# A Note on the -Euler Measures

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

**2009**:956910

**DOI: **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.

## 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

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

Thus, we have

Therefore, we obtain the following theorem.

Theorem 3.1.

## Declarations

### Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

## Authors’ Affiliations

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