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

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

(1.1)

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

(1.2)

(see [1–8]). From (1.2), we have

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

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

(1.10)

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

(1.11)

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

(2.1)

where

(2.2)

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

(2.3)

We can show that

(2.4)

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

(2.5)

Thus, we have

(2.6)

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

(2.7)

By the same method, we also see that

(2.8)

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

(2.9)

Let and in (2.9). Then we have

(2.10)

From (2.10), we note that

(2.11)

If we take in (2.11), then we have

(2.12)

From (2.3), we note that

(2.13)

By the symmetric property of in , we also see that

(2.14)

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

(2.15)

Let and , we have

(2.16)

From (2.16), we can derive

(2.17)