Theory and Modern Applications

# A Note on the -Euler Measures

## 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)

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## Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

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Correspondence to Kyung-Won Hwang.

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Kim, T., Hwang, KW. & Lee, B. A Note on the -Euler Measures. Adv Differ Equ 2009, 956910 (2009). https://doi.org/10.1155/2009/956910

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

### Keywords

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