- Lee-Chae Jang
^{1}Email author, - Kyung-Won Hwang
^{2}and - Young-Hee Kim
^{3}

**2010**:309480

**DOI: **10.1155/2010/309480

© Lee-Chae Jang et al. 2010

**Received: **10 July 2009

**Accepted: **11 February 2010

**Published: **16 February 2010

## Abstract

## 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

for all (see [5–12]). Hence, .

Let be a fixed positive integer with . We now set

where lies in . For any , we set

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. [13–23]).

For , the -adic invariant integral on is defined as

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

The -adic integral has been used in many areas such as mathematics, physics, probability theory, dynamical systems, and biological models. Especially, Khrennikov [24–26] applied to many areas using ingenious technique. The Genocchi numbers and polynomials are defined by the generating functions as follows:

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

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

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

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

Theorem 2.1.

From Theorem 2.1, if we take , then

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

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

Theorem 2.2.

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

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

Theorem 2.3.

It is easily to check that

where with . Thus we have the following theorem.

Theorem 2.4.

We note that if we take , then we have

where . By (2.10), we easily see that

Note that , where are the th Genocchi numbers defined as

From (2.11), we can see that

Let be the generating function of as follows:

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

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

Theorem 2.5.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Kim T:
**On the multiple**-Genocchi and Euler numbers.*Russian Journal of Mathematical Physics*2008,**15**(4):481-486. 10.1134/S1061920808040055MathSciNetView ArticleMATHGoogle Scholar - 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 - Kim T:
**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51-57.MathSciNetView ArticleMATHGoogle Scholar - 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.Google Scholar - Kim T:
**Note on Dedekind type DC sums.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):249-260.MathSciNetMATHGoogle Scholar - Kim T:
**A note on some formulae for the**-Euler numbers and polynomials.*Proceedings of the Jangjeon Mathematical Society*2006,**9**(2):227-232.MathSciNetMATHGoogle Scholar - Kim T:
**A note on the generalized**-Euler numbers.*Proceedings of the Jangjeon Mathematical Society*2009,**12**(1):45-50.MathSciNetMATHGoogle Scholar - Kim T:
**Note on the**-Euler numbers of higher order.*Advanced Studies in Contemporary Mathematics*2009,**19**(1):25-29.MathSciNetMATHGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - Ozden H, Simsek Y, Rim S-H, Cangul IN:
**A note on**-adic -Euler measure.*Advanced Studies in Contemporary Mathematics*2007,**14:**233-239.MathSciNetGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - 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.2007.14.1.5MathSciNetView ArticleMATHGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - Kim T:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288-299.MathSciNetMATHGoogle Scholar - Kim T:
**On Euler-Barnes multiple zeta functions.***Russian Journal of Mathematical Physics*2003,**10**(3):261-267.MathSciNetMATHGoogle Scholar - Kim T:
**Analytic continuation of multiple**-zeta functions and their values at negative integers.*Russian Journal of Mathematical Physics*2004,**11**(1):71-76.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar - Kim T:
**Multiple**-adic -function.*Russian Journal of Mathematical Physics*2006,**13**(2):151-157. 10.1134/S1061920806020038MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**A note on**-adic -integral on associated with -Euler numbers.*Advanced Studies in Contemporary Mathematics*2007,**15:**133-138.MATHGoogle Scholar - 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.2007.07.027MathSciNetView ArticleMATHGoogle 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.2006.09.027MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**A note on**-adic -integral on associated with -Euler numbers.*Advanced Studies in Contemporary Mathematics*2007,**15**(2):133-137.MathSciNetMATHGoogle Scholar - Kim T:
-Euler numbers and polynomials associated with
-adic
**-integrals.***Journal of Nonlinear Mathematical Physics*2007,**14**(1):15-27. 10.2991/jnmp.2007.14.1.3MathSciNetView ArticleMATHGoogle Scholar - Khrennikov AYu:
*p-adic Valued Distributions and Their Applications to the Mathematical Physics*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1994.View ArticleGoogle Scholar - 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.Google Scholar - Khrennikov AYu:
*Interpretations of Probability*. VSP, Utrecht, The Netherlands; 1999:ii+228.MATHGoogle Scholar

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