- Research Article
- Open Access

- Taekyun Kim
^{1}, - Kyung-Won Hwang
^{2}Email author and - Byungje Lee
^{3}

**2009**:956910

https://doi.org/10.1155/2009/956910

© Taekyun Kim et al. 2009

**Received: **6 March 2009

**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.

## Keywords

- Partial Differential Equation
- Ordinary Differential Equation
- Functional Equation
- Complex Number
- Prime Number

## 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 [1–37]):

The ordinary Euler numbers are defined as (see [1–37])

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

with the usual convention of replacing by

Remark 1.1.

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

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

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.

## 2. -adic -Euler Measures

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

Theorem 2.1.

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

Proof.

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:

Therefore, we obtain the following theorem.

Theorem 2.2.

Therefore, we obtain the following theorem and corollary.

Theorem 2.3.

Corollary 2.4.

## 3. -adic - -Series

## Declarations

### Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.