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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