- 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

## Keywords

## 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.

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.

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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