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A Note on
-Genocchi Polynomials and Numbers of Higher Order
Advances in Difference Equations volume 2010, Article number: 309480 (2010)
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

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.
Let and
. Then for all
,

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.
Let and
. Then one has

where .
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.
Let . Then one has

It is easily to check that

where with
. Thus we have the following theorem.
Theorem 2.4.
Let with
. Then for all
,

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.
Let . Then for all
,

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Acknowledgment
This paper was supported by KOSEF (2009-0073396, 2009-A419-0065).
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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
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DOI: https://doi.org/10.1155/2010/309480
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation