- Research Article
- Open Access
A Study on the -Adic -Integral Representation on Associated with the Weighted -Bernstein and -Bernoulli Polynomials
© T. Kim et al. 2011
- 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 .
- Continuous Function
- Generate Function
- Natural Number
- Complex Number
- Induction Hypothesis
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]).
with the usual convention about replacing by .
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.
In this section, we assume that and with .
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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