# 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

**DOI: **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.

## 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

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