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

*Journal of Inequalities and Applications*
**volume 2011**, Article number: 513821 (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 [1–13]).

Let be the set of continuous functions on . For and , the weighted -Bernstein operator of order for is defined by

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

(see [10]).

The Carlitz's -Bernoulli numbers are defined by

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:

with the usual convention about replacing by .

The weighted -Bernoulli numbers are constructed in previous paper [6] as follows: for ,

with the usual convention about replacing by . Let . By the definition (1.2) of -adic -integral on , we easily get

Continuing this process, we obtain easily the relation

where and (see [6]).

Then by (1.2), applying to the function , we can see that

The weighted -Bernoulli polynomials are also defined by the generating function as follows:

(see[6]). Thus, we note that

From (1.2) and the previous equalities, we obtain the Witt's formula for the weighted -Bernoulli polynomials as follows:

By using (1.2) and the weighted -Bernoulli polynomials, we easily get

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:

The -adic -Bernstein polynomials with weight of degree are given by

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

By the definition of -adic -integral on , we get

From (2.3) and (2.4), we have

Therefore, we obtain the following lemma.

Lemma 2.1.

For , one has

By (2.2), (2.3), and (2.4), we get

Thus, we have

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

Proposition 2.2.

For with , one has

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

Corollary 2.3.

For with , one has

Taking the bosonic -integral on for one weighted -Bernstein polynomials in (2.1), we have

By the symmetry of -Bernstein polynomials, we get

For , by (2.11) and (2.13), we have

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

In particular, when , one has

Let with . Then we see that

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

Theorem 2.5.

For with , one has

For , we have

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

Theorem 2.6.

For with , one has

Furthermore, for , one has

By the induction hypothesis, we obtain the following theorem.

Theorem 2.7.

For and with , one has

For , let with . Then we show that

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

Theorem 2.8.

For , let with . Then one sees that for

For , one has

## References

Acikgoz M, Simsek Y:

**On multiple interpolation functions of the Nörlund-type****-Euler polynomials.***Abstract and Applied Analysis*2009,**2009:**-14.Bayad A, Choi J, Kim T, Kim Y-H, Jang LC:

**-extension of Bernstein polynomials with weighted (;).***Journal of Computational and Applied Mathematics*. In pressCarlitz L:

**Expansions of****-Bernoulli numbers.***Duke Mathematical Journal*1958,**25:**355–364. 10.1215/S0012-7094-58-02532-8Hegazi AS, Mansour M:

**A note on****-Bernoulli numbers and polynomials.***Journal of Nonlinear Mathematical Physics*2006,**13**(1):9–18. 10.2991/jnmp.2006.13.1.2Jang 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.Kim T:

**On the weighted -Bernoulli numbers and polynomials.**http://arxiv.org/abs/1011.5305Kim T:

**A note on -Bernstein polynomials.***Russian Journal of Mathematical Physics*2011.,**18**(1):Kim T:

**Barnes-type multiple -zeta functions and -Euler polynomials.***Journal of Physics A*2010,**43**(25):-11.Kim T:

**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51–57.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.1999.2373Kupershmidt BA:

**Reflection symmetries of****-Bernoulli polynomials.***Journal of Nonlinear Mathematical Physics*2005,**12**(supplement 1):412–422. 10.2991/jnmp.2005.12.s1.34Ozden H, Cangul IN, Simsek Y:

**Remarks on****-Bernoulli numbers associated with Daehee numbers.***Advanced Studies in Contemporary Mathematics*2009,**18**(1):41–48.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.

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Kim, T., Bayad, A. & Kim, YH. A Study on the -Adic -Integral Representation on Associated with the Weighted -Bernstein and -Bernoulli Polynomials.
*J Inequal Appl* **2011**, 513821 (2011). https://doi.org/10.1155/2011/513821

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

### Keywords

- Continuous Function
- Generate Function
- Natural Number
- Complex Number
- Induction Hypothesis