Open Access

On the Generalized -Genocchi Numbers and Polynomials of Higher-Order

Advances in Difference Equations20112011:424809

https://doi.org/10.1155/2011/424809

Received: 10 August 2010

Accepted: 18 February 2011

Published: 10 March 2011

Abstract

We first consider the -extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to . The purpose of this paper is to present a systemic study of some families of higher-order generalized -Genocchi numbers and polynomials attached to by using the generating function of those numbers and polynomials.

1. Introduction

As a well known definition, the Genocchi polynomials are defined by
(11)
where we use the technical method's notation by replacing by , symbolically, (see [1, 2]). In the special case , are called the th Genocchi numbers. From the definition of Genocchi numbers, we note that , and even coefficients are given by (see [3]), where is a Bernoulli number and is an Euler polynomial. The first few Genocchi numbers for are . The first few prime Genocchi numbers are given by and . It is known that there are no other prime Genocchi numbers with . For a real or complex parameter , the higher-order Genocchi polynomials are defined by
(12)
(see [1, 4]). In the special case , are called the th Genocchi numbers of order . From (1.1) and (1.2), we note that . For with , let be the Dirichlet character with conductor . It is known that the generalized Genocchi polynomials attached to are defined by
(13)

(see [1]). In the special case , are called the th generalized Genocchi numbers attached to (see [1, 46]).

For a real or complex parameter , the generalized higher-order Genocchi polynomials attached to are also defined by
(14)

(see [7]). In the special case , are called the th generalized Genocchi numbers attached to of order (see [1, 49]). From (1.3) and (1.4), we derive .

Let us assume that with as an indeterminate. Then we, use the notation
(15)
The -factorial is defined by
(16)
and the Gaussian binomial coefficient is also defined by
(17)
(see [5, 10]). Note that
(18)
It is known that
(19)
(see [5, 10]). The -binomial formula are known that
(110)

(see[10, 11]).

There is an unexpected connection with -analysis and quantum groups, and thus with noncommutative geometry -analysis is a sort of -deformation of the ordinary analysis. Spherical functions on quantum groups are -special functions. Recently, many authors have studied the -extension in various areas (see [115]). Govil and Gupta [10] have introduced a new type of -integrated Meyer-König-Zeller-Durrmeyer operators, and their results are closely related to the study of -Bernstein polynomials and -Genocchi polynomials, which are treated in this paper. In this paper, we first consider the -extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to . The purpose of this paper is to present a systemic study of some families of higher-order generalized -Genocchi numbers and polynomials attached to by using the generating function of those numbers and polynomials.

2. Generalized -Genocchi Numbers and Polynomials

For , let us consider the -extension of the generalized Genocchi polynomials of order attached to as follows:
(21)
Note that
(22)
By (2.1) and (1.4), we can see that . From (2.1), we note that
(23)

In the special case , are called the th generalized -Genocchi numbers of order attached to . Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has
(24)
Note that
(25)

Thus we obtain the following corollary.

Corollary 2.2.

For , we have
(26)
For and , one also considers the extended higher-order generalized -Genocchi polynomials as follows:
(27)
From (2.7), one notes that
(28)

where .

Therefore, we obtain the following theorem.

Theorem 2.3.

For , one has
(29)
Note that
(210)
By (2.10), one sees that
(211)

By (2.10) and (2.11), we obtain the following corollary.

Corollary 2.4.

For , we have
(212)

By (2.7), we can derive the following corollary.

Corollary 2.5.

For with , we have
(213)

For in Theorem 2.3, we obtain the following corollary.

Corollary 2.6.

For , one has
(214)
In particular,
(215)
Let in Corollary 2.6. Then one has
(216)
Let . Then, one has defines Barnes' type generalized -Genocchi polynomials attached to as follows:
(217)
By (2.17), one sees that
(218)
It is easy to see that
(219)

Therefore, we obtain the following theorem.

Theorem 2.7.

For , one has
(220)

Authors’ Affiliations

(1)
Department of Mathematics, Hannam University
(2)
Division of General Education-Mathematics, Kwangwoon University
(3)
Department of Wireless Communications Engineering, Kwangwoon University

References

  1. Jang L-C, Hwang K-W, Kim Y-H:A note on -Genocchi polynomials and numbers of higher order. Advances in Difference Equations 2010, 2010:-6.Google Scholar
  2. Kurt V: A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Applied Mathematical Sciences 2009,3(53–56):2757-2764.MathSciNetMATHGoogle Scholar
  3. 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.2006.03.037MathSciNetView ArticleMATHGoogle Scholar
  4. Jang L-C:A study on the distribution of twisted -Genocchi polynomials. Advanced Studies in Contemporary Mathematics (Kyungshang) 2009,18(2):181-189.MathSciNetMATHGoogle Scholar
  5. Kim T: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Advanced Studies in Contemporary Mathematics (Kyungshang) 2010,20(1):23-28.MathSciNetMATHGoogle Scholar
  6. Kim T:A note on the -Genocchi numbers and polynomials. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
  7. Rim S-H, Lee SJ, Moon EJ, Jin JH:On the -Genocchi numbers and polynomials associated with -zeta function. Proceedings of the Jangjeon Mathematical Society 2009,12(3):261-267.MathSciNetMATHGoogle Scholar
  8. Rim S-H, Park KH, Moon EJ: On Genocchi numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-7.Google Scholar
  9. Ryoo CS: Calculating zeros of the twisted Genocchi polynomials. Advanced Studies in Contemporary Mathematics (Kyungshang) 2008,17(2):147-159.MathSciNetMATHGoogle Scholar
  10. Govil NK, Gupta V:Convergence of -Meyer-König-Zeller-Durrmeyer operators. Advanced Studies in Contemporary Mathematics (Kyungshang) 2009,19(1):97-108.MathSciNetMATHGoogle Scholar
  11. Kim T:Barnes-type multiple -zeta functions and -Euler polynomials. Journal of Physics 2010,43(25):-11.Google Scholar
  12. Cangul IN, Kurt V, Ozden H, Simsek Y:On the higher-order - -Genocchi numbers. Advanced Studies in Contemporary Mathematics (Kyungshang) 2009,19(1):39-57.MathSciNetGoogle Scholar
  13. Kim T:On the multiple -Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481-486. 10.1134/S1061920808040055MathSciNetView ArticleMATHGoogle Scholar
  14. Kim T:Note on the Euler -zeta functions. Journal of Number Theory 2009,129(7):1798-1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar
  15. Cenkci M, Can M, Kurt V: -extensions of Genocchi numbers. Journal of the Korean Mathematical Society 2006,43(1):183-198.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© C. S. Ryoo et al. 2011

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