# 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

**DOI: **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 .

## 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

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