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 [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

(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.