On Carlitz's Type -Euler Numbers Associated with the Fermionic -Adic Integral on

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:

(1.1)

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

(1.2)

From (1.2), we know the recurrence formula of Euler numbers is given by

(1.3)

with the usual convention of replacing by (see [7, 18]).

In [17], the -extension of Euler numbers are defined as

(1.4)

with the usual convention of replacing by

As the same motivation of the construction in [18], Carlitz's type -Euler numbers are defined as

(1.5)

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

(1.6)

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

(1.7)

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

(2.1)

(cf. [5–17, 19, 20, 22]). The bosonic -adic integral on is considered as the limit that is,

(2.2)

From (2.1), we have the fermionic -adic integral on as follows:

(2.3)

Using (2.3), we can readily derive the classical Euler polynomials, namely

(2.4)

In particular, when is the well-known the Euler numbers (cf. [7, 16, 19]).

By definition of we show that

(2.5)

where (see [7]). By (2.5) and induction, we obtain

(2.6)

where and From (2.6), we note that

(2.7)

For and any integer we define

(2.8)

It is easy to see that (see [23, page 172]). We put with and We define for by

(2.9)

If we set in (2.7), we have

(2.10)

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

(2.11)

Also, if in (2.5), then

(2.12)

On the other hand, by (2.12), we obtain that

(2.13)

is equivalent to

(2.14)

From the definition of fermionic -adic integral on and (2.11), we can derive

(2.15)

is equivalent to

(2.16)

From (2.12), (2.13), (2.14), (2.15), and (2.16), it is easy to show that

(2.17)

where are Carlitz's type -Euler numbers defined by (see [18])

(2.18)

Therefore, we obtain the recurrence formula for the Carlitz's type -Euler numbers as follows:

(2.19)

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

(2.20)

Carlitz's type -Euler numbers can be determined inductively by

(2.21)

with the usual convention of replacing by

Carlitz type -Euler polynomials are defined by means of the generating function as follows:

(2.22)

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

(2.23)

From (2.22), one gets the following.

Lemma 2.2.

(1)

(2)

It is clear from (1) and (2) of Lemma 2.2 that

(2.24)

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

(2.25)

which leads to

(2.26)

The result now follows by using (1) of Proposition 2.3.

Corollary 2.5.

If then

(2.27)

Let with and be a fixed odd prime number. One sets

(2.28)

where with (cf. [7, 9]). Note that the natural map induces

(2.29)

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

(2.30)

where and Then, one has the generating function of generalized Carlitz type -Euler polynomials attached to

(2.31)

Now, fixed any with and From (2.31), one has

(2.32)

where and with By (2.31) and (2.32), one can derive

(2.33)

where and with Therefore, one obtains the following.

Theorem 2.6.

(2.34)

where and

Let denote the Teichmüller character mod For one sets

(2.35)

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

(2.36)

Then, is analytic for

The values of this function at nonpositive integers are given by

Theorem 2.7.

For integers

(2.37)

where In particular, if then

Proof.

(2.38)

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

(2.39)

Therefore, we obtain the following

(2.40)

If then so that is a multiple of From (2.40), we derive

(2.41)

Thus, we have

(2.42)

By Corollary 2.5, we easily see that

(2.43)

From (2.42) and (2.43), we have

(2.44)

since From Theorem 2.7 and (2.44), we have

(2.45)

for Therefore, we have the following theorem.

Theorem 2.8.

Let be a positive integer multiple of and and let

(2.46)

Then, is analytic for and

(2.47)

Furthermore, for

(2.48)