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

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

(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)