proposition.

Proposition 2.1.

For ,

From (2.4), we note that

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

for , since . Therefore we obtain the following theorem.

Theorem 2.2.

For ,

Also we obtain

Therefore we obtain the following result.

Corollary 2.3.

For ,

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

Continuing this process, we obtain the following theorem.

Theorem 2.4.

The multiplication of the sequence of Bernstein polynomials

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

We put

Theorem 2.5.

The multiplication of

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

Theorem 2.6.

The multiplication of

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

Problem 2.

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

## References

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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.
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**A study on the p-adic integral representation on**associated with Bernstein and Bernoulli polynomials.*Advances in Difference Equations*2010,**2010:**-6. - 6.
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**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.
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**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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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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### Keywords

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