- Min-Soo Kim
^{1}, - Taekyun Kim
^{2}Email author and - Cheon-Seoung Ryoo
^{3}

**2010**:358986

https://doi.org/10.1155/2010/358986

© Min-Soo Kim et al. 2010

**Received: **2 August 2010

**Accepted: **28 September 2010

**Published: **11 October 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
.

## Keywords

## 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

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]).

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

with the usual convention of replacing by

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.

## 2. Carlitz's Type -Euler Numbers in the -Adic Case

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

with the usual convention of replacing by

From (2.22), one gets the following.

Lemma 2.2.

From (2.24), we may state the following.

Proposition 2.3.

Proposition 2.4.

For the value of is times the coefficient of in the formal expansion of in powers of That is,

Proof.

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

Corollary 2.5.

Hereafter, if is a function on one denotes by the same the function on Namely one considers as a function on

where and with Therefore, one obtains the following.

Theorem 2.6.

Note that since is defined by for (cf. [10, 12, 21]). One notes that is analytic for

The values of this function at nonpositive integers are given by

Theorem 2.7.

Proof.

Therefore by (2.30), the theorem is proved.

for Therefore, we have the following theorem.

Theorem 2.8.

## Declarations

### 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.

## Authors’ Affiliations

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