Some Identities on Bernstein Polynomials Associated with q -Euler Polynomials

Abstract

We investigate some interesting properties of the q -Euler polynomials. The purpose of this paper is to give some relationships between Bernstein and q -Euler polynomials, which are derived by the p -adic integral representation of the Bernstein polynomials associated with q -Euler polynomials.

(ProQuest: ... denotes non-US-ASCII text omitted.)

A. Bayad 1 and T. Kim 2 and B. Lee 3 and S.-H. Rim 4

Recommended by John Rassias

1, Département de Mathématiques, Université d'Evry Val d'Essonne, Bboulevard F. Mitterrand, 91025 Evry Cedex, France
2, Division of General Education-Mathematics, Kwang-Woon University, Seoul 139-701, Republic of Korea
3, Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea
4, Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

Received 11 January 2011; Accepted 11 February 2011

1. Introduction

Let p be a fixed odd prime number. Throughout this paper _...p , _...p , and _...p denote the ring of p -adic integers, the field of p -adic numbers, and the field of p -adic completion of the algebraic closure of _...p , respectively (see [1-15]). Let ... be the set of natural numbers and _...+ =...∪{0} . The normalized p -adic absolute value is defined by |p_|p =1/p . As an indeterminate, we assume that q∈_...p with |1-q_|p <1 . Let UD(_...p ) be the space of uniformly differentiable function on _...p . For f∈UD(_...p ) , the p -adic invariant integral on _...p is defined by [figure omitted; refer to PDF] (see [7-10]). For n∈... , we can derive the following integral equation from (1.1): [figure omitted; refer to PDF] where _fn (x)=f(x+n) (see [7-11]). As well-known definition, the Euler polynomials are given by the generating function as follows: [figure omitted; refer to PDF] (see [3, 5, 7-15]), with usual convention about replacing ^En (x) by _En (x) . In the special case x=0 , _En (0)=_En are called the n th Euler numbers. From (1.3), we can derive the following recurrence formula for Euler numbers: [figure omitted; refer to PDF] (see [12]), with usual convention about replacing ^En by _En . By the definitions of Euler numbers and polynomials, we get [figure omitted; refer to PDF] (see [3, 5, 7-15]). Let C[0,1] denote the set of continuous functions on [0,1] . For f∈C[0,1] , Bernstein introduced the following well-known linear positive operator in the field of real numbers ... : [figure omitted; refer to PDF] where (nk)=n(n-1)...(n-k+1)/k!=n!/k!(n-k)! (see [1, 2, 7, 11, 12, 14]). Here, _...n (f|"x) is called the Bernstein operator of order n for f . For k,n∈_...+ , the Bernstein polynomials of degree n are defined by [figure omitted; refer to PDF] In this paper, we study the properties of q -Euler numbers and polynomials. From these properties, we investigate some identities on the q -Euler numbers and polynomials. Finally, we give some relationships between Bernstein and q -Euler polynomials, which are derived by the p -adic integral representation of the Bernstein polynomials associated with q -Euler polynomials.

2. q -Euler Numbers and Polynomials

In this section, we assume that q∈_...p with |1-q_|p <1 . Let f(x)=^qxext . From (1.1) and (1.2), we have [figure omitted; refer to PDF] Now, we define the q -Euler numbers as follows: [figure omitted; refer to PDF] with the usual convention about replacing ^Eqn by _En,q .

By (2.2), we easily get [figure omitted; refer to PDF] with usual convention about replacing ^Eqn by _En,q .

We note that [figure omitted; refer to PDF] where _Hn (-^q-1 ) is the n th Frobenius-Euler numbers.

From (2.1), (2.2), and (2.4), we have [figure omitted; refer to PDF] Now, we consider the q -Euler polynomials as follows: [figure omitted; refer to PDF] with the usual convention ^Eqn (x) by _En,q (x) .

From (1.2), (2.1), and (2.6), we get [figure omitted; refer to PDF] By comparing the coefficients on the both sides of (2.6) and (2.7), we get the following Witt's formula for the q -Euler polynomials as follows: [figure omitted; refer to PDF] From (2.6) and (2.8), we can derive the following equation: [figure omitted; refer to PDF] By (2.6) and (2.9), we obtain the following reflection symmetric property for the q -Euler polynomials.

Theorem 2.1.

For n∈_...+ , one has [figure omitted; refer to PDF]

From (2.5), (2.6), (2.7), and (2.8), we can derive the following equation: for n∈... , [figure omitted; refer to PDF] by using recurrence formula (2.3). Therefore, we obtain the following theorem.

Theorem 2.2.

For n∈... , one has [figure omitted; refer to PDF]

By using (2.5) and (2.8), we get [figure omitted; refer to PDF] Therefore, we obtain the following theorem.

Theorem 2.3.

For n∈... , one has [figure omitted; refer to PDF]

By using Theorem 2.3, we will study for the p -adic integral representation on _...p of the Bernstein polynomials associated with q -Euler polynomials in Section 3.

3. Bernstein Polynomials Associated with q -Euler Numbers and Polynomials

Now, we take the p -adic integral on _...p for the Bernstein polynomials in (1.7) as follows: [figure omitted; refer to PDF] By the definition of Bernstein polynomials, we see that [figure omitted; refer to PDF] Let n,k∈_...+ with n>k . Then, by (3.2), we get [figure omitted; refer to PDF] Thus, we obtain the following theorem.

Theorem 3.1.

For n,k∈_...+ with n>k , one has [figure omitted; refer to PDF]

By (3.1) and Theorem 3.1, we get the following corollary.

Corollary 3.2.

For n,k∈_...+ with n>k , one has [figure omitted; refer to PDF]

For m,n,k∈_...+ with m+n>2k . Then, we get [figure omitted; refer to PDF] Therefore, we obtain the following theorem.

Theorem 3.3.

For m,n,k∈_...+ with m+n>2k , one has [figure omitted; refer to PDF]

By using binomial theorem, for m,n,k∈_...+ , we get [figure omitted; refer to PDF] [figure omitted; refer to PDF] By comparing the coefficients on the both sides of (3.8) and Theorem 3.3, we obtain the following corollary.

Corollary 3.4.

Let m,n,k∈_...+ with m+n>2k . Then, we get [figure omitted; refer to PDF]

For s∈... , let _n1 ,_n2 ,...,_ns , k∈_...+ with _n1 +_n2 +...+_ns >sk . By induction, we get [figure omitted; refer to PDF] Therefore, we obtain the following theorem.

Theorem 3.5.

Let s∈... . For _n1 ,_n2 ,...,_ns ,k∈_...+ with _n1 +_n2 +...+_ns >sk , one has [figure omitted; refer to PDF]

For _n1 ,_n2 ,...,_ns ,k∈_...+ by binomial theorem, we get [figure omitted; refer to PDF]

By using (3.13) and Theorem 3.5, we obtain the following corollary.

Corollary 3.6.

Let s∈... . For _n1 ,_n2 ,...,_ns ,k∈_...+ with _n1 +_n2 +...+_ns >sk , one has [figure omitted; refer to PDF]

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