Open Access

Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials

Advances in Difference Equations20102010:305018

Received: 30 August 2010

Accepted: 27 October 2010

Published: 31 October 2010


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 [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
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.



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.

Authors’ Affiliations

Department of Mathematics, KAIST
Division of General Education-Mathematics, Kwangwoon University
Department of Wireless Communications Engineering, Kwangwoon University
Department of Mathematics, Hannam University


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© Min-Soo Kim et al. 2010

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