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  • Research Article
  • Open Access

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

Advances in Difference Equations20102010:875098

https://doi.org/10.1155/2010/875098

  • Received: 12 April 2010
  • Accepted: 28 June 2010
  • Published:

Abstract

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

Keywords

  • Prime Number
  • Analytic Continuation
  • Cyclic Group
  • Number Field
  • Euler Number

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, Chungju, 138-701, Republic of Korea
(2)
Department of Wireless Communications Engineering, Kwangwoon University, Seoul, 139-701, Republic of Korea
(3)
Division of General Education-Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea

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