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

*Advances in Difference Equations*
**volume 2011**, Article number: 424809 (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

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

(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

(see [1]). In the special case , are called the th generalized Genocchi numbers attached to (see [1, 4–6]).

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

(see [7]). In the special case , are called the th generalized Genocchi numbers attached to of order (see [1, 4–9]). From (1.3) and (1.4), we derive .

Let us assume that with as an indeterminate. Then we, use the notation

The -factorial is defined by

and the Gaussian binomial coefficient is also defined by

It is known that

(see [5, 10]). The -binomial formula are known that

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 [1–15]). 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:

Note that

By (2.1) and (1.4), we can see that . From (2.1), we note that

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

Note that

Thus we obtain the following corollary.

Corollary 2.2.

For , we have

For and , one also considers the extended higher-order generalized -Genocchi polynomials as follows:

From (2.7), one notes that

where .

Therefore, we obtain the following theorem.

Theorem 2.3.

For , one has

Note that

By (2.10), one sees that

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

Corollary 2.4.

For , we have

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

Corollary 2.5.

For with , we have

For in Theorem 2.3, we obtain the following corollary.

Corollary 2.6.

For , one has

In particular,

Let in Corollary 2.6. Then one has

Let . Then, one has defines Barnes' type generalized -Genocchi polynomials attached to as follows:

By (2.17), one sees that

It is easy to see that

Therefore, we obtain the following theorem.

Theorem 2.7.

For , one has

## 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. - 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. - 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.2006.03.037 - 4.
Jang L-C:

**A study on the distribution of twisted****-Genocchi polynomials.***Advanced Studies in Contemporary Mathematics (Kyungshang)*2009,**18**(2):181-189. - 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. - 6.
Kim T:

**A note on the****-Genocchi numbers and polynomials.***Journal of Inequalities and Applications*2007,**2007:**-8. - 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. - 8.
Rim S-H, Park KH, Moon EJ:

**On Genocchi numbers and polynomials.***Abstract and Applied Analysis*2008,**2008:**-7. - 9.
Ryoo CS:

**Calculating zeros of the twisted Genocchi polynomials.***Advanced Studies in Contemporary Mathematics (Kyungshang)*2008,**17**(2):147-159. - 10.
Govil NK, Gupta V:

**Convergence of****-Meyer-König-Zeller-Durrmeyer operators.***Advanced Studies in Contemporary Mathematics (Kyungshang)*2009,**19**(1):97-108. - 11.
Kim T:

**Barnes-type multiple***-zeta functions and**-Euler polynomials.**Journal of Physics*2010,**43**(25):-11. - 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. - 13.
Kim T:

**On the multiple***-Genocchi and Euler numbers.**Russian Journal of Mathematical Physics*2008,**15**(4):481-486. 10.1134/S1061920808040055 - 14.
Kim T:

**Note on the Euler****-zeta functions.***Journal of Number Theory*2009,**129**(7):1798-1804. 10.1016/j.jnt.2008.10.007 - 15.
Cenkci M, Can M, Kurt V:

**-extensions of Genocchi numbers.***Journal of the Korean Mathematical Society*2006,**43**(1):183-198.

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Ryoo, C., Kim, T., Choi, J. *et al.* On the Generalized -Genocchi Numbers and Polynomials of Higher-Order.
*Adv Differ Equ* **2011, **424809 (2011). https://doi.org/10.1155/2011/424809

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### Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation