1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols , , , and denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers. The normalized valuation in is denoted by with . Let be the space of uniformly differentiable function on . For , the fermionic -adic invariant integral on is defined as

(1.1)

(see [1–25]). For , we note that

(1.2)

Let be a fixed odd positive integer. For , we set

(1.3)

where lies in (see [1–13]). It is well known that for ,

(1.4)

For , let be the cyclic group of order . That is, . The -adic locally constant space, , is defined by

Let be Dirichlet's character with conductor and let . Then the generalized twisted Bernoulli polynomials attached to are defined as

(1.5)

In [4, 7, 10–12], the generalized twisted Bernoulli polynomials of order attached to are also defined as follows:

(1.6)

Recently, the symmetry identities for the generalized twisted Bernoulli polynomials and the generalized twisted Bernoulli polynomials of order are studied in [4, 12].

In this paper, we study the symmetric properties of the multivariate -adic invariant integral on . From these symmetric properties, we derive the symmetry identities for the twisted generalized Euler polynomials of higher order. In [14], Kim gave the relation between the power sum polynomials and the generalized higher-order Euler polynomials. The main purpose of this paper is to give the symmetry identities for the twisted generalized Euler polynomials of higher order using the symmetric properties of the multivariate -adic invariant integral on .

2. Symmetry Identities for the Twisted Generalized Euler Polynomials of Higher Order

Let be Dirichlet's character with an odd conductor . That is, with . For , the twisted generalized Euler polynomials attached to , , are defined as

(2.1)

In the special case , are called the th twisted generalized Euler numbers attached to .

From (2.1), we note that

(2.2)

For with , we have

(2.3)

Let . Then we see that

(2.4)

Now we define the twisted generalized Euler polynomials of order attached to as follows:

(2.5)

In the special case , are called the th twisted generalized Euler numbers of order .

Let with , and . Then we set

(2.6)

where

(2.7)

From (2.6), we note that

(2.8)

From (2.4), we can easily derive the following equation:

(2.9)

It is not difficult to show that

(2.10)

By (2.8), (2.9), and (2.10), we see that

(2.11)

In the viewpoint of the symmetry of for and , we have

(2.12)

Comparing the coefficients on both sides of (2.11) and (2.12), we obtain the following theorem.

Theorem 2.1.

Let with , and . For and , one has

(2.13)

Let and in Theorem 2.1. Then we also have the following corollary.

Corollary 2.2.

For with , and , one has

(2.14)

Let be the trivial character and . Then we also have the following corollary.

Corollary 2.3.

Let with . Then one has

(2.15)

where are the th twisted Euler polynomials.

If we take in Corollary 2.3, then we obtain the following corollary.

Corollary 2.4 (Distribution for the twisted Euler polynomials).

For with , one has

(2.16)

From (2.6), we can derive that

(2.17)

By the symmetry property of in and , we also see that

(2.18)

Comparing the coefficients on both sides of (2.17) and (2.18), we obtain the following theorem which shows the relationship between the power sums and the twisted generalized Euler polynomials of higher order.

Theorem 2.5.

Let with , and . For and , one has

(2.19)

If we take , , and in Theorem 2.5, then we have the following identity:

(2.20)