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# A Note on the -Euler Measures

DOI: 10.1155/2009/956910

Accepted: 20 May 2009

Published: 28 June 2009

## Abstract

Properties of -extensions of Euler numbers and polynomials which generalize those satisfied by and are used to construct -extensions of -adic Euler measures and define -adic - -series which interpolate -Euler numbers at negative integers. Finally, we give Kummer Congruence for the -extension of ordinary Euler numbers.

## 1. Introduction

Let be a fixed prime number. Throughout this paper and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the normalized exponential valuation of with . When one talks of -extension, is variously considered as an indeterminate, a complex number or -adic numbers . If , one normally assumes . If , one normally assumes . In this paper, we use the notations of -number as follows (see [137]):

(11)

The ordinary Euler numbers are defined as (see [137])

(12)

where is written as when is replaced by . From the definition of Euler number, we can derive

(13)

with the usual convention of replacing by

Remark 1.1.

The second kind Euler numbers are also defined as follows (see [25]):
(14)
The Euler polynomials are also defined by
(15)
Thus, we have
(16)

In [7], -Euler numbers, , can be determined inductively by

(17)
where must be replaced by , symbolically. The -Euler polynomials are given by that is,
(18)

Let be a fixedoddpositive integer. Then we have (see [7])

(19)

We use (1.9) to get bounded -adic -Euler measures and finally take the Mellin transform to define -adic - -series which interpolate -Euler numbers at negative integers.

Let be a fixed odd positive integer, and let be a fixed odd prime number. Define

(21)

where lies in , (see [137]).

Theorem 2.1.

Let be given by
(22)

Then extends to a -valued measure on the compact open sets . Note that , where is fermionic measure on (see [7])

Proof.

It is sufficient to show that
(23)
By (1.9) and (2.2), we see that
(24)

and we easily see that for some constant .

Let be a Dirichlet character with conductor with . Then we define the generalized -Euler numbers attached to as follows:

(25)
The locally constant function on can be integrated by the -adic bounded -Euler measure as follows:
(26)

Therefore, we obtain the following theorem.

Theorem 2.2.

Let be the Dirichlet character with conductor with . Then one has
(27)
Let . From (2.2), we note that
(28)
Thus, we have
(29)

Therefore, we obtain the following theorem and corollary.

Theorem 2.3.

For , one has
(210)

Corollary 2.4.

For , one has
(211)

In this section, we assume that with . Let denote the Teichmüller character . For , we set . Note that , and is defined by , for . For ,we define

(31)

Thus, we have

(32)
Since for , we have Let . Then we have
(33)

Therefore, we obtain the following theorem.

Theorem 3.1.

Let . Then one has
(34)

## Declarations

### Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

## Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University
(2)
General Education Department, Kookmin University
(3)
Department of Wireless Communications Engineering, Kwangwoon University

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