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On Carlitz's Type
-Euler Numbers Associated with the Fermionic
-Adic Integral on ![](//media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_IEq3_HTML.gif)
Journal of Inequalities and Applications volume 2010, Article number: 358986 (2010)
Abstract
We consider the following problem in the paper of Kim et al. (2010): "Find Witt's formula for Carlitz's type -Euler numbers." We give Witt's formula for Carlitz's type
-Euler numbers, which is an answer to the above problem. Moreover, we obtain a new p-adic q-l-function
for Dirichlet's character
, with the property that
for
using the fermionic p-adic integral on
.
1. Introduction
Throughout this paper, let be an odd prime number. The symbol,
and
denote the rings of
-adic integers, the field of
-adic numbers, and the field of
-adic completion of the algebraic closure of
respectively. The
-adic absolute value in
is normalized in such way that
Let
be the set of natural numbers and
As the definition of -number, we use the following notations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ1_HTML.gif)
Note that for
where
tends to 1 in the region
When one talks of -analogue,
is variously considered as an indeterminate, a complex number
or a
-adic number
If
one normally assumes
We will further suppose that
so that
for
If
then we assume that
After Carlitz [1, 2] gave -extensions of the classical Bernoulli numbers and polynomials, the
-extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors (cf. [1–21]). The Euler numbers and polynomials have been studied by researchers in the field of number theory, mathematical physics, and so on (cf. [1, 2, 9, 11, 13–16, 22, 23]). Recently, various
-extensions of these numbers and polynomials have been studied by many mathematicians (cf. [6–8, 10, 12, 17, 18, 20]). Also, some authors have studied in the several area of
-theory (cf. [3, 4, 16, 19, 24]).
It is known that the generating function of Euler numbers is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ2_HTML.gif)
From (1.2), we know the recurrence formula of Euler numbers is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ3_HTML.gif)
with the usual convention of replacing by
(see [7, 18]).
In [17], the -extension of Euler numbers
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ4_HTML.gif)
with the usual convention of replacing by
As the same motivation of the construction in [18], Carlitz's type -Euler numbers
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ5_HTML.gif)
with the usual convention of replacing by
It was shown that
where
is the
th Euler number. In the complex case, the generating function of Carlitz's type
-Euler numbers
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ6_HTML.gif)
where is a complex number with
(see [18]). The remark point is that the series on the right-hand side of (1.6) is uniformly convergent in the wider sense. In
-adic case, Kim et al. [18] could not determine the generating function of Carlitz's type
-Euler numbers and Witt's formula for Carlitz's type
-Euler numbers.
In this paper, we obtain the generating function of Carlitz's type -Euler numbers in the
-adic case. Also, we give Witt's formula for Carlitz's type
-Euler numbers, which is a partial answer to the problem in [18]. Moreover, we obtain a new
-adic
-
-function
for Dirichlet's character
with the property that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ7_HTML.gif)
for using the fermionic
-adic integral on
2. Carlitz's Type
-Euler Numbers in the
-Adic Case
Let be the space of uniformly differentiable functions on
Then, the
-adic
-integral of a function
on
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ8_HTML.gif)
(cf. [5–17, 19, 20, 22]). The bosonic -adic integral on
is considered as the limit
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ9_HTML.gif)
From (2.1), we have the fermionic -adic integral on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ10_HTML.gif)
Using (2.3), we can readily derive the classical Euler polynomials, namely
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ11_HTML.gif)
In particular, when is the well-known the Euler numbers (cf. [7, 16, 19]).
