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Note on
-Nasybullin's Lemma Associated with the Modified
-Adic
-Euler Measure
Journal of Inequalities and Applications volume 2010, Article number: 482717 (2010)
Abstract
We derive the modified -adic
-measures related to
-Nasybullin's type lemma.
1. Introduction
Let be a fixed prime number. Throughout this paper, the symbols
,
,
, and
denote the ring of rational integers, the ring of
-adic rational integers, the field of
-adic rational numbers, and the completion of algebraic closure of
, respectively. Let
be the set of natural numbers and
. The
-adic absolute value in
is normalized in such a way that
(see [1–17]). For
with
, let
be the least common multiple of
and
. We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ1_HTML.gif)
where lies in
.
When one talks of -extension,
is variously considered as an indeterminate, a complex number
or a
-adic number
. In this paper, we assume that
with
(see [1–6, 18–23]). As the definition of
-number, we use the following notations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ2_HTML.gif)
Let be the space of uniformly differentiable function on
. For
, the
-adic
-invariant integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ3_HTML.gif)
The -Euler numbers,
, can be determined inductively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ4_HTML.gif)
with the usual convention of replacing by
(see [11]). The modified
-Euler numbers
of
are defined in [2] as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ5_HTML.gif)
with the usual convention of replacing by
. For any positive integer
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ6_HTML.gif)
is known as a measure on (see [9]). In [2], the Witt's type formulas for
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ7_HTML.gif)
The modified -Euler polynomials are also defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ8_HTML.gif)
with the usual convention of replacing by
(see [2]). Thus, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ9_HTML.gif)
Recently Govil and Gupta [22] have introduced a new type of q-integrated Meyer-König-Zeller-Durrmeyer (q-MKZD) operators, obtained moments for these operators, and estimated the convergence of these integrated q-MKZD operators. In this paper, we consider the q-extension which is in a direction different than that of Govil and Gupta [22].
Let be a field over
. Then we call a function
a
-measure on
if
is finitely additive function defined on open-closed subsets in
, whose values are in the field
. Any open-closed subset in
is a disjoint union of some finite intervals
in
, where
is prime to
, and therefore a
-measure
is determined by its values on all intervals in
. Let
denote the set of all rational numbers, whose denominator is a divisor of
for some
. In Section 2, we derive the modified
-adic
-measures related to
-Nasybullin's type lemma.
2. The Modified
-Adic
-Measure
Let be a
-valued function defined on
with the following property.
There exist two constants such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ10_HTML.gif)
for any number . Suppose that
is a root of the equation
. Then we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ11_HTML.gif)
for any interval . From (2.2), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ12_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ13_HTML.gif)
Therefore we obtain the following theorem.
Theorem 2.1.
For with
and
, let
be a
-valued function defined on
with the following properties.
There exist two constants such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ14_HTML.gif)
for any . Suppose that
is a root of the equation
. Then there exists a
-measure
on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ15_HTML.gif)
for any interval .
From (1.9), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ16_HTML.gif)
Let be the
th
-Euler polynomials and let
be the
th
-Euler functions, that is, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ17_HTML.gif)
Note that is the Euler function. By (2.7), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ18_HTML.gif)
Thus, the -Euler function
satisfies the properties of Theorem 2.1 with constants
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ19_HTML.gif)
Then is equal to
, as
reduces simply to
. Therefore, we obtain the following theorem.
Theorem 2.2.
For , let the function
be defined on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ20_HTML.gif)
Then is a
-measure on
.
For with
and
, let
be a primitive Dirichlet character modulo
. Then the generalized
-Euler numbers are defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ21_HTML.gif)
From (2.12) and (2.7), we can easily derive the following Witt's formula:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ22_HTML.gif)
We can compute a -analogue of the
-adic
-
-function by the following
-adic
-Mellin Mazur transform with respect to
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ23_HTML.gif)
Since the character is constant on the interval
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ24_HTML.gif)
where are the
th generalized
-Euler numbers attached to
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ25_HTML.gif)
Assume that with
. Let
be the Teichmüller character mod
. For
, we set
. Note that
and
are defined by
for
. For
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ26_HTML.gif)
For (2.14), (2.16) and (2.17), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ27_HTML.gif)
Since for
, we have
. Let
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ28_HTML.gif)
Therefore, we obtain the following theorem.
Theorem 2.3.
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482717/MediaObjects/13660_2009_Article_2164_Equ29_HTML.gif)
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Kim, T., Kim, YH., Jang, LC. et al. Note on -Nasybullin's Lemma Associated with the Modified
-Adic
-Euler Measure.
J Inequal Appl 2010, 482717 (2010). https://doi.org/10.1155/2010/482717
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DOI: https://doi.org/10.1155/2010/482717