- Research Article
- Open Access
© T. Kim et al. 2011
Received: 6 December 2010
Accepted: 15 January 2011
Published: 20 January 2011
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]).
where and (see ).
where and (see ).
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.
Therefore, we obtain the following lemma.
Therefore, by (2.8), we obtain the following proposition.
By using Proposition 2.2 and Lemma 2.1, we obtain the following corollary.
By comparing the coefficients on the both sides of (2.12) and (2.14), we obtain the following theorem.
Therefore, by (2.17), we obtain the following theorem.
Therefore, by (2.18) and (2.19), we obtain the following theorem.
By the induction hypothesis, we obtain the following theorem.
Therefore, by Theorem 2.7 and (2.23), we obtain the following theorem.
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