- Research Article
- Open Access

- 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

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Kim T:
**Symmetry -adic invariant integral on for Bernoulli and Euler polynomials.***Journal of Difference Equations and Applications*2008,**14**(12):1267–1277. 10.1080/10236190801943220MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**Note on the Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(2):131–136.MATHMathSciNetGoogle Scholar - Kim T:
**Note on -Genocchi numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(1):9–15.MATHMathSciNetGoogle Scholar - Kim T:
**The modified -Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161–170.MATHMathSciNetGoogle Scholar - Kim T:
**On a -analogue of the -adic log gamma functions and related integrals.***Journal of Number Theory*1999,**76**(2):320–329. 10.1006/jnth.1999.2373MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288–299.MATHMathSciNetGoogle Scholar - Kim T:
**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51–57.MATHMathSciNetView ArticleGoogle Scholar - Kim T, Choi JY, Sug JY:
**Extended -Euler numbers and polynomials associated with fermionic -adic -integral on .***Russian Journal of Mathematical Physics*2007,**14**(2):160–163. 10.1134/S1061920807020045MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on .***Russian Journal of Mathematical Physics*2009,**16**(1):93–96. 10.1134/S1061920809010063MATHMathSciNetView ArticleGoogle Scholar - Kim T:
**On -adic interpolating function for -Euler numbers and its derivatives.***Journal of Mathematical Analysis and Applications*2008,**339**(1):598–608. 10.1016/j.jmaa.2007.07.027MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Ryoo CS:
**Numerical computations of the roots of the generalized twisted -Bernoulli polynomials.***Neural, Parallel & Scientific Computations*2007,**15**(2):193–206.MATHMathSciNetGoogle Scholar - Cenkci M, Can M, Kurt V:
**-adic interpolation functions and Kummer-type congruences for -twisted and -generalized twisted Euler numbers.***Advanced Studies in Contemporary Mathematics*2004,**9**(2):203–216.MATHMathSciNetGoogle Scholar - Howard FT:
**Applications of a recurrence for the Bernoulli numbers.***Journal of Number Theory*1995,**52**(1):157–172. 10.1006/jnth.1995.1062MATHMathSciNetView ArticleGoogle Scholar - Kupershmidt BA:
**Reflection symmetries of -Bernoulli polynomials.***Journal of Nonlinear Mathematical Physics*2005,**12:**412–422. 10.2991/jnmp.2005.12.s1.34MathSciNetView ArticleGoogle Scholar - Ozden H, Simsek Y:
**Interpolation function of the -extension of twisted Euler numbers.***Computers & Mathematics with Applications*2008,**56**(4):898–908. 10.1016/j.camwa.2008.01.020MathSciNetView ArticleGoogle Scholar - Jang L-C:
**A study on the distribution of twisted -Genocchi polynomials.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):181–189.MATHMathSciNetGoogle Scholar - Schork M:
**Ward's "calculus of sequences", -calculus and the limit .***Advanced Studies in Contemporary Mathematics*2006,**13**(2):131–141.MATHMathSciNetGoogle Scholar - Tuenter HJH:
**A symmetry of power sum polynomials and Bernoulli numbers.***The American Mathematical Monthly*2001,**108**(3):258–261. 10.2307/2695389MATHMathSciNetView ArticleGoogle Scholar

## 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.