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 [5–12]). 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. [13–23]).

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 [24–26] 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)