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On the Twisted
-Analogs of the Generalized Euler Numbers and Polynomials of Higher Order
Advances in Difference Equations volume 2010, Article number: 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 [1–16].
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