#### Abstract

Using bosonic -adic -integral on , we give some interesting relationships between -Bernoulli numbers with weight (,) and -Bernstein polynomials with weight . Also, using -Bernstein polynomials with two variables, we derive some interesting properties associated with -Bernoulli numbers with weight (,).

#### 1. Introduction

In many areas the Bernstein polynomials have many applications: in statistics, in computer applications, in approximations of functions, in numerical analysis, in -adic analysis, and in the solution of differential equations. In particular, recognition of human speech has long been a hot issue among artificial intelligence and signal processing researchers, and -Bernstein polynomials, which are mathematical operators, have been applied for pattern recognition, and a new method has thus been developed. For -Bernoulli numbers and polynomials, several results have been studied by Carlitz (see [1, 2]), Kim (see [3–7]), Ozden et al. (see [8]), and M.-S. Kim et al. (see [9]). Bernoulli numbers and polynomials possess many interesting properties and arise in many areas of mathematics, mathematical physics, and statistical physics. Recently, many mathematicians have studied in the area of Bernoulli numbers and polynomials. First, we introduce the -Bernoulli numbers and polynomials with weight as extended measure. And we investigate some identities due to the -Bernoulli numbers and polynomials with weight . Second, we consider the -adic analogue of the extended -Bernstein polynomials on and investigate some properties of several extended -Bernstein polynomials by using the bosonic -adic -integral. Finally, we investigate the relation of Bernstein polynomials and Bernoulli numbers and polynomials using -adic -integral on .

Throughout this paper, we use the following notations. By we denote the ring of -adic rational integers, denotes the field of -adic rational numbers, denotes the completion of algebraic closure of , denotes the set of natural numbers, denotes the ring of rational integers, denotes the field of rational numbers, denotes the set of complex numbers, and . Let be the normalized exponential valuation of with . When one talks of -extension, is considered in many ways such as an indeterminate, a complex number , or -adic number . If one normally assumes that . If , we normally assume that so that for . Throughout this paper we use the following notation: (see [1–22]). The -numbers satisfy many simple relations, all easily verified, as below:

For a fixed positive integer with , let where and (see [3]). We say that is a uniformly differential function at a point and denotes this property by if the difference quotients have a . For , let us begin with the expression representing a -analogue of the Riemann sum for . The integral of on is defined as the limit () of the sums (if exists). The -adic -integral (or -Volkenborn integrals of is defined by T. Kim (see [4]) as follows (see [4]). If we take in (6), then we easily see that From (6), we obtain where (cf. [1–19]).

Carlitz’s -Bernoulli numbers can be defined recursively by and by the value that with the usual convention of replacing .

Carliz’s -Bernoulli polynomials are also defined by

It is well known that where are called the th Carlitz’s -Bernoulli polynomials.

In this paper, we define the -Bernoulli numbers and polynomials with weight as below.

For and with , the -Bernoulli numbers with weight are defined by

For and with , the -Bernoulli polynomials with weight are also defined by

#### 2. The Properties of the -Bernoulli Numbers and Polynomials with Weight

Our primary goal of this section is to find some properties of the -Bernoulli numbers with weight , respectively, , . This is a very important process to find the relationship between Bernoulli numbers and polynomials and Bernstein polynomials. In this section, by using the bosonic -adic -integral on , we obtain some properties.

For and with , -Bernoulli numbers with weight are represented as follows by simple calculus:

We set By (14) and (15), we have

Note that for , where are the th Carliz’s -Bernoulli polynomials. In particular, , are the th Carliz’s -Bernoulli numbers.

For and with , since , we get the formula as below: with usual convention about replacing by . Also, by (6) and the bosonic -adic -integral on , we get the finite summation formula as below:

Therefore, by (18) and (19), we have the following theorem.

Theorem 1. *For and with , one has
**
and if , .*

We also get from Theorem 1 that for ,

From (7) we get the following formula: and by comparison of coefficients we obtain as below:

From (8), we get the following formula: Specially, we consider when and , then we obtain the following Theorem.

Theorem 2. *For and with , one has
*

By the bosonic -adic -integral on and (21), for

Therefore, from Theorem 2 and (26), we obtain the following theorem.

Theorem 3. * For and with , one has
*

#### 3. Properties of -Bernstein Polynomials

In this section, we introduce the -Bernstein polynomials and we get some properties to use. 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 ). In recent years, Gupta et al. have studied the generation function for -Bernstein polynomials (see [19]). Their generating function for is given by where . Observe that

Also, -Bernstein polynomials with weight , are studied by T. Kim (see [7]), and the formula is as below:

Let be continuous functions on . -Bernstein operator is defined as below:

From now on, we introduce some properties of -Bernstein polynomials with weight and -Bernstein operator. It is easy to show that these properties are true by some calculus are(1)(2), (3), (4)In the above property 4, we consider special cases , , as below.

If ,

If ,

If , then one has that .

From definition of with , and , for , we have that

It is possible to write as a linear combination of by using the degree evaluation formula and mathematical induction: By the same method, we get Continuing this process, we have the following:

For , , , By some calculus, we obtain the following: Hence, from (41) and (42) we have the following theorem.

Theorem 4. *For , with and , one has
*

#### 4. Relations of -Bernoulli Numbers and Polynomials with Weight and -Bernstein Polynomials

In this section, assume that with . With bosonic -adic -integral on , we get from Theorem 2 and (26) that

Therefore, we have the following theorem.

Theorem 5. *For , with and , one has
*

Using , we get the other identities as below:

From Theorem 5 and (46), we obtain the following:

Specially, if , then If , then

Note that for , Also,

Hence, one has the theorem as below.

Theorem 6. *For , with and , one has
**
Also, from Theorem 6, we get the identity as below:
**
Continuing this process, we obtain the following theorem.*

Theorem 7. * For , with and , one has
**
where .*

Hence, from Theorem 7, we get the corollary as below.

Corollary 8. *For , with and , one has
**
where .*

#### 5. Extension of -Bernstein Polynomials with Weight

In this section, we introduce the extended Kim’s -Bernstein polynomials of order and the extended Kim’s -Bernstein operator of order . Also, we define the extended Kim’s -Bernstein polynomials of order with weight and the extended Kim’s -Bernstein operator of order with weight and we investigate the properties of these. Finally, we investigate the relation of -Bernoulli numbers with weight and extension of -Bernstein polynomials with weight . We assume that with . Let be the set of continuous function on and . For and , the extended Kim’s -Bernstein polynomials of order are as below:

For , and , the extended Kim’s -Bernstein operator of order is as below:

For , , and , we define the extended Kim’s -Bernstein polynomials of order with weight and the extended Kim’s -Bernstein operator of order with weight as below:

Properties of and are same to and , respectively. So we will enumerate these properties without proof as below.

For , , (1), (2), (3), (4), (5). Taking double bosonic -adic -integral on , we get from (26) and (27) that Thus, from (59), we have the following theorem.

Theorem 9. *For , with and , one has *

We get from the -symmetric properties of the -Bernstein polynomials that for

Note that for

Note also that for

Therefore, from (63) and (65), we have the following theorem.

Theorem 10. *For , with and , one has
*

Note that for and where .

Note also from the binomial theorem that and , where .

Hence, from (65) and (66), we see that the following theorem holds.

Theorem 11. *For , with and , one has
*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.