- Research Article
- Open Access

# On the Identities of Symmetry for the -Euler Polynomials of Higher Order

- Taekyun Kim
^{1}Email author, - KyoungHo Park
^{2}and - Kyung-won Hwang
^{3}

**2009**:273545

https://doi.org/10.1155/2009/273545

© Taekyun Kim et al. 2009

**Received:**19 February 2009**Accepted:**18 June 2009**Published:**20 July 2009

## Abstract

The main purpose of this paper is to investigate several further interesting properties of symmetry for the multivariate -adic fermionic integral on . From these symmetries, we can derive some recurrence identities for the -Euler polynomials of higher order, which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the -Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation

## 1. Introduction/Definition

(cf. [1–8, 11–18]). For each positive integer , let . Then we have

Then the values of at are called the -Euler numbers of order . When , the polynomials or numbers are called the -Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariate -adic fermionic integral on . From the properties of symmetry for the multivariate -adic fermionic integral on , we derive some identities of symmetry for the -Euler polynomials of higher order. By using our identities of symmetry for the -Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.

## 2. On the Symmetry for the -Euler Polynomials of Higher Order

By comparing the coefficients on both sides of (2.7) and (2.8), we obtain the following.

Theorem 2.1.

By the symmetric property of in , we also see that

By comparing the coefficients on both sides of (2.13) and (2.14), we obtain the following theorem.

Theorem 2.2.

## Declarations

### Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

## 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.