Discrete Dynamics in Nature and Society

VolumeΒ 2011Β (2011), Article IDΒ 176296, 11 pages

http://dx.doi.org/10.1155/2011/176296

## Some Relations between Twisted ()-Euler Numbers with Weight *Ξ±* and -Bernstein Polynomials with Weight *Ξ±*

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 19 July 2011; Accepted 26 August 2011

Academic Editor: JohnΒ Rassias

Copyright Β© 2011 N. S. Jung et al. 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.

#### Abstract

By using fermionic *p*-adic *q*-integral on , we give some
interesting relationship between the twisted (*h, q*)-Euler numbers with weight *Ξ±* and the
*q*-Bernstein polynomials.

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, we always make use of the following notations: denotes the ring of rational integers, denotes the ring of -adic rational integers, denotes the field of -adic rational numbers, and denotes the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the cyclic group of order , and let (see [1β22]), be the locally constant space. For , we denote by the locally constant function . The -adic absolute value is defined by , where . In this paper, we assume that with as an indeterminate. The -number is defined by (see [1β22]). Note that . For the fermionic -adic -integral on is defined by Kim as follows: (see [1β7]). From (1.4), we note that where for .

For and , Kim defined -Bernstein polynomials, which are different -Bernstein polynomials of Phillips, as follows: (see [5]). In [9], the -adic extension of (1.6) is given by For , , , and with , twisted -Euler numbers with weight are defined by In the special case, , are called the -th twisted -Euler numbers with weight .

In this paper, we investigate some relations between the -Bernstein polynomials and the twisted -Euler numbers with weight . From these relations, we derive some interesting identities on the twisted -Euler numbers and polynomials with weight .

#### 2. Twisted -Euler Numbers and Polynomials with Weight

By using -adic -integral and (1.8), we obtain We set By (2.1) and (2.2), we have Since , we obtain

Therefore, we obtain the following theorem.

Theorem 2.1. *For and , we have
**
Furthermore,
**
with usual convention about replacing with .**Let . Then we see that
**
In the special case, , let .**By (2.1), we get
**
From (2.3) and (2.7), we note that
**
By (2.9), we get the following recurrence formula:
*

By (2.10) and Theorem 2.1, we obtain the following theorem.

Theorem 2.2. *For and , we have
**
with usual convention about replacing with .**By (2.4), Theorem 2.1, and Theorem 2.2, we have
*

Therefore, we obtain the following theorem.

Theorem 2.3. *For , we have
**By (2.8), we see that
*

Therefore, we obtain the following theorem.

Theorem 2.4. *For , we have
**
Let . By Theorems 2.3 and 2.4, we get
**From (2.16), we have
*

Therefore, we obtain the following corollary.

Corollary 2.5. *For , we have
**Kim defined the -Bernstein polynomials with weight of degree as below.**For , the -adic -Bernstein polynomials with weight of degree are given by
**
compare [5, 10, 22] By (2.19), we get the symmetry of -Bernstein polynomials as follows:
**
see [8]. Thus, by Corollary 2.5, (2.19), and (2.20), we see that
**For with , we have
**
Let us take the fermionic -integral on for the -Bernstein polynomials with weight of degree as follows:
*

Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 2.6. *Let with . Then we have
**
Moreover,
**Let with . Then we get
*

Therefore, by (2.26), we obtain the following theorem.

Theorem 2.7. *For with , we have
**From the binomial theorem, we can derive the following equation:
*

Thus, by (2.28) and Theorem 2.7, we obtain the following corollary.

Corollary 2.8. *Let with . Then we have
**For and with , let with . Then we take the fermionic -adic -integral on for the -Bernstein polynomials with weight of degree as follows:
*

Therefore, by (2.30), we obtain the following theorem.

Theorem 2.9. *For with , let with . Then we get
**By the definition of -Bernstein polynomials with weight and the binomial theorem, we easily get
*

Therefore, we have the following corollary.

Corollary 2.10. *For with , let with . Then we have
*

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