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 [1–37]):

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

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

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with the usual convention of replacing by

Remark 1.1.

The second kind Euler numbers are also defined as follows (see [25]):

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The Euler polynomials are also defined by

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Thus, we have

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

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

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

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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

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where lies in , (see [1–37]).

Theorem 2.1.

Let be given by

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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

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By (1.9) and (2.2), we see that

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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:

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

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Therefore, we obtain the following theorem.

Theorem 2.2.

Let be the Dirichlet character with conductor with . Then one has

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Let . From (2.2), we note that

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Thus, we have

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Therefore, we obtain the following theorem and corollary.

Theorem 2.3.

For , one has

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Corollary 2.4.

For , one has

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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

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Thus, we have

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Since for , we have Let . Then we have

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Therefore, we obtain the following theorem.

Theorem 3.1.

Let . Then one has

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