- Research Article
- Open access
- Published:
Interpolation Functions of
-Extensions of Apostol's Type Euler Polynomials
Journal of Inequalities and Applications volume 2009, Article number: 451217 (2009)
Abstract
The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic
-adic integral on
. We define the
-
-Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define
-extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic
-adic integral on
. We have the interpolation functions of these
-
-Euler polynomials. We also give
-extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these
-
-Euler polynomials.
1. Introduction, Definitions, and Notations
After Carlitz [1] gave -extensions of the classical Bernoulli numbers and polynomials, the
-extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors. Many authors have studied on various kinds of
-analogues of the Euler numbers and polynomials (cf., [1–39]).T Kim [7–23] has published remarkable research results for
-extensions of the Euler numbers and polynomials and their interpolation functions. In [13], T Kim presented a systematic study of some families of multiple
-Euler numbers and polynomials. By using the
-Volkenborn integration on
, he constructed the
-adic
-Euler numbers and polynomials of higher order and gave the generating function of these numbers and the Euler
-
-function. In [20], Kim studied some families of multiple
-Genocchi and
-Euler numbers using the multivariate
-adic
-Volkenborn integral on
, and gave interesting identities related to these numbers. Recently, Kim [21] studied some families of
-Euler numbers and polynomials of Nölund's type using multivariate fermionic
-adic integral on
.
Many authors have studied the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, and their -extensions