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On the Symmetric Properties of Higher-Order Twisted
-Euler Numbers and Polynomials
Advances in Difference Equations volume 2010, Article number: 765259 (2010)
Abstract
In 2009, 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 Kim et al. (2009).
1. Introduction
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:

For a fixed positive integer with
, set

where satisfies the condition
For any

(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

have a limit as
For the fermionic
-adic invariant
-integral on
is defined as

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

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

For , let
be the
adic locally constant space defined by

where for some
is the cyclic group of order
It is well known that the twisted Euler polynomials of order
are defined as

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].
2. Some Identities of the Higher-Order Twisted
Euler Numbers and Polynomials
Let with
and
.
For and
, we set

where

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

From the definition of integral, we also see that

It is easy to see that

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

From the symmetry of in
and
, we also see that

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

Remark 2.2.
Taking and
in Theorem 2.1, we can derive the following identity:

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

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

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

From the symmetry of in
and
, we see that

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

Remark 2.5.
Let and
in Theorem
Then it follows that

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

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

Remark 2.7.
If we can observe the result of [15].
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Moon, EJ., Rim, SH., Jin, JH. et al. On the Symmetric Properties of Higher-Order Twisted -Euler Numbers and Polynomials.
Adv Differ Equ 2010, 765259 (2010). https://doi.org/10.1155/2010/765259
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Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation