- Kyoung Ho Park^{1}Email author
https://doi.org/10.1155/2009/946569
© Kyoung Ho Park. 2009
Received: 5 November 2008
Accepted: 14 January 2009
Published: 20 January 2009
Abstract
The aim of this paper is to construct -adic twisted two-variable Euler-( , )- -functions, which interpolate generalized twisted ( , )-Euler polynomials at negative integers. In this paper, we treat twisted ( , )-Euler numbers and polynomials associated with -adic invariant integral on . We will construct two-variable twisted ( , )-Euler-zeta function and two-variable ( , )- -function in Complex -plane.
1. Introduction
Tsumura and Young treated the interpolation functions of the Bernoulli and Euler polynomials in [1, 2]. Kim and Simsek studied on -adic interpolation functions of these numbers and polynomials [3–48]. In [49], Carlitz originally constructed -Bernoulli numbers and polynomials. Many authors studied these numbers and polynomials [4, 28, 38, 41, 50]. After that, twisted -Bernoulli and Euler numbers(polynomials) were studied by several authors [1–32, 32–65]. In [62], Whashington constructed one-variable -adic- -function which interpolates generalized classical Bernoulli numbers at negative integers. Fox introduced the two-variable -adi -functions [53]. Young defined -adic integral representation for the two-variable -adic -functions [64]. Furthermore, Kim constructed the two-variable -adic - -function, which is interpolation function of the generalized -Bernoulli polynomials [8]. This function is the -extension of the two-variable -adic -function. Kim constructed -extension of the generalized formula for two-variable of Diamond and Ferrero and Greenberg formula for two-variable -adic -function in the terms of the -adic gamma and log-gamma functions [8]. Kim and Rim introduced twisted -Euler numbers and polynomials associated with basic twisted - -functions [28]. Also, Jang et al. investigated the -adic analogue twisted - -function, which interpolates generalized twisted -Euler numbers attached to Dirichlet's character [55]. Kim et al. have studied two-variable -adic -functions, which interpolate the generalized Bernoulli polynomials at negative integers. In this paper, we will construct two-variale -adic twisted Euler - -functions. This functions interpolation functions of the generalized twisted -Euler polynomials.
Hence, , for any with in the present -adic case.
We say that is uniformly differential function at a point , and we write , if the difference quotients, have a limit as .
2. Twisted -Euler Numbers and Polynomials
Remark 2.1.
where and . In particular, if we take , then . These numbers are called -Euler numbers.
Let be the Dirichlet's character with conductor with . Then the generalized twisted -Euler polynomials attached to is given by as follows:
where is any multiple of with and .
3. Two-Variable Twisted -Euler-Zeta Function and - -Function
In this section, we will construct two-variable twisted -Euler-zeta function and two-variable - -function in Complex -plane. We assume with .
where and is th root of unity. In particular, if we take , then we have . These numbers are called twisted Euler numbers. By using derivative operator, we have .
From now on, we will define the two-variable - -functions which interpolates the generalized -Euler polynomials.
Definition 3.1.
Thus, we see the function which interpolates the generalized -Euler polynomials as follows.
Theorem 3.2.
Thus, we have the following theorem.
Theorem 3.3.
Remark 3.4.
From (3.9) and (3.10), we have the following corollary.
Corollary 3.5.
Secondly, we will define two-variable twisted Euler - -function as follows.
Definition 3.6.
We will investigate the relations between and as follows.
Thus we obtain the following theorem.
Theorem 3.7.
Thus, we see that the function interpolates generalized -Euler polynomials attached to at negative integer values of as followings.
Theorem 3.8.
Note that if we take , then Theorem 3.8 reduces to Theorem 3.3.
Equation (3.20) means that the function interpolates polynomials at negative integers.
From (3.16) and (3.20), we have the following theorem.
Theorem 3.9.
Remark 3.10.
From (2.12), if we take , then we have the following corollary.
Corollary 3.11.
4. -Adic Twisted Two-Variable Euler -L-Functions
In [62], Washington constructed one-variable -adic- -function which interpolates generalized classical Bernoulli numbers negative integers. Kim [22] investigated the -adic analogues of two-variables Euler - -function. In this section, we will construct -adic twisted two-variable Euler- - -functions, which interpolate generalized twisted -Euler polynomials at negative integers. Our notations and methods are essentially due to Kim and Washington (cf. [22, 62]).
be a sequence of power series, each of which converges in a fixed subset such that
Then for all (cf. [2, 22, 50, 51, 60, 62]).
Let be the Dirichlet's character with conductor with and let be a positive multiple of and .
Therefore we obtain the following theorem.
Theorem 4.1.
Thus we note that for all , where is twisted -adic Euler - -function, (cf. [15, 22]).
Thus we have the following theorem.
Theorem 4.2.
Authors’ Affiliations
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