- Research Article
- Open Access
Note on the -Extension of Barnes' Type Multiple Euler Polynomials
- Leechae Jang^{1},
- Taekyun Kim^{2},
- Young-Hee Kim^{2}Email author and
- Kyung-Won Hwang^{3}Email author
https://doi.org/10.1155/2009/136532
© Leechae Jang et al. 2009
- Received: 30 August 2009
- Accepted: 28 September 2009
- Published: 8 October 2009
Abstract
We construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler polynomials. We also derive many properties and formulae for our -Euler polynomials of higher order by using the multiple integral equations on .
Keywords
- Integral Equation
- Inversion Formula
- Euler Polynomial
1. Introduction
Let be a fixed odd prime number. Throughout this paper, symbols , , , and will denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of 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 number or a -adic number . If one normally assumes If then one normally assumes We use the following notations:
Let a fixed positive odd integer with . For , we set
where lies in . The fermionic -adic -measures on are defined as
(see [5]).
We say that is a uniformly differentiable function at a point and write , if the difference quotients have a limit as . For , let us begin with expression
which represents a -analogue of Riemann sums for in the fermionic sense (see [4, 5]). The integral of on is defined by the limit of these sums (as ) if this limit exists. The fermionic invariant -adic -integral of function is defined as
Note that if in , then
The Barnes' type Euler polynomials are considered as follows:
where (cf. [7]).
From (1.5), we can derive the fermionic invariant integral on as follows:
For , let , one has
By (1.9), we see that
From (1.10), we note that
In the view point of (1.11), we try to study the -extension of Barnes' type Euler polynomials by using the -extension of fermionic -adic invariant integral on .
The purpose of this paper is to construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler numbers and polynomials. Also, we give many properties and formulae for our -Euler polynomials of higher order. Finally, we give the generating function for these -Euler polynomials of higher order.
2. Barnes' Type Multiple -Euler Polynomials
Let . For and with , we define the Barnes' type multiple -Euler polynomials as follows:
where
In the special case , are called the Barnes' type multiple -Euler numbers. From (2.1), one has
Therefore, we obtain the following theorem.
Theorem 2.1.
By (1.7), we easily see that
From (1.7), we can derive
By (2.6), one has
Hence we obtain the following theorem.
Theorem 2.2.
It is not difficult to show that the following integral equation is satisfied:
where with . By (2.9), we obtain the following theorem.
Theorem 2.3.
For the special case in Theorem 2.3, one has
By (2.1), (2.3), and (2.9), we obtain the following corollary.
Corollary 2.4.
From (2.3), we note that
where is an odd positive integer. By (2.13), we obtain the following theorem.
Theorem 2.5.
Remark 2.6.
Declarations
Acknowledgment
This paper was supported by Konkuk University (2009).
Authors’ Affiliations
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Copyright
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