- Yilmaz Simsek^{1}Email author,
- Ismail Naci Cangul^{2},
- Veli Kurt^{1} and
- Daeyeoul Kim^{3}
https://doi.org/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
References
- Jang L-C, Kim T, Lee D-H, Park D-W: An application of polylogarithms in the analogs of Genocchi numbers. Notes on Number Theory and Discrete Mathematics 2001, 7(3):65-69.MathSciNetGoogle Scholar
- Kim T, Jang L-C, Pak HK:A note on -Euler and Genocchi numbers. Proceedings of the Japan Academy. Series A 2001, 77(8):139-141. 10.3792/pjaa.77.139MATHMathSciNetView ArticleGoogle Scholar
- Kim T:A note on the -Genocchi numbers and polynomials. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
- Rim S-H, Kim T, Ryoo CS:On the alternating sums of powers of consecutive -integers. Bulletin of the Korean Mathematical Society 2006, 43(3):611-617.MATHMathSciNetView ArticleGoogle Scholar
- Kim T:Sums of powers of consecutive -integers. Advanced Studies in Contemporary Mathematics 2004, 9(1):15-18.MATHMathSciNetGoogle Scholar
- Kim T:Power series and asymptotic series associated with the -analog of the two-variable -adic -function. Russian Journal of Mathematical Physics 2005, 12(2):186-196.MATHMathSciNetGoogle Scholar
- Kim T, Rim S-H, Simsek Y:A note on the alternating sums of powers of consecutive -integers. Advanced Studies in Contemporary Mathematics 2006, 13(2):159-164.MATHMathSciNetGoogle Scholar
- Simsek Y: -analogue of twisted -series and -twisted Euler numbers. Journal of Number Theory 2005, 110(2):267-278. 10.1016/j.jnt.2004.07.003MATHMathSciNetView ArticleGoogle Scholar
- Simsek Y, Kim D, Kim T, Rim S-H:A note on the sums of powers of consecutive -integers. Journal of Applicable Functional Differential Equations 2006, 1(1):81-88.MATHMathSciNetGoogle Scholar
- Kim T:A note on -Volkenborn integration. Proceedings of the Jangjeon Mathematical Society 2005, 8(1):13-17.MATHMathSciNetGoogle Scholar
- Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007, 14(1):15-27. 10.2991/jnmp.2007.14.1.3MATHMathSciNetView ArticleGoogle Scholar
- Kim T:On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007, 326(2):1458-1465. 10.1016/j.jmaa.2006.03.037MATHMathSciNetView ArticleGoogle Scholar
- Kim T, et al.: Introduction to Non-Archimedian Analysis, Kyo Woo Sa. Korea, 2004, http://www.kyowoo.co.kr/
- Shiratani K, Yamamoto S:On a -adic interpolation function for the Euler numbers and its derivatives. Memoirs of the Faculty of Science, Kyushu University. Series A 1985, 39(1):113-125. 10.2206/kyushumfs.39.113MATHMathSciNetView ArticleGoogle Scholar
- Srivastava HM, Kim T, Simsek Y: -Bernoulli numbers and polynomials associated with multiple -zeta functions and basic -series. Russian Journal of Mathematical Physics 2005, 12(2):241-268.MATHMathSciNetGoogle Scholar
- Waldschmidt M, Moussa P, Luck JM, Itzykson C (Eds): From Number Theory to Physics. Springer, Berlin, Germany; 1995.Google Scholar
- Kim T:On the analogs of Euler numbers and polynomials associated with -adic -integral on at . Journal of Mathematical Analysis and Applications 2007, 331(2):779-792. 10.1016/j.jmaa.2006.09.027MATHMathSciNetView ArticleGoogle Scholar
- Kim T, Rim S-H:A note on two variable Dirichlet's -function. Advanced Studies in Contemporary Mathematics 2005, 10(1):1-6.MATHMathSciNetGoogle Scholar
- Simsek Y, Kim D, Rim S-H:On the two-variable Dirichlet - -series. Advanced Studies in Contemporary Mathematics 2005, 10(2):131-142.MATHMathSciNetGoogle Scholar
- Simsek Y: -Dedekind type sums related to -zeta function and basic -series. Journal of Mathematical Analysis and Applications 2006, 318(1):333-351. 10.1016/j.jmaa.2005.06.007MATHMathSciNetView ArticleGoogle Scholar
- Simsek Y: -Hardy-Berndt type sums associated with -Genocchi type zeta and -functions. http://arxiv.org/abs/0710.5681v1
- Cenkci M, Can M, Kurt V: -adic interpolation functions and Kummer-type congruences for -twisted and -generalized twisted Euler numbers. Advanced Studies in Contemporary Mathematics 2004, 9(2):203-216.MATHMathSciNetGoogle Scholar
- Cangul IN, Kurt V, Simsek Y, Pak HK, Rim S-H:An invariant -adic -integral associated with -Euler numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007, 14(1):8-14. 10.2991/jnmp.2007.14.1.2MathSciNetView ArticleGoogle Scholar
- Tsumura H:A note on -analogues of the Dirichlet series and -Bernoulli numbers. Journal of Number Theory 1991, 39(3):251-256. 10.1016/0022-314X(91)90048-GMATHMathSciNetView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.