Open Access

On the Twisted -Analogs of the Generalized Euler Numbers and Polynomials of Higher Order

Advances in Difference Equations20102010:875098

DOI: 10.1155/2010/875098

Received: 12 April 2010

Accepted: 28 June 2010

Published: 13 July 2010

Abstract

We consider the twisted -extensions of the generalized Euler numbers and polynomials attached to .

1. Introduction and Preliminaries

Let be an odd prime number. For , let be the cyclic group of order , and let be the space of locally constant functions in the -adic number field . When one talks of -extension, is variously considered as an indeterminate, a complex number , or -adic number . If , one normally assumes that . If , one normally assumes that . In this paper, we use the notation
(1.1)
Let be a fixed positive odd integer. For , we set
(1.2)

where lies in compared to [116].

Let be the Dirichlet's character with an odd conductor . Then the generalized -Euler polynomials attached to , , are defined as
(1.3)
In the special case , are called the th -Euler numbers attached to . For , the -adic fermionic integral on is defined by
(1.4)
Let . Then, we see that
(1.5)
For , let . Then, we have
(1.6)
Thus, we have
(1.7)
By (1.7), we see that
(1.8)
From (1.8), we can derive the Witt's formula for as follows:
(1.9)
The th generalized -Euler polynomials of order , , are defined as
(1.10)

In the special case , are called the th -Euler numbers of order attached to .

Now, we consider the multivariate -adic invariant integral on as follows:
(1.11)
By (1.10) and (1.11), we see the Witt's formula for as follows:
(1.12)

The purpose of this paper is to present a systemic study of some formulas of the twisted -extension of the generalized Euler numbers and polynomials of order attached to .

2. On the Twisted -Extension of the Generalized Euler Polynomials

In this section, we assume that with and . For with , let be the Dirichlet's character with conductor . For , let us consider the twisted -extension of the generalized Euler numbers and polynomials of order attached to . We firstly consider the twisted -extension of the generalized Euler polynomials of higher order as follows:
(2.1)
By (2.1), we see that
(2.2)
From the multivariate fermionic -adic invariant integral on , we can derive the twisted -extension of the generalized Euler polynomials of order attached to as follows:
(2.3)
Thus, we have
(2.4)
Let be the generating function for . By (2.3), we easily see that
(2.5)

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has
(2.6)
Let . Then we define the extension of as follows:
(2.7)
Then, are called the th generalized -Euler polynomials of order attached to . In the special case , are called the th generalized -Euler numbers of order . By (1.7), we obtain the Witt's formula for as follows:
(2.8)

where .

Let where . From (2.8), we note that
(2.9)
Let be the generating function for . From (2.8), we can easily derive
(2.10)

By (2.10), we obtain the following theorem.

Theorem 2.2.

For , , one has
(2.11)
Let . Then we see that
(2.12)
It is easy to see that
(2.13)
Thus, we have
(2.14)

By (2.14), we obtain the following theorem.

Theorem 2.3.

For with , one has
(2.15)
By (1.7), we easily see that
(2.16)
Thus,we have
(2.17)

By (2.17), we obtain the following theorem.

Theorem 2.4.

For with , one has
(2.18)
It is easy to see that
(2.19)
Let . Then we note that
(2.20)
From (2.20), we can derive
(2.21)

3. Further Remark

In this section, we assume that with . Let be the Dirichlet's character with an odd conductor . From the Mellin transformation of in (2.10), we note that
(3.1)

where , and , . By (3.1), we can define the Dirichlet's type multiple - -function as follows.

Definition 3.1.

For , with , one defines the Dirichlet's type multiple - -function related to higher order -Euler polynomials as
(3.2)

where , , , and .

Note that is analytic continuation in whole complex -plane. In (2.10), we note that
(3.3)

By Laurent series and Cauchy residue theorem in (3.1) and (3.3), we obtain the following theorem.

Theorem 3.2.

Let be Dirichlet's character with odd conductor and let . For , , and , one has
(3.4)

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, KonKuk University
(2)
Department of Wireless Communications Engineering, Kwangwoon University
(3)
Division of General Education-Mathematics, Kwangwoon University

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Copyright

© Lee Chae Jang et al. 2010

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.