- Research Article
- Open Access
On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials
- T Kim^{1}Email author,
- J Choi^{1},
- YH Kim^{1} and
- CS Ryoo^{2}
https://doi.org/10.1155/2010/864247
© T. Kim et al. 2010
- Received: 30 August 2010
- Accepted: 3 December 2010
- Published: 14 December 2010
Abstract
The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.
Keywords
- Continuous Function
- Real Number
- Simple Calculation
- Differentiable Function
- Positive Operator
1. Introduction
Throughout this paper, let be an odd prime number. The symbol, , , and denote the ring of -adic integers, the field of -adic rational numbers, the complex number field and the completion of algebraic closure of , respectively.
Let be the set of natural numbers and . Let be the normalized exponential valuation of with . Note that .
When one talks of -extension, is variously considered as an indeterminate, a complex number , or -adic number . If , we normally assume , and if , we always assume .
where .
(see [1–15]). In the special case, , and are called the th Euler numbers.
where (see [3, 4, 7, 10, 11, 14]). Here, is called the Bernstein operator of order for .
For example, , , , , , and for , .
In this paper, we study the properties of Bernstein polynomials in the -adic number field. For , we give some properties of several type Bernstein polynomials to represent the fermionic -adic invariant integral on . From those properties, we derive some interesting identities on the Euler polynomials.
2. Fermionic -adic Integral Representation of Bernstein Polynomials
Therefore, we obtain the following theorem.
Theorem 2.1.
Theorem 2.1 is important to derive our main result in this paper.
Therefore, we obtain the following proposition.
Proposition 2.2.
From (2.9), we obtain the following theorem.
Theorem 2.3.
By Proposition 2.2 and Theorem 2.3, we obtain the following corollary.
Corollary 2.4.
Therefore, we obtain the following theorem.
Theorem 2.5.
Thus, we obtain the following proposition.
Proposition 2.6.
By Theorem 2.5 and Proposition 2.6, we obtain the following corollary.
Corollary 2.7.
where with .
Therefore, we obtain the following theorem.
Theorem 2.8.
By (2.17) and Theorem 2.8, we obtain the following corollary.
Corollary 2.9.
Using the above theorems and mathematical induction, we obtain the following theorem.
Theorem 2.10.
By Theorem 2.10 and (2.22), we obtain the following corollary.
Corollary 2.11.
Therefore, we obtain the following theorem.
Theorem 2.12.
where for . By Theorem 2.12 and (2.26), we obtain the following corollary.
Corollary 2.13.
where with .
Therefore, we obtain the following theorem.
Theorem 2.14.
Let .
(i)For with , one has
(ii)For , one has
By Theorem 2.14, we obtain the following corollary.
Corollary 2.15.
Authors’ Affiliations
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