By definition of we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ12_HTML.gif)
where (see [7]). By (2.5) and induction, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ13_HTML.gif)
where and
From (2.6), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ14_HTML.gif)
For and any integer
we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ15_HTML.gif)
It is easy to see that (see [23, page 172]). We put
with
and
We define
for
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ16_HTML.gif)
If we set in (2.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ17_HTML.gif)
From (2.10), we note that if then
hence there is no need to consider both (odd and even) cases. Thus, for each
we obtain
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ18_HTML.gif)
Also, if in (2.5), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ19_HTML.gif)
On the other hand, by (2.12), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ20_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ21_HTML.gif)
From the definition of fermionic -adic integral on
and (2.11), we can derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ22_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ23_HTML.gif)
From (2.12), (2.13), (2.14), (2.15), and (2.16), it is easy to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ24_HTML.gif)
where are Carlitz's type
-Euler numbers defined by (see [18])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ25_HTML.gif)
Therefore, we obtain the recurrence formula for the Carlitz's type -Euler numbers as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ26_HTML.gif)
with the usual convention of replacing by
Therefore, by (2.16), (2.18), and (2.19), we obtain the following theorem, which is a partial answer to the problem in [18].
Theorem 2.1 (Witt's formula for ).
For
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ27_HTML.gif)
Carlitz's type -Euler numbers
can be determined inductively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ28_HTML.gif)
with the usual convention of replacing by
Carlitz type -Euler polynomials
are defined by means of the generating function
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ29_HTML.gif)
In the cases will be called Carlitz type
-Euler numbers (cf. [8, 19]). One also can see that the generating functions
are determined as solutions of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ30_HTML.gif)
From (2.22), one gets the following.
Lemma 2.2.
(1)
(2)
It is clear from (1) and (2) of Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ31_HTML.gif)
From (2.24), we may state the following.
Proposition 2.3.
If and
then
-
(1)
,
-
(2)
Proposition 2.4.
For the value of
is
times the coefficient of
in the formal expansion of
in powers of
That is,
Proof.
From (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ32_HTML.gif)
which leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ33_HTML.gif)
The result now follows by using (1) of Proposition 2.3.
Corollary 2.5.
If then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ34_HTML.gif)
Let with
and
be a fixed odd prime number. One sets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ35_HTML.gif)
where with
(cf. [7, 9]). Note that the natural map
induces
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ36_HTML.gif)
Hereafter, if is a function on
one denotes by the same
the function
on
Namely one considers
as a function on
Let be the Dirichlet character with an odd conductor
Then, the generalized Carlitz type
-Euler polynomials attached to
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ37_HTML.gif)
where and
Then, one has the generating function of generalized Carlitz type
-Euler polynomials attached to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ38_HTML.gif)
Now, fixed any with
and
From (2.31), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ39_HTML.gif)
where and
with
By (2.31) and (2.32), one can derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ40_HTML.gif)
where and
with
Therefore, one obtains the following.
Theorem 2.6.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ41_HTML.gif)
where and
Let denote the Teichmüller character mod
For
one sets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ42_HTML.gif)
Note that since is defined by
for
(cf. [10, 12, 21]). One notes that
is analytic for
One defines an interpolation function for Carlitz type -Euler numbers. For
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ43_HTML.gif)
Then, is analytic for
The values of this function at nonpositive integers are given by
Theorem 2.7.
For integers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ44_HTML.gif)
where In particular, if
then
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ45_HTML.gif)
Therefore by (2.30), the theorem is proved.
Let be the Dirichlet character with an odd conductor
Let
be a positive integer multiple of
and
Then, by (2.22) and (2.31), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ46_HTML.gif)
Therefore, we obtain the following
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ47_HTML.gif)
If then
so that
is a multiple of
From (2.40), we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ48_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ49_HTML.gif)
By Corollary 2.5, we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ50_HTML.gif)
From (2.42) and (2.43), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ51_HTML.gif)
since From Theorem 2.7 and (2.44), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ52_HTML.gif)
for Therefore, we have the following theorem.
Theorem 2.8.
Let be a positive integer multiple of
and
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ53_HTML.gif)
Then, is analytic for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ54_HTML.gif)
Furthermore, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F358986/MediaObjects/13660_2010_Article_2130_Equ55_HTML.gif)
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Acknowledgments
The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0001654). The second author was supported by the research grant of Kwangwoon University in 2010.
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Kim, MS., Kim, T. & Ryoo, CS. On Carlitz's Type -Euler Numbers Associated with the Fermionic
-Adic Integral on
.
J Inequal Appl 2010, 358986 (2010). https://doi.org/10.1155/2010/358986
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DOI: https://doi.org/10.1155/2010/358986