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A Note on the
-Euler Measures
Advances in Difference Equations volume 2009, Article number: 956910 (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.
The second kind Euler numbers are also defined as follows (see [25]):

The Euler polynomials are also defined by

Thus, we have

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

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

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

where lies in
, (see [1–37]).
Theorem 2.1.
Let be given by

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

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

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:

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

Therefore, we obtain the following theorem.
Theorem 2.2.
Let be the Dirichlet character with conductor
with
. Then one has

Let . From (2.2), we note that

Thus, we have

Therefore, we obtain the following theorem and corollary.
Theorem 2.3.
For , one has

Corollary 2.4.
For , one has

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

Since for
, we have
Let
. Then we have

Therefore, we obtain the following theorem.
Theorem 3.1.
Let . Then one has

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Acknowledgments
This paper was supported by Jangjeon Mathematical Society.
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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