© Min-Soo Kim et al. 2010
Received: 2 August 2010
Accepted: 28 September 2010
Published: 11 October 2010
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 .
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
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]).
where is a complex number with (see ). 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.  could not determine the generating function of Carlitz's type -Euler numbers and Witt's formula for Carlitz's type -Euler numbers.
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 .
From (2.22), one gets the following.
From (2.24), we may state the following.
The result now follows by using (1) of Proposition 2.3.
The values of this function at nonpositive integers are given by
Therefore by (2.30), the theorem is proved.
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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