1. Introduction and Preliminaries

Let be an odd prime number. For , let be the cyclic group of order , and let be the space of locally constant functions in the -adic number field . When one talks of -extension, is variously considered as an indeterminate, a complex number , or -adic number . If , one normally assumes that . If , one normally assumes that . In this paper, we use the notation

(1.1)

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

(1.2)

where lies in compared to [1–16].

Let be the Dirichlet's character with an odd conductor . Then the generalized -Euler polynomials attached to , , are defined as

(1.3)

In the special case , are called the th -Euler numbers attached to . For , the -adic fermionic integral on is defined by

(1.4)

Let . Then, we see that

(1.5)

For , let . Then, we have

(1.6)

Thus, we have

(1.7)

By (1.7), we see that

(1.8)

From (1.8), we can derive the Witt's formula for as follows:

(1.9)

The th generalized -Euler polynomials of order , , are defined as

(1.10)

In the special case , are called the th -Euler numbers of order attached to .

Now, we consider the multivariate -adic invariant integral on as follows:

(1.11)

By (1.10) and (1.11), we see the Witt's formula for as follows:

(1.12)

The purpose of this paper is to present a systemic study of some formulas of the twisted -extension of the generalized Euler numbers and polynomials of order attached to .

2. On the Twisted -Extension of the Generalized Euler Polynomials

In this section, we assume that with and . For with , let be the Dirichlet's character with conductor . For , let us consider the twisted -extension of the generalized Euler numbers and polynomials of order attached to . We firstly consider the twisted -extension of the generalized Euler polynomials of higher order as follows:

(2.1)

By (2.1), we see that

(2.2)

From the multivariate fermionic -adic invariant integral on , we can derive the twisted -extension of the generalized Euler polynomials of order attached to as follows:

(2.3)

Thus, we have

(2.4)

Let be the generating function for . By (2.3), we easily see that

(2.5)

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has

(2.6)

Let . Then we define the extension of as follows:

(2.7)

Then, are called the th generalized -Euler polynomials of order attached to . In the special case , are called the th generalized -Euler numbers of order . By (1.7), we obtain the Witt's formula for as follows:

(2.8)

where .

Let where . From (2.8), we note that

(2.9)

Let be the generating function for . From (2.8), we can easily derive

(2.10)

By (2.10), we obtain the following theorem.

Theorem 2.2.

For , , one has

(2.11)

Let . Then we see that

(2.12)

It is easy to see that

(2.13)

Thus, we have

(2.14)

By (2.14), we obtain the following theorem.

Theorem 2.3.

For with , one has

(2.15)

By (1.7), we easily see that

(2.16)

Thus,we have

(2.17)

By (2.17), we obtain the following theorem.

Theorem 2.4.

For with , one has

(2.18)

It is easy to see that

(2.19)

Let . Then we note that

(2.20)

From (2.20), we can derive

(2.21)

3. Further Remark

In this section, we assume that with . Let be the Dirichlet's character with an odd conductor . From the Mellin transformation of in (2.10), we note that

(3.1)

where , and , . By (3.1), we can define the Dirichlet's type multiple --function as follows.

Definition 3.1.

For , with , one defines the Dirichlet's type multiple --function related to higher order -Euler polynomials as

(3.2)

where , , , and .

Note that is analytic continuation in whole complex -plane. In (2.10), we note that

(3.3)

By Laurent series and Cauchy residue theorem in (3.1) and (3.3), we obtain the following theorem.

Theorem 3.2.

Let be Dirichlet's character with odd conductor and let . For , , and , one has

(3.4)