- Research Article
- Open Access

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

- Lee Chae Jang
^{1}Email author, - Byungje Lee
^{2}and - Taekyun Kim
^{3}

**2010**:875098

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

© Lee Chae Jang et al. 2010

**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 .

## Keywords

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

## 1. Introduction and Preliminaries

where lies in compared to [1–16].

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

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

Therefore, we obtain the following theorem.

Theorem 2.1.

where .

By (2.10), we obtain the following theorem.

Theorem 2.2.

By (2.14), we obtain the following theorem.

Theorem 2.3.

By (2.17), we obtain the following theorem.

Theorem 2.4.

## 3. Further Remark

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

Definition 3.1.

where , , , and .

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

Theorem 3.2.

## Authors’ Affiliations

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## Copyright

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.