Open Access

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

Journal of Inequalities and Applications20112011:513821

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

Received: 6 December 2010

Accepted: 15 January 2011

Published: 20 January 2011

Abstract

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
(1.1)

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
(1.2)

(see [10]).

The Carlitz's -Bernoulli numbers are defined by
(1.3)
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:
(1.4)

with the usual convention about replacing by .

The weighted -Bernoulli numbers are constructed in previous paper [6] as follows: for ,
(1.5)
with the usual convention about replacing by . Let . By the definition (1.2) of -adic -integral on , we easily get
(1.6)
Continuing this process, we obtain easily the relation
(1.7)

where and (see [6]).

Then by (1.2), applying to the function , we can see that
(1.8)
The weighted -Bernoulli polynomials are also defined by the generating function as follows:
(1.9)
(see[6]). Thus, we note that
(1.10)
From (1.2) and the previous equalities, we obtain the Witt's formula for the weighted -Bernoulli polynomials as follows:
(1.11)
By using (1.2) and the weighted -Bernoulli polynomials, we easily get
(1.12)

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:
(2.1)
The -adic -Bernstein polynomials with weight of degree are given by
(2.2)

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
(2.3)
By the definition of -adic -integral on , we get
(2.4)
From (2.3) and (2.4), we have
(2.5)

Therefore, we obtain the following lemma.

Lemma 2.1.

For , one has
(2.6)
By (2.2), (2.3), and (2.4), we get
(2.7)
Thus, we have
(2.8)

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

Proposition 2.2.

For with , one has
(2.9)

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

Corollary 2.3.

For with , one has
(2.10)
(2.11)
Taking the bosonic -integral on for one weighted -Bernstein polynomials in (2.1), we have
(2.12)
By the symmetry of -Bernstein polynomials, we get
(2.13)
For , by (2.11) and (2.13), we have
(2.14)

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
(2.15)
In particular, when , one has
(2.16)
Let with . Then we see that
(2.17)

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

Theorem 2.5.

For with , one has
(2.18)
For , we have
(2.19)

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

Theorem 2.6.

For with , one has
(2.20)
Furthermore, for , one has
(2.21)

By the induction hypothesis, we obtain the following theorem.

Theorem 2.7.

For and with , one has
(2.22)
For , let with . Then we show that
(2.23)

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

Theorem 2.8.

For , let with . Then one sees that for
(2.24)
For , one has
(2.25)

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University
(2)
Département de Mathématiques, Université d'Evry Val d'Essonne

References

  1. Acikgoz M, Simsek Y: On multiple interpolation functions of the Nörlund-type -Euler polynomials. Abstract and Applied Analysis 2009, 2009:-14.Google Scholar
  2. 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
  3. Carlitz L: Expansions of -Bernoulli numbers. Duke Mathematical Journal 1958, 25: 355–364. 10.1215/S0012-7094-58-02532-8MathSciNetView ArticleMATHGoogle Scholar
  4. 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
  5. 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
  6. Kim T: On the weighted -Bernoulli numbers and polynomials. http://arxiv.org/abs/1011.5305
  7. Kim T: A note on -Bernstein polynomials. Russian Journal of Mathematical Physics 2011.,18(1):Google Scholar
  8. Kim T: Barnes-type multiple -zeta functions and -Euler polynomials. Journal of Physics A 2010,43(25):-11.Google Scholar
  9. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar
  10. 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
  11. 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
  12. 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
  13. 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

Copyright

© T. Kim et al. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.