On the Symmetric Properties of Higher-Order Twisted -Euler Numbers and Polynomials

Let be a fixed odd prime number. Throughout this paper, the symbols and will denote the ring of rational integers, the ring of adic integers, the field of adic rational numbers, the complex number field, and the completion of the algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with

When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . If one normally assumes that . If , then we assume that so that for each We use the following notation:

(1.1)

For a fixed positive integer with , set

(1.2)

where satisfies the condition For any

(1.3)

(see [1–13]) is known to be a distribution on .

We say that is a uniformly differentiable function at and denote this property by if the difference quotients

(1.4)

have a limit as

For the fermionic -adic invariant -integral on is defined as

(1.5)

(see [14]). Let us define the fermionic -adic invariant integral on as follows:

(1.6)

(see [1–12, 14–20]). From the definition of integral, we have

(1.7)

For , let be the adic locally constant space defined by

(1.8)

where for some is the cyclic group of order

It is well known that the twisted Euler polynomials of order are defined as

(1.9)

and are called the twisted Euler numbers of order When the polynomials and numbers are called the twisted Euler polynomials and numbers, respectively. When and , the polynomials and numbers are called the twisted Euler polynomials and numbers, respectively. When , and , the polynomials and numbers are called the ordinary Euler polynomials and numbers, respectively.

In [15], Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher order, recently. In this paper, we extend our result to the higher-order twisted Euler numbers and polynomials.

The purpose of this paper is to establish various identities concerning higher-order twisted -Euler numbers and polynomials by the properties of adic invariant integral on . Especially, if , we derive the result of [15].

Let with and .

For and , we set

(2.1)

where

(2.2)

In (2.1), we note that is symmetric in and .

From (2.1), we derive that

(2.3)

From the definition of integral, we also see that

(2.4)

It is easy to see that

(2.5)

where .

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

(2.6)

From the symmetry of in and , we also see that

(2.7)

Comparing the coefficients on the both sides of (2.6) and (2.7), we obtain an identity for the twisted Euler polynomials of higher order as follows.

Theorem 2.1.

Let with and .

For and we have

(2.8)

Remark 2.2.

Taking and in Theorem 2.1, we can derive the following identity:

(2.9)

Moreover, if we take and in Theorem 2.1, then we have the following identity for the twisted Euler numbers of higher order.

Corollary 2.3.

Let with and . For and we have

(2.10)

We also note that taking in Corollary shows the following identity:

(2.11)

Now we will derive another interesting identities for the twisted -Euler numbers and polynomials of higher order. From (2.3), we can derive that

(2.12)

From the symmetry of in and , we see that

(2.13)

Comparing the coefficients on the both sides of (2.12) and (2.13), we obtain the following theorem which shows the relationship between the power sums and the twisted Euler polynomials.

Theorem 2.4.

Let with and For and we have

(2.14)

Remark 2.5.

Let and in Theorem Then it follows that

(2.15)

Moreover, if we take and in Theorem 2.4, then we have the following identity for the twisted Euler numbers of higher order.

Corollary 2.6.

Let with For and we have

(2.16)

If we take in Corollary 2.3, we derive the following identity for the twisted Euler polynomials: for with and

(2.17)

Remark 2.7.

If we can observe the result of [15].