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On Carlitz's Type Euler Numbers Associated with the Fermionic Adic Integral on
Journal of Inequalities and Applications volumeÂ 2010, ArticleÂ number:Â 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 padic qlfunction for Dirichlet's character , with the property that for using the fermionic padic 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:
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. [1â€“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, 13â€“16, 22, 23]). Recently, various extensions of these numbers and polynomials have been studied by many mathematicians (cf. [6â€“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
From (1.2), we know the recurrence formula of Euler numbers is given by
with the usual convention of replacing by (see [7, 18]).
In [17], the extension of Euler numbers are defined as
with the usual convention of replacing by
As the same motivation of the construction in [18], Carlitz's type Euler numbers are defined as
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
where is a complex number with (see [18]). The remark point is that the series on the righthand 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
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
(cf. [5â€“17, 19, 20, 22]). The bosonic adic integral on is considered as the limit that is,
From (2.1), we have the fermionic adic integral on as follows:
Using (2.3), we can readily derive the classical Euler polynomials, namely
In particular, when is the wellknown the Euler numbers (cf. [7, 16, 19]).
By definition of we show that
where (see [7]). By (2.5) and induction, we obtain
where and From (2.6), we note that
For and any integer we define
It is easy to see that (see [23, page 172]). We put with and We define for by
If we set in (2.7), we have
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
Also, if in (2.5), then
On the other hand, by (2.12), we obtain that
is equivalent to
From the definition of fermionic adic integral on and (2.11), we can derive
is equivalent to
From (2.12), (2.13), (2.14), (2.15), and (2.16), it is easy to show that
where are Carlitz's type Euler numbers defined by (see [18])
Therefore, we obtain the recurrence formula for the Carlitz's type Euler numbers as follows:
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
Carlitz's type Euler numbers can be determined inductively by
with the usual convention of replacing by
Carlitz type Euler polynomials are defined by means of the generating function as follows:
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
From (2.22), one gets the following.
Lemma 2.2.
(1)
(2)
It is clear from (1) and (2) of Lemma 2.2 that
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
which leads to
The result now follows by using (1) of Proposition 2.3.
Corollary 2.5.
If then
Let with and be a fixed odd prime number. One sets
where with (cf. [7, 9]). Note that the natural map induces
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
where and Then, one has the generating function of generalized Carlitz type Euler polynomials attached to
Now, fixed any with and From (2.31), one has
where and with By (2.31) and (2.32), one can derive
where and with Therefore, one obtains the following.
Theorem 2.6.
where and
Let denote the TeichmÃ¼ller character mod For one sets
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
Then, is analytic for
The values of this function at nonpositive integers are given by
Theorem 2.7.
For integers
where In particular, if then
Proof.
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
Therefore, we obtain the following
If then so that is a multiple of From (2.40), we derive
Thus, we have
By Corollary 2.5, we easily see that
From (2.42) and (2.43), we have
since From Theorem 2.7 and (2.44), we have
for Therefore, we have the following theorem.
Theorem 2.8.
Let be a positive integer multiple of and and let
Then, is analytic for and
Furthermore, for
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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 (20100001654). 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