- Research Article
- Open Access

- Eun-Jung Moon
^{1}, - Seog-Hoon Rim
^{2}Email author, - Jeong-Hee Jin
^{1}and - Sun-Jung Lee
^{1}

**2010**:765259

https://doi.org/10.1155/2010/765259

© Eun-Jung Moon et al. 2010

**Received: **14 December 2009

**Accepted: **19 March 2010

**Published: **30 March 2010

## Abstract

In 2009, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher-order, recently. In this paper, we extend our result to the higher-order twisted -Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher-order twisted -Euler numbers and polynomials by the properties of -adic invariant integral on . Especially, if , we derive the result of Kim et al. (2009).

## Keywords

## 1. Introduction

Let be a fixed odd prime number. Throughout this paper, the symbols and will denote the ring of rational integers, the ring of adic integers, the field of adic rational numbers, the complex number field, and the completion of the algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with

When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . If one normally assumes that . If , then we assume that so that for each We use the following notation:

For a fixed positive integer with , set

where satisfies the condition For any

(see [1–13]) is known to be a distribution on .

We say that is a uniformly differentiable function at and denote this property by if the difference quotients

For the fermionic -adic invariant -integral on is defined as

(see [14]). Let us define the fermionic -adic invariant integral on as follows:

(see [1–12, 14–20]). From the definition of integral, we have

For , let be the adic locally constant space defined by

where for some is the cyclic group of order

It is well known that the twisted Euler polynomials of order are defined as

and are called the twisted Euler numbers of order When the polynomials and numbers are called the twisted Euler polynomials and numbers, respectively. When and , the polynomials and numbers are called the twisted Euler polynomials and numbers, respectively. When , and , the polynomials and numbers are called the ordinary Euler polynomials and numbers, respectively.

In [15], Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher order, recently. In this paper, we extend our result to the higher-order twisted Euler numbers and polynomials.

The purpose of this paper is to establish various identities concerning higher-order twisted -Euler numbers and polynomials by the properties of adic invariant integral on . Especially, if , we derive the result of [15].

## 2. Some Identities of the Higher-Order Twisted Euler Numbers and Polynomials

In (2.1), we note that is symmetric in and .

From (2.1), we derive that

From the definition of integral, we also see that

It is easy to see that

From (2.3), (2.4), and (2.5), we can derive

From the symmetry of in and , we also see that

Comparing the coefficients on the both sides of (2.6) and (2.7), we obtain an identity for the twisted Euler polynomials of higher order as follows.

Theorem 2.1.

Remark 2.2.

Moreover, if we take and in Theorem 2.1, then we have the following identity for the twisted Euler numbers of higher order.

Corollary 2.3.

We also note that taking in Corollary shows the following identity:

Now we will derive another interesting identities for the twisted -Euler numbers and polynomials of higher order. From (2.3), we can derive that

From the symmetry of in and , we see that

Comparing the coefficients on the both sides of (2.12) and (2.13), we obtain the following theorem which shows the relationship between the power sums and the twisted Euler polynomials.

Theorem 2.4.

Remark 2.5.

Moreover, if we take and in Theorem 2.4, then we have the following identity for the twisted Euler numbers of higher order.

Corollary 2.6.

Remark 2.7.

If we can observe the result of [15].

## Authors’ Affiliations

## References

- Kim T:
**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51-57.MathSciNetView ArticleMATHGoogle 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/S1061920807020045MathSciNetView ArticleMATHGoogle 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/S1061920809010063MathSciNetView ArticleMATHGoogle 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.2007.07.027MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar - Howard FT:
**Applications of a recurrence for the Bernoulli numbers.***Journal of Number Theory*1995,**52**(1):157-172. 10.1006/jnth.1995.1062MathSciNetView ArticleMATHGoogle Scholar - Kupershmidt BA:
**Reflection symmetries of****-Bernoulli polynomials.***Journal of Nonlinear Mathematical Physics*2005,**12**(supplement 1):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.MathSciNetMATHGoogle Scholar - Schork M:
**Ward's "calculus of sequences",****-calculus and the limit**.*Advanced Studies in Contemporary Mathematics*2006,**13**(2):131-141.MathSciNetMATHGoogle Scholar - Tuenter HJH:
**A symmetry of power sum polynomials and Bernoulli numbers.***The American Mathematical Monthly*2001,**108**(3):258-261. 10.2307/2695389MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**Note on the Euler****-zeta functions.***Journal of Number Theory*2009,**129**(7):1798-1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**Symmetry****-adic invariant integral on****for Bernoulli and Euler polynomials.***Journal of Difference Equations and Applications*2008,**14**(12):1267-1277. 10.1080/10236190801943220MathSciNetView ArticleMATHGoogle Scholar - Kim T, Park KH, Hwang K-W:
**On the identities of symmetry for the****-Euler polynomials of higher order.***Advances in Difference Equations*2009,**2009:**-9.Google Scholar - Kim T:
**Note on the Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(2):131-136.MathSciNetMATHGoogle Scholar - Kim T:
**Note on****-Genocchi numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(1):9-15.MathSciNetMATHGoogle Scholar - Kim T:
**The modified****-Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161-170.MathSciNetMATHGoogle 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.1999.2373MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288-299.MathSciNetMATHGoogle 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.