# On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

## Abstract

The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.

## 1. Introduction

Throughout this paper, let be an odd prime number. The symbol, , , and denote the ring of -adic integers, the field of -adic rational numbers, the complex number field and the completion of algebraic closure of , respectively.

Let be the set of natural numbers and . Let be the normalized exponential valuation of with . Note that .

When one talks of -extension, is variously considered as an indeterminate, a complex number , or -adic number . If , we normally assume , and if , we always assume .

We say that is uniformly differentiable function at a point and write , if the difference quotient has a limit as . For , the fermionic -adic -integral on is defined as

(1.1)

(see [1]). In the special case in (1.1), the integral

(1.2)

is called the fermionic -adic invariant integral on (see [2]). From (1.2), we note

(1.3)

where .

Moreover, for , let . Then we note that

(1.4)

It is well known that the Euler polynomials are defined by

(1.5)

(see [1â€“15]). In the special case, , and are called the th Euler numbers.

Let . Then, by (1.3), (1.4), and (1.5), we see that

(1.6)

Let denote the set of continuous functions on . For , Bernstein introduced the following well-known linear positive operator in the field of real numbers :

(1.7)

where (see [3, 4, 7, 10, 11, 14]). Here, is called the Bernstein operator of order for .

For , the Bernstein polynomial of degree is defined by

(1.8)

For example, , , , , , and for , .

In this paper, we study the properties of Bernstein polynomials in the -adic number field. For , we give some properties of several type Bernstein polynomials to represent the fermionic -adic invariant integral on . From those properties, we derive some interesting identities on the Euler polynomials.

## 2. Fermionic -adic Integral Representation of Bernstein Polynomials

By (1.5) and (1.6), we see that

(2.1)

We also have that

(2.2)

From (2.1) and (2.2), we note that . It is easy to show that

(2.3)

By (1.5), (1.6), (2.1), (2.2), and (2.3), we see that for ,

(2.4)

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has

(2.5)

Theorem 2.1 is important to derive our main result in this paper.

Taking the fermionic -adic integral on for one Bernstein polynomial in (1.8), we get

(2.6)

Therefore, we obtain the following proposition.

Proposition 2.2.

For , one is

(2.7)

It is known that . Thus, one has

(2.8)

By (2.8) and Theorem 2.1, we see that for ,

(2.9)

From (2.9), we obtain the following theorem.

Theorem 2.3.

For with , we have

(2.10)

By Proposition 2.2 and Theorem 2.3, we obtain the following corollary.

Corollary 2.4.

For with , we have

(2.11)

For with , fermionic -adic invariant integral for multiplication of two Bernstein polynomials on can be given by the following relation:

(2.12)

Therefore, we obtain the following theorem.

Theorem 2.5.

For with , one has

(2.13)

For , one has

(2.14)

Thus, we obtain the following proposition.

Proposition 2.6.

For , one has

(2.15)

By Theorem 2.5 and Proposition 2.6, we obtain the following corollary.

Corollary 2.7.

For with , one has

(2.16)

In the same manner, multiplication of three Bernstein polynomials can be given by the following relation:

(2.17)

where with .

For with , by the symmetry of Bernstein polynomals, we see that

(2.18)

Therefore, we obtain the following theorem.

Theorem 2.8.

For with , one has

(2.19)

By (2.17) and Theorem 2.8, we obtain the following corollary.

Corollary 2.9.

For with , one has

(2.20)

Using the above theorems and mathematical induction, we obtain the following theorem.

Theorem 2.10.

Let . For with , the multiplication of the sequence of Bernstein polynomials with different degrees under fermionic -adic invariant integral on can be given as

(2.21)

We also easily see that

(2.22)

By Theorem 2.10 and (2.22), we obtain the following corollary.

Corollary 2.11.

Let . For with , one has

(2.23)

Let with . By the definition of , we easily get

(2.24)

Therefore, we obtain the following theorem.

Theorem 2.12.

Let . For with , one has

(2.25)

By simple calculation, we easily get

(2.26)

where for . By Theorem 2.12 and (2.26), we obtain the following corollary.

Corollary 2.13.

Let . For with , one has

(2.27)

The fermionic -adic invariant integral of multiplication of Bernstein polynomials, the th degree Bernstein polynomials with and with multiplicity on , respectively, can be given by

(2.28)

where with .

Assume that . Then one has

(2.29)

Therefore, we obtain the following theorem.

Theorem 2.14.

Let .

(i)For with , one has

(2.30)

(ii)For , one has

(2.31)

By Theorem 2.14, we obtain the following corollary.

Corollary 2.15.

For with , one has

(2.32)

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Kim, T., Choi, J., Kim, Y. et al. On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials. J Inequal Appl 2010, 864247 (2010). https://doi.org/10.1155/2010/864247