#### Abstract

By using the ordinary fermionic -adic invariant integral on , we derive some interesting identities related to the Frobenius-Euler polynomials.

#### 1. Introduction

Let be a fixed prime. Throughout this paper and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of When one talks about -extension, is variously considered as an indeterminate, a complex , or a -adic number ; see [1–14]. If , then we assume If , then we assume For we use the notation and ; see [15, 16]. The normalized valuation in is denoted by with We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as . For , let us start with the expressionrepresenting a -analogue of Riemann sums for ; see [15, 16]. The integral of on will be defined as a limit () of those sums, when it exists. The -deformed bosonic -adic integral of the function is defined by see [15]. Thus, we note thatwhere

The fermionic -adic invariant integral on is defined as see [15].

In this paper, we prove an identity of symmetry for the Frobenius-Euler polynomials. Finally we investigate the several further interesting properties of the symmetry for the fermionic -adic invariant integral on related to the Frobenius-Euler polynomials and numbers.

#### 2. Some Identities of the Frobenius-Euler Polynomials

Let (or ) be algebraic. Then the th Frobenius-Euler numbers are defined as with the usual convention about replacing by

The th Frobenius-Euler polynomials are also defined asFrom (1.4), we can easily derive By continuing this process, we see thatWhen is an odd positive integer, we obtainIf with , then we haveFrom (1.4) and (2.3), we deriveThus, we note that Let with . Then we obtain For with , we have By substituting into (2.5), we can easily see thatLet . Then is called the alternating sums of powers of consecutive -integers. From the definition of the fermionic -adic invariant integral on , we can derive

By (2.12), we easily see that

Let be odd. By using double fermionic -adic invariant integral on , we obtain

Now we also consider the following fermionic -adic invariant integral on associated with Frobenius-Euler polynomials:

From (2.15) and (2.12), we can derive Let

By (2.15), (2.16), and (2.17), we see that From (2.17) we derive By (2.16) and (2.19), we see that By the symmetry of -adic invariant integral on , we also see thatwhere are the th Frobenius-Euler polynomials.

By comparing the coefficients on the both sides of (2.20) and (2.21), we obtain the following theorem.

Theorem 2.1. * For with , , , one has **where are the th
Frobenius-Euler polynomials.*

If we take in Theorem 2.1, then we have

From (2.11) and (2.12), we deriveFrom the symmetry of , we note that By comparing the coefficients on the both sides of (2.24) and (2.25), we obtain the following theorem.

Theorem 2.2. * Let be odd, and let with . Then, one has*

By setting in Theorem 2.2, we get the multiplication theorem for the Frobenius-Euler polynomials as follows:

*Remark 2.3. *By using the fermionic -adic invariant -integral on , the symmetric properties related to Frobenius-Euler
polynomials are studied in [17]. In this paper, we have studied the symmetric
properties of Frobenius-Euler polynomials, which are different from the symmetric
properties treated in a previous paper [17]. To derive the symmetric properties
of Frobenius-Euler polynomials, we used the ordinary fermionic -adic invariant
integrals on in this paper.

#### Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2008.