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 [1–15]). 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)