- Research Article
- Open Access
- Published:
Multiple Twisted
-Euler Numbers and Polynomials Associated with
-Adic
-Integrals
Advances in Difference Equations volume 2008, Article number: 738603 (2008)
Abstract
By using -adic
-integrals on
, we define multiple twisted
-Euler numbers and polynomials. We also find Witt's type formula for multiple twisted
-Euler numbers and discuss some characterizations of multiple twisted
-Euler Zeta functions. In particular, we construct multiple twisted Barnes' type
-Euler polynomials and multiple twisted Barnes' type
-Euler Zeta functions. Finally, we define multiple twisted Dirichlet's type
-Euler numbers and polynomials, and give Witt's type formula for them.
1. Introduction
Let be a fixed odd prime number. Throughout this paper,
,
, and
are, respectively, the ring of
-adic rational integers, the field of
-adic rational numbers, and the
-adic completion of the algebraic closure of
. The
-adic absolute value in
is normalized so that
. When one talks about
-extension,
is variously considered as an indeterminate, a complex number,
or a
-adic number
. If
, one normally assumes that
. If
, one normally assumes that
so that
for each
. We use the notations

  (cf. [1–14]), for all . For a fixed odd positive integer
with
, set

where lies in
. For any
,

is known to be a distribution on (cf. [1–28]).
We say that is uniformly differentiable function at a point
and denote this property by
if the difference quotients

have a limit as
(cf. [25]).
The -adic
-integral of a function
was defined as


(cf. [4, 24, 25, 28]), from (1.6), we derive

where . If we take
, then we have
. From (1.7), we obtain that

In Section 2, we define the multiple twisted -Euler numbers and polynomials on
and find Witt's type formula for multiple twisted
-Euler numbers. We also have sums of consecutive multiple twisted
-Euler numbers. In Section 3, we consider multiple twisted
-Euler Zeta functions which interpolate new multiple twisted
-Euler polynomials at negative integers and investigate some characterizations of them. In Section 4, we construct the multiple twisted Barnes' type
-Euler polynomials and multiple twisted Barnes' type
-Euler Zeta functions which interpolate new multiple twisted Barnes' type
-Euler polynomials at negative integers. In Section 5, we define multiple twisted Dirichlet's type
-Euler numbers and polynomials and give Witt's type formula for them.
2. Multiple Twisted
-Euler Numbers and Polynomials
In this section, we assume that with
. For
, by the definition of
-adic
-integral on
, we have

where . If
is odd positive integer, we have

Let be the locally constant space, where
is the cyclic group of order
. For
, we denote the locally constant function by

(cf. [5, 7–14, 16, 18]). If we take , then we have

Now we define the twisted -Euler numbers
as follows:

We note that by substituting ,
are the familiar Euler numbers. Over five decades ago, Carlitz defined
-extension of Euler numbers (cf. [15]). From (2.4) and (2.5), we note that Witt's type formula for a twisted
-Euler number is given by

for each and
.
Twisted -Euler polynomials
are defined by means of the generating function

where . By using the
th iterative fermionic
-adic
-integral on
, we define multiple twisted
-Euler number as follows:

Thus we give Witt's type formula for multiple twisted -Euler numbers as follows.
Theorem 2.1.
For each and
,

where

From (2.8) and (2.9), we obtain the following theorem.
Theorem 2.2.
For and
,

From these formulas, we consider multivariate fermionic -adic
-integral on
as follows:

Then we can define the multiple twisted -Euler polynomials
as follows:

From (2.12) and (2.13), we note that

Then by the th differentiation on both sides of (2.14), we obtain the following.
Theorem 2.3.
For each and
,

Note that

Then we see that

From (2.15) and (2.17), we obtain the sums of powers of consecutive -Euler numbers as follows.
Theorem 2.4.
For each and
,

3. Multiple Twisted
-Euler Zeta Functions
For with
and
, the multiple twisted
-Euler numbers can be considered as follows:

