Theory and Modern Applications

# A Note on -Genocchi Polynomials and Numbers of Higher Order

## Abstract

We investigate several arithmetic properties of -Genocchi polynomials and numbers of higher order.

## 1. Introduction and Preliminaries

Recently, Kim [1] studied -Genocchi and Euler numbers using Fermionic -integral and introduced related applications. Kim [2] also gives the -extensions of the Euler numbers which can be viewed as interpolating of -analogue of Euler zeta function at negative integers and gives Bernoulli numbers at negative integers by interpolating Riemann zeta function. These numbers are very useful for number theory and mathematical physics. Kim [3, 4] studied -Bernoulli numbers and polynomials related to Gaussian binomial coefficient and studied also some identities of -Euler polynomials and -stirling numbers. Kim [5] made Dedekind DC sum in the meaning of extension of Dedekind sum or Hardy sum and introduced lots of interesting results. The purpose of this paper is to investigate several arithmetic properties of -Genocchi polynomials and numbers of higher order.

Let be a fixed odd prime. Throughout this paper , , , and will, respectively, denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then we assume so that for . We also use the notations

(1.1)

for all (see [512]). Hence, .

Let be a fixed positive integer with . We now set

(1.2)

where lies in . For any , we set

(1.3)

and this can be extended to a distribution on .

We say that is a uniformly differentiable function at a point and write , if the difference quotients have a limit as (cf. [1323]).

For , the -adic invariant integral on is defined as

(1.4)

(see [14, 23]). Let and . From (1.4), we have

(1.5)

The -adic integral has been used in many areas such as mathematics, physics, probability theory, dynamical systems, and biological models. Especially, Khrennikov [2426] applied to many areas using ingenious technique. The Genocchi numbers and polynomials are defined by the generating functions as follows:

(1.6)

(see [5, 7, 15]). The -extension of Genocchi numbers are defined by

(1.7)

(see [1, 2]), and the -extension of Genocchi polynomials is also given by

(1.8)

In Section 2, we investigate several arithmetic properties of -Genocchi polynomials and numbers of higher order.

## 2. -Genocchi Numbers of Higher Order

Let and with . The -Genocchi polynomials of order are defined as

(2.1)

where . It is easily to see that for each and . From (2.1), we can obtain the following theorem.

Theorem 2.1.

Let and . Then for all ,

(2.2)

From Theorem 2.1, if we take , then

(2.3)

Now, we define -Genocchi number of higher order as follows:

(2.4)

From (2.4), we can derive the following theorem.

Theorem 2.2.

Let and . Then one has

(2.5)

where .

Note that , where are the ordinary Genocchi numbers of order defined as

(2.6)

By (2.4) and (2.5), we can obtain the following theorem.

Theorem 2.3.

Let . Then one has

(2.7)

It is easily to check that

(2.8)

where with . Thus we have the following theorem.

Theorem 2.4.

Let with . Then for all ,

(2.9)

We note that if we take , then we have

(2.10)

where . By (2.10), we easily see that

(2.11)

Note that , where are the th Genocchi numbers defined as

(2.12)

From (2.11), we can see that

(2.13)

Let be the generating function of as follows:

(2.14)

By (2.7) and (2.14), we see that

(2.15)

By (2.14) and (2.15), we can obtain the following theorem.

Theorem 2.5.

Let . Then for all ,

(2.16)

## References

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

2. Kim T:Note on the Euler -zeta functions. Journal of Number Theory 2009,129(7):1798-1804. 10.1016/j.jnt.2008.10.007

3. Kim T:-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51-57.

4. Kim T:Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on . Russian Journal of Mathematical Physics 2009, 16: 501-508.

5. Kim T: Note on Dedekind type DC sums. Advanced Studies in Contemporary Mathematics 2009,18(2):249-260.

6. Kim T:A note on some formulae for the -Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2006,9(2):227-232.

7. Kim T:A note on the generalized -Euler numbers. Proceedings of the Jangjeon Mathematical Society 2009,12(1):45-50.

8. Kim T:Note on the -Euler numbers of higher order. Advanced Studies in Contemporary Mathematics 2009,19(1):25-29.

9. Kim Y-H, Kim W, Ryoo CS:On the twisted -Euler zeta function associated with twisted -Euler numbers. Proceedings of the Jangjeon Mathematical Society 2009,12(1):93-100.

10. Ozden H, Simsek Y, Rim S-H, Cangul IN:A note on -adic -Euler measure. Advanced Studies in Contemporary Mathematics 2007, 14: 233-239.

11. Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Advanced Studies in Contemporary Mathematics 2008,16(2):251-278.

12. Simsek Y, Kurt V, Kim D:New approach to the complete sum of products of the twisted -Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007,14(1):44-56. 10.2991/jnmp.2007.14.1.5

13. Cenkci M, Simsek Y, Kurt V:Further remarks on multiple -adic --function of two variables. Advanced Studies in Contemporary Mathematics 2007,14(1):49-68.

14. Kim T:-Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288-299.

15. Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003,10(3):261-267.

16. Kim T:Analytic continuation of multiple -zeta functions and their values at negative integers. Russian Journal of Mathematical Physics 2004,11(1):71-76.

17. 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.

18. Kim T:Multiple -adic -function. Russian Journal of Mathematical Physics 2006,13(2):151-157. 10.1134/S1061920806020038

19. Kim T:A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007, 15: 133-138.

20. Kim T:On -adic interpolating function for -Euler numbers and its derivatives. Journal of Mathematical Analysis and Applications 2008,339(1):598-608. 10.1016/j.jmaa.2007.07.027

21. 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.2006.09.027

22. Kim T:A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):133-137.

23. Kim T:-Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007,14(1):15-27. 10.2991/jnmp.2007.14.1.3

24. Khrennikov AYu: p-adic Valued Distributions and Their Applications to the Mathematical Physics. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1994.

25. Khrennikov AYu: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Mathematics and Its Applications. Volume 427. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xviii+371.

26. Khrennikov AYu: Interpretations of Probability. VSP, Utrecht, The Netherlands; 1999:ii+228.

## Acknowledgment

This paper was supported by KOSEF (2009-0073396, 2009-A419-0065).

## Author information

Authors

### Corresponding author

Correspondence to Lee-Chae Jang.

## Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

Jang, LC., Hwang, KW. & Kim, YH. A Note on -Genocchi Polynomials and Numbers of Higher Order. Adv Differ Equ 2010, 309480 (2010). https://doi.org/10.1155/2010/309480

• Revised:

• Accepted:

• Published:

• DOI: https://doi.org/10.1155/2010/309480

### Keywords

• Differential Equation
• Partial Differential Equation
• Ordinary Differential Equation
• Functional Analysis
• Functional Equation