- Research Article
- Open Access

# On the Symmetric Properties of the Multivariate -Adic Invariant Integral on Associated with the Twisted Generalized Euler Polynomials of Higher Order

- Taekyun Kim
^{1}, - Byungje Lee
^{2}and - Young-Hee Kim
^{1}Email author

**2010**:826548

https://doi.org/10.1155/2010/826548

© Taekyun Kim et al. 2010

**Received:**6 November 2009**Accepted:**14 March 2010**Published:**30 March 2010

## Abstract

We study the symmetric properties for the multivariate -adic invariant integral on related to the twisted generalized Euler polynomials of higher order.

## Keywords

- Positive Integer
- Natural Number
- Prime Number
- Rational Number
- Differentiable Function

## 1. Introduction

For , let be the cyclic group of order . That is, . The -adic locally constant space, , is defined by

Recently, the symmetry identities for the generalized twisted Bernoulli polynomials and the generalized twisted Bernoulli polynomials of order are studied in [4, 12].

In this paper, we study the symmetric properties of the multivariate -adic invariant integral on . From these symmetric properties, we derive the symmetry identities for the twisted generalized Euler polynomials of higher order. In [14], Kim gave the relation between the power sum polynomials and the generalized higher-order Euler polynomials. The main purpose of this paper is to give the symmetry identities for the twisted generalized Euler polynomials of higher order using the symmetric properties of the multivariate -adic invariant integral on .

## 2. Symmetry Identities for the Twisted Generalized Euler Polynomials of Higher Order

In the special case , are called the th twisted generalized Euler numbers attached to .

In the special case , are called the th twisted generalized Euler numbers of order .

Comparing the coefficients on both sides of (2.11) and (2.12), we obtain the following theorem.

Theorem 2.1.

Let and in Theorem 2.1. Then we also have the following corollary.

Corollary 2.2.

Let be the trivial character and . Then we also have the following corollary.

Corollary 2.3.

where are the th twisted Euler polynomials.

If we take in Corollary 2.3, then we obtain the following corollary.

Corollary 2.4 (Distribution for the twisted Euler polynomials).

Comparing the coefficients on both sides of (2.17) and (2.18), we obtain the following theorem which shows the relationship between the power sums and the twisted generalized Euler polynomials of higher order.

Theorem 2.5.

## Declarations

### Acknowledgment

The present research has been conducted by the research grant of Kwangwoon University in 2010.

## 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.