From (3.1), we note that

By the th differentiation on both sides of (3.2) at
, we obtain that

From (3.3), we derive multiple twisted -Euler Zeta function as follows:

for all . We also obtain the following theorem in which multiple twisted
-Euler Zeta functions interpolate multiple twisted
-Euler polynomials.
Theorem 3.1.
For and
,

4. Multiple Twisted Barnes' Type
-Euler Polynomials
In this section, we consider the generating function of multiple twisted -Euler polynomials:

We note that

By the th differentiation on both sides of (4.2) at
, we obtain that

Thus we can consider multiple twisted Hurwitz's type -Euler Zeta function as follows:

for all and
. We note that
is analytic function in the whole complex
-plane and
. We also remark that if
and
, then
is Hurwitz's type
-Euler Zeta function (see [7, 27]). The following theorem means that multiple twisted
-Euler Zeta functions interpolate multiple twisted
-Euler polynomials at negative integers.
Theorem 4.1.
For ,
,
, and
,

Let us consider

where and
. Then
will be called multiple twisted Barnes' type
-Euler polynomials. We note that

By the th differentiation of both sides of (4.6), we obtain the following theorem.
Theorem 4.2.
For each ,
,
, and
,

where

From (4.8), we consider multiple twisted Barnes' type -Euler Zeta function defined as follows: for each
,
,
, and
,

We note that is analytic function in the whole complex
-plane. We also see that multiple twisted Barnes' type
-Euler Zeta functions interpolate multiple twisted Barnes' type
-Euler polynomials at negative integers as follows.
Theorem 4.3.
For each ,
,
, and
,

5. Multiple Twisted Dirichlet's Type
-Euler Numbers and Polynomials
Let be a Dirichlet's character with conductor
and
. If we take
, then we have
. From (2.2), we derive

In view of (5.1), we can define twisted Dirichlet's type -Euler numbers as follows:

(cf. [17, 19, 21, 22]). From (5.1) and (5.2), we can give Witt's type formula for twisted Dirichlet's type -Euler numbers as follows.
Theorem 5.1.
Let be a Dirichlet's character with conductor
. For each
,
, we have

We note that if , then
is the generalized
-Euler numbers attached to
(see [18, 26]). From (5.2), we also see that

By (5.2) and (5.4), we obtain that

From (5.5), we can define the -function as follows:

for all . We note that
is analytic function in the whole complex
-plane. From (5.5) and (5.6), we can derive the following result.
Theorem 5.2.
Let be a Dirichlet's character with conductor
. For each
,
, we have

Now, in view of (5.1), we can define multiple twisted Dirichlet's type -Euler numbers by means of the generating function as follows:

where . We note that if
, then
is a multiple generalized
-Euler number (see [22]).
By using the same method used in (2.8) and (2.9),

From (5.9), we can give Witt's type formula for multiple twisted Dirichlet's type -Euler numbers.
Theorem 5.3.
Let be a Dirichlet's character with conductor
. For each
,
, and
, we have

where and

From (5.10), we also obtain the sums of powers of consecutive multiple twisted Dirichlet's type -Euler numbers as follows.
Theorem 5.4.
Let be a Dirichlet's character with conductor
. For each
,
, and
, we have

Finally, we consider multiple twisted Dirichlet's type -Euler polynomials defined by means of the generating functions as follows:

where and
. From (5.13), we note that

Clearly, we obtain the following two theorems.
Theorem 5.5.
Let be a Dirichlet's character with conductor
. For each
,
,
, and
, we have

where

Theorem 5.6.
Let be a Dirichlet's character with conductor
. For each
,
,
, and
, we have

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Jang, LC. Multiple Twisted -Euler Numbers and Polynomials Associated with
-Adic
-Integrals.
Adv Differ Equ 2008, 738603 (2008). https://doi.org/10.1155/2008/738603
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DOI: https://doi.org/10.1155/2008/738603
Keywords
- Ordinary Differential Equation
- Functional Equation
- Prime Number
- Rational Number
- Constant Function