© 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.
- Acikgoz M, Simsek Y: On multiple interpolation functions of the Nörlund-type -Euler polynomials. Abstract and Applied Analysis 2009, 2009:-14.Google Scholar
- Bayad A, Choi J, Kim T, Kim Y-H, Jang LC: -extension of Bernstein polynomials with weighted (;). Journal of Computational and Applied Mathematics. In pressGoogle Scholar
- Carlitz L: Expansions of -Bernoulli numbers. Duke Mathematical Journal 1958, 25: 355–364. 10.1215/S0012-7094-58-02532-8MathSciNetView ArticleMATHGoogle Scholar
- Hegazi AS, Mansour M: A note on -Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2006,13(1):9–18. 10.2991/jnmp.2006.13.1.2MathSciNetView ArticleMATHGoogle Scholar
- Jang L-C, Kim W-J, Simsek Y: A study on the -adic integral representation on associated with Bernstein and Bernoulli polynomials. Advances in Difference Equations 2010, 2010:-6.Google Scholar
- Kim T: On the weighted -Bernoulli numbers and polynomials. http://arxiv.org/abs/1011.5305
- Kim T: A note on -Bernstein polynomials. Russian Journal of Mathematical Physics 2011.,18(1):Google Scholar
- Kim T: Barnes-type multiple -zeta functions and -Euler polynomials. Journal of Physics A 2010,43(25):-11.Google Scholar
- Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar
- Kim T: On a -analogue of the -adic log gamma functions and related integrals. Journal of Number Theory 1999,76(2):320–329. 10.1006/jnth.1999.2373MathSciNetView ArticleMATHGoogle Scholar
- Kupershmidt BA: Reflection symmetries of -Bernoulli polynomials. Journal of Nonlinear Mathematical Physics 2005,12(supplement 1):412–422. 10.2991/jnmp.2005.12.s1.34MathSciNetView ArticleGoogle Scholar
- Ozden H, Cangul IN, Simsek Y: Remarks on -Bernoulli numbers associated with Daehee numbers. Advanced Studies in Contemporary Mathematics 2009,18(1):41–48.MathSciNetMATHGoogle Scholar
- Rim S-H, Jin J-H, Moon E-J, Lee S-J: On multiple interpolation functions of the -Genocchi polynomials. Journal of Inequalities and Applications 2010, 2010:-13.Google Scholar
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