On Some Arithmetical Properties of the Genocchi Numbers and Polynomials
- Kyoung Ho Park^{1} and
- Young-Hee Kim^{2}Email author
DOI: 10.1155/2008/195049
© K. H. Park and Y.-H. Kim. 2008
Received: 31 October 2008
Accepted: 25 December 2008
Published: 11 February 2009
Abstract
We investigate the properties of the Genocchi functions and the Genocchi polynomials. We obtain the Fourier transform on the Genocchi function. We have the generating function of -Genocchi polynomials. We define the Cangul-Ozden-Simsek's type twisted -Genocchi polynomials and numbers. We also have the generalized twisted -Genocchi numbers attached to the Dirichlet's character . Finally, we define zeta functions related to -Genocchi polynomials and have the generating function of the generalized -Genocchi numbers attached to .
1. Introduction
After Carlitz introduced an interesting -analogue of Frobenius-Euler numbers in [1], -Bernoulli and -Euler numbers and polynomials have been studied by several authors. Recently, many authors have an interest in the -extension of the Genocchi numbers and polynomials(cf. [2–5]). Kim et al. [5] defined the -Genocchi numbers and the -Genocchi polynomials. In [3], Kim derived the -analogs of the Genocchi numbers and polynomials by constructing -Euler numbers. He also gave some interesting relations between -Euler and -Genocchi numbers. The first author et al. [6] obtained the distribution relation for the Genocchi polynomials.
The main aim of this paper is to derive the Fourier transform for the Genocchi function. Recently, Kim [7] investigated the properties of the Euler functions and derived the interesting formula related to the infinite series by using the Fourier transform for the Euler function. In this paper, we investigate some arithmetical properties of the Genocchi functions and the Genocchi polynomials.
In [8], Cangul-Ozden-Simsek constructed new generating functions of the twisted -extension of twisted Euler polynomials and numbers attached to the Dirichlet character . Cangul et al. [8] also defined the twisted -extension of zeta functions, which interpolate the twisted -extension of Euler numbers at negative integers. In this paper, we define the Cangul-Ozden-Simsek type twisted -Genocchi polynomials and numbers. We have the generating function of -Genocchi polynomials. We have the generalized twisted -Genocchi numbers attached to the Dirichlet character . We define zeta functions related to -Genocchi polynomials and we have the generating function of the generalized -Genocchi numbers attached to .
We say that is uniformly differential function at a point , and we write , if the difference quotients, have a limit as .
where . (For details see [1–44].)
In this paper, we investigate arithmetical properties of the Genocchi functions and the Genocchi polynomials. In Section 2, we derive the Fourier transform on the Genocchi function. In Section 3, we define the Cangul-Ozden-Simsek type twisted -Genocchi polynomials and numbers. We have the generating function of -Genocchi polynomials. We also have the generalized twisted -Genocchi numbers attached to . In Section 4, we define zeta functions related to -Genocchi polynomials and we have the generating function of the generalized -Genocchi numbers attached to .
2. Genocchi Numbers and Functions
where is the Kronecker symbol.
Thus, we obtain the following lemma.
Lemma 2.1.
For , one has .
By (2.7), we obtain the following proposition.
Proposition 2.2.
From now on, we assume that is the Genocchi function. Let us consider the Fourier transform for the Genocchi function as follows.
Therefore, we obtain the following theorem.
Theorem 2.3.
By (2.17) and Lemma 2.1, we obtain the following corollary.
Corollary 2.4.
By (2.20), we obtain the following corollary.
Corollary 2.5.
3. -Extension of Twisted Genocchi Numbers and Polynomials
In this section, we will define the -extensions of twisted Genocchi numbers and polynomials which are the Cangul-Ozden-Simsek type twisted -Genocchi numbers and polynomials, respectively. We will have the generating function of -Genocchi polynomials and the generalized twisted -Genocchi numbers attached to .
By using the multivariate integral, we can also consider the multiple Genocchi numbers and polynomials.
and then is a -adic locally constant space.
where is the twisted -Euler polynomials.
where .
Now, we also consider the Cangul-Ozden-Simsek type twisted -Genocchi numbers attached to as follows.
where are called the generalized twisted -Euler numbers attached to .
4. Zeta Functions Related to the Genocchi Polynomials
where .
for . Therefore, we obtain the following proposition.
Proposition 4.1.
From Proposition 4.1, we can derive the Genocchi zeta function which interpolates Genocchi polynomials related to -Genocchi polynomials at negative integers.
For , we define the Hurwitz-type Genocchi zeta functions related to -Genocchi polynomials and numbers as follows.
Definition 4.2.
By Proposition 4.1 and Definition 4.2, we obtain the following theorem.
Theorem 4.3.
From (4.10) and (4.11), we obtain the following theorem.
Theorem 4.4.
Authors’ Affiliations
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