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# On the Twisted -Analogs of the Generalized Euler Numbers and Polynomials of Higher Order

*Advances in Difference Equations*
**volumeÂ 2010**, ArticleÂ number:Â 875098 (2010)

## Abstract

We consider the twisted -extensions of the generalized Euler numbers and polynomials attached to .

## 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

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

where lies in compared to [1â€“16].

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

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

Let . Then, we see that

For , let . Then, we have

Thus, we have

By (1.7), we see that

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

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

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:

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

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:

By (2.1), we see that

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:

Thus, we have

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

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has

Let . Then we define the extension of as follows:

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:

where .

Let where . From (2.8), we note that

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

By (2.10), we obtain the following theorem.

Theorem 2.2.

For , , one has

Let . Then we see that

It is easy to see that

Thus, we have

By (2.14), we obtain the following theorem.

Theorem 2.3.

For with , one has

By (1.7), we easily see that

Thus,we have

By (2.17), we obtain the following theorem.

Theorem 2.4.

For with , one has

It is easy to see that

Let . Then we note that

From (2.20), we can derive

## 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

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

where , , , and .

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

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

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Jang, L., Lee, B. & Kim, T. On the Twisted -Analogs of the Generalized Euler Numbers and Polynomials of Higher Order.
*Adv Differ Equ* **2010**, 875098 (2010). https://doi.org/10.1155/2010/875098

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DOI: https://doi.org/10.1155/2010/875098