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

*Advances in Difference Equations*
**volume 2009**, Article number: 273545 (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

Let be a fixed odd prime number. Throughout this paper, and will, respectively, denote the ring of -adic rational integer, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the normalized exponential valuation of with . Let be the space of uniformly differentiable functions on . For , with , the fermionic -adic -integral on is defined as

(see [1]). Let us define the fermionic -adic invariant integral on as follows:

(see [1–8]). From (1.2), we have

(see [9, 10]), where . For with , let . Then, we define the -Euler numbers as follows:

where are called the -Euler numbers. We can show that

where are the Frobenius-Euler numbers. By comparing the coefficients on both sides of (1.4) and (1.5), we see that

Now, we also define the -Euler polynomials as follows:

In the viewpoint of (1.5), we can show that

where are the th Frobenius-Euler polynomials. From (1.7) and (1.8), we note that

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

The -Euler polynomials of order , denoted , are defined as

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

Let with (mod 2) and . Then we set

where

Thus, we note that this expression for is symmetry in and . From (2.1), we have

We can show that

By (1.4) and (1.11), we see that

Thus, we have

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

By the same method, we also see that

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

Theorem 2.1.

For with , , and , one has

Let and in (2.9). Then we have

From (2.10), we note that

If we take in (2.11), then we have

From (2.3), we note that

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.

For with and , one has

Let and , we have

From (2.16), we can derive

## References

- 1.
Kim T:

**Symmetry -adic invariant integral on for Bernoulli and Euler polynomials.***Journal of Difference Equations and Applications*2008,**14**(12):1267–1277. 10.1080/10236190801943220 - 2.
Kim T:

**Note on the Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(2):131–136. - 3.
Kim T:

**Note on -Genocchi numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(1):9–15. - 4.
Kim T:

**The modified -Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161–170. - 5.
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.1999.2373 - 6.
Kim T:

**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288–299. - 7.
Kim T:

**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51–57. - 8.
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/S1061920807020045 - 9.
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/S1061920809010063 - 10.
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.2007.07.027 - 11.
Agarwal RP, Ryoo CS:

**Numerical computations of the roots of the generalized twisted -Bernoulli polynomials.***Neural, Parallel & Scientific Computations*2007,**15**(2):193–206. - 12.
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. - 13.
Howard FT:

**Applications of a recurrence for the Bernoulli numbers.***Journal of Number Theory*1995,**52**(1):157–172. 10.1006/jnth.1995.1062 - 14.
Kupershmidt BA:

**Reflection symmetries of -Bernoulli polynomials.***Journal of Nonlinear Mathematical Physics*2005,**12:**412–422. 10.2991/jnmp.2005.12.s1.34 - 15.
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.2008.01.020 - 16.
Jang L-C:

**A study on the distribution of twisted -Genocchi polynomials.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):181–189. - 17.
Schork M:

**Ward's "calculus of sequences", -calculus and the limit .***Advanced Studies in Contemporary Mathematics*2006,**13**(2):131–141. - 18.
Tuenter HJH:

**A symmetry of power sum polynomials and Bernoulli numbers.***The American Mathematical Monthly*2001,**108**(3):258–261. 10.2307/2695389

## Acknowledgment

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

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kim, T., Park, K. & Hwang, Kw. On the Identities of Symmetry for the -Euler Polynomials of Higher Order.
*Adv Differ Equ* **2009, **273545 (2009). https://doi.org/10.1155/2009/273545

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### Keywords

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