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).
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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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DOI: https://doi.org/10.1155/2010/765259