Open Access

A Study on the -Adic -Integral Representation on Associated with the Weighted -Bernstein and -Bernoulli Polynomials

Journal of Inequalities and Applications20112011:513821

Received: 6 December 2010

Accepted: 15 January 2011

Published: 20 January 2011


We investigate some interesting properties of the weighted -Bernstein polynomials related to the weighted -Bernoulli numbers and polynomials by using -adic -integral on .

1. Introduction and Preliminaries

Let be a fixed prime number. Throughout this paper, , , and will denote the ring of -adic integers, the field of -adic rational numbers, and the completion of the algebraic closure of , respectively. Let be the set of natural numbers, and let . Let be the normalized exponential valuation of with . Let be regarded as either a complex number or a -adic number . If , then we always assume . If , we assume that . In this paper, we define the -number as (see [113]).

Let be the set of continuous functions on . For and , the weighted -Bernstein operator of order for is defined by

Here is called the weighted -Bernstein polynomials of degree (see [2, 5, 6]).

Let be the space of uniformly differentiable functions on . For , the -adic -integral on , which is called the bosonic -integral on , is defined by

(see [10]).

The Carlitz's -Bernoulli numbers are defined by
with the usual convention about replacing by (see [3, 9, 10]). In [3], Carlitz also defined the expansion of Carlitz's -Bernoulli numbers as follows:

with the usual convention about replacing by .

The weighted -Bernoulli numbers are constructed in previous paper [6] as follows: for ,
with the usual convention about replacing by . Let . By the definition (1.2) of -adic -integral on , we easily get
Continuing this process, we obtain easily the relation

where and (see [6]).

Then by (1.2), applying to the function , we can see that
The weighted -Bernoulli polynomials are also defined by the generating function as follows:
(see[6]). Thus, we note that
From (1.2) and the previous equalities, we obtain the Witt's formula for the weighted -Bernoulli polynomials as follows:
By using (1.2) and the weighted -Bernoulli polynomials, we easily get

where and (see [6]).

In this paper, we consider the weighted -Bernstein polynomials to express the bosonic -integral on and investigate some properties of the weighted -Bernstein polynomials associated with the weighted -Bernoulli polynomials by using the expression of -adic -integral on of those polynomials.

2. Weighted -Bernstein Polynomials and -Bernoulli Polynomials

In this section, we assume that and with .

Now we consider the -adic weighted -Bernstein operator as follows:
The -adic -Bernstein polynomials with weight of degree are given by

where , , and (see [6, 7]). Note that . That is, the weighted -Bernstein polynomials are symmetric.

From the definition of the weighted -Bernoulli polynomials, we have
By the definition of -adic -integral on , we get
From (2.3) and (2.4), we have

Therefore, we obtain the following lemma.

Lemma 2.1.

For , one has
By (2.2), (2.3), and (2.4), we get
Thus, we have

Therefore, by (2.8), we obtain the following proposition.

Proposition 2.2.

For with , one has

By using Proposition 2.2 and Lemma 2.1, we obtain the following corollary.

Corollary 2.3.

For with , one has
Taking the bosonic -integral on for one weighted -Bernstein polynomials in (2.1), we have
By the symmetry of -Bernstein polynomials, we get
For , by (2.11) and (2.13), we have

By comparing the coefficients on the both sides of (2.12) and (2.14), we obtain the following theorem.

Theorem 2.4.

For with , one has
In particular, when , one has
Let with . Then we see that

Therefore, by (2.17), we obtain the following theorem.

Theorem 2.5.

For with , one has
For , we have

Therefore, by (2.18) and (2.19), we obtain the following theorem.

Theorem 2.6.

For with , one has
Furthermore, for , one has

By the induction hypothesis, we obtain the following theorem.

Theorem 2.7.

For and with , one has
For , let with . Then we show that

Therefore, by Theorem 2.7 and (2.23), we obtain the following theorem.

Theorem 2.8.

For , let with . Then one sees that for
For , one has

Authors’ Affiliations

Division of General Education-Mathematics, Kwangwoon University
Département de Mathématiques, Université d'Evry Val d'Essonne


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© T. Kim et al. 2011

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