- Yilmaz Simsek^{1}Email author,
- Ismail Naci Cangul^{2},
- Veli Kurt^{1} and
- Daeyeoul Kim^{3}
DOI: 10.1155/2008/815750
© Yilmaz Simsek et al. 2008
Received: 19 March 2007
Accepted: 14 December 2007
Published: 25 December 2007
Abstract
The main purpose of this paper is to study on generating functions of the -Genocchi numbers and polynomials. We prove new relation for the generalized -Genocchi numbers which is related to the -Genocchi numbers and -Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define -Genocchi zeta and -functions, which are interpolated -Genocchi numbers and polynomials at negative integers. We also give some applications of generalized -Genocchi numbers.
1. Introduction Definitions and Notations
(cf. [21]), where is Euler's gamma function and (cf. [1], [13, page 108, equation (2.43)]). The first author defined -analogue of the Genocchi zeta functions as follows [21].
Definition 1.1.
Remark 1.2.
Definition 1.3 ({see [21]}).
where , , and , cf. [13].
where and cf. [13].
We summarize our work as follows. In Section 2, we study on generating functions of the -Genocchi numbers and polynomials. By using infinite and finite series, we give some definitions of the -Genocchi numbers and polynomials. We find new relations between generalized -Genocchi numbers with attached to -Genocchi numbers and Barnes' type Changhee -Bernoulli numbers. In Section 3, by applying Mellin transformation and derivative operator to the generating functions of the -Genocchi numbers, we construct -Genocchi zeta and -functions, which are interpolated -Genocchi numbers and polynomials at negative integers. We also give some new relations related to these numbers and polynomials.
2. -genocchi Number and Polynomials
(cf.[3, 10, 11, 23]), where denotes -Genocchi numbers.
We note that -Genocchi numbers, were defined by Kim [3, 10, 11].
Then by comparing coefficients of on both sides of the above equation, for we obtain the following result.
Theorem 2.1.
By comparing coefficients of on both sides of the above equation, we arrive at the following corollary.
Corollary 2.2.
We give some of -Genocchi polynomials as follows:
From the generating function we have the following.
Corollary 2.3.
Proof of the Corollary 2.3 was given by Kim [3, 12]. We give some of -Genocchi numbers as follows: ,
After some elementary calculations, we arrive at the following corollary.
Corollary 2.4.
Corollary 2.5.
Proof.
Proof of this corollary is easily obtained from (2.4).
where denotes the Dirichlet character with conductor the set of positive integers.
Observe that when (2.13) reduces to (2.3).
In [15], Srivastava et al. defined the following generalized Barnes-type Changhee -Bernoulli numbers.
By substituting (2.9) and (2.19) into (2.16), after some calculations, we arrive at the following theorem.
Theorem 2.6.
Remark 2.7.
In Theorem 2.6, we give new relations between generalized -Genocchi numbers, with attached to , -Genocchi numbers, , and Barnes-type Changhee -Bernoulli numbers. For detailed information about generalized Barnes-type Changhee -Bernoulli numbers with attached to see [15].
Theorem 2.8.
Remark 2.9.
3. -genocchi Zeta and -Functions
Thus, Hurwitz-type -Genocchi zeta function is defined by the following definition.
Definition 3.1.
Hurwitz-type -Genocchi zeta function interpolates -Genocchi polynomials at negative integers. For , , and by applying Cauchy residue theorem to (3.1), we can obtain the following theorem.
Theorem 3.2.
Remark 3.3.
Thus we obtained the desired result.
Thus we can define Dirichlet-type -Genocchi -function as follows.
Definition 3.4.
Relation between and is given by the following theorem.
Theorem 3.5.
Proof.
After some elementary calculations, we arrive at the desired result of the theorem.
The function interpolates generalized -Genocchi numbers, which are given by the following theorem.
Theorem 3.6.
Proof.
Proof of this theorem is similar to that of Theorem 3.2. So we omit the proof.
By comparing both sides of the above equation and Theorem 3.6, we obtain distributions relation of the generalized Genocchi numbers as follows.
Corollary 3.7.
where and is the -Genocchi polynomial.
By substituting (2.5) into (3.12), we have the following corollary.
Corollary 3.8.
Thus we arrive at the following corollary.
Corollary 3.9.
Declarations
Acknowledgments
The first and third authors have been supported by the Scientific Research Project, Administration Akdeniz University. The second author has been supported by Uludag University Research Fund, Projects no. F2004/40 and F2008-31. The fourth author has been supported by National Institute for Mathematical Sciences Doryong-dong, Yuseong-gu, Daejeon. The authors express their sincere gratitude to referees for their suggestions and comments.
Authors’ Affiliations
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