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

- T Kim
^{1}, - A Bayad
^{2}Email author and - Y-H Kim
^{1}

**2011**:513821

https://doi.org/10.1155/2011/513821

© T. Kim et al. 2011

**Received: **6 December 2010

**Accepted: **15 January 2011

**Published: **20 January 2011

## Abstract

## 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 [1–13]).

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

(see [10]).

with the usual convention about replacing by .

where and (see [6]).

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 .

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

Therefore, we obtain the following lemma.

Lemma 2.1.

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

Proposition 2.2.

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

Corollary 2.3.

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

Theorem 2.4.

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

Theorem 2.5.

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

Theorem 2.6.

By the induction hypothesis, we obtain the following theorem.

Theorem 2.7.

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

Theorem 2.8.

## Authors’ Affiliations

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## Copyright

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