Abstract

The purpose of this paper is to give some identities on the Frobenius-Euler numbers and polynomials by using the fermionic -adic -integral equation on .

1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . The -adic absolute value on is normalized so that . Assume that with .

Let be a continuous function on . Then the fermionic -adic -integral on for is defined by Kim as follows: From (1.1), we note that where and (see [1]). The ordinary Euler polynomials are defined by with the usual convention about replacing by (see [1–10]). In the special case, , is called the th Euler number.

As the extension of (1.3), the Frobenius-Euler polynomials are defined by In the special case, , is called the th Frobenius-Euler number.

By (1.3) and (1.4), we easily get .

From (1.4), we note that with the usual convention about replacing by .

In this paper, we consider the fermionic -adic -integral on for the Frobenius-Euler numbers and polynomials. From these -adic -integrals on , we derive some new and interesting identities on the Frobenius-Euler numbers and polynomials.

2. Identities on the Frobenius-Euler Numbers

From (1.2) and (1.4), we can derive the following: Thus, by (2.1), we get Witt's formula for as follows: By (1.5) and (2.1), we get with the usual convention about replacing by .

From (1.5) and (2.3), we note that By (2.1) and (2.2), we get Therefore, by (2.5), we obtain the following lemma.

Lemma 2.1. For , one has

From (2.3), we can derive the following: where is the Kronecker symbol.

Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For , one has

First we consider the fermionic -adic -integral on for the th Frobenius-Euler polynomials as follows: On the other hand, by (2.5) and Lemma 2.1, we get From Lemma 2.1, Theorem 2.2, and (2.10), we note that Therefore, by (2.10) and (2.11), we obtain the following theorem.

Theorem 2.3. For , one has In particular, for , one has

Let us consider the following fermionic -adic -integral on for the product of Bernoulli and Frobenius-Euler polynomials as follows: It is known that .

On the other hand, by Lemma 2.1, we get Therefore, by (2.14) and (2.15), we obtain the following theorem.

Theorem 2.4. For , one has In particular, for , , one has

It is known that . Let us consider the following fermionic -adic -integral on : Therefore by (2.18), we obtain the following theorem.

Theorem 2.5. For , one has

From (1.3), we can derive the following: Let us take the fermionic -adic -integral on in (2.20) as follows: Therefore, by (2.21), we obtain the following theorem.

Theorem 2.6. For , one has

By Theorems 2.5 and 2.6, we obtain the following corollary.

Corollary 2.7. For , one has

By (1.3), we easily get .

Thus, we have Therefore, by (2.24), we obtain the following theorem.

Theorem 2.8. For , one has

Acknowledgment

The second author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2009-0072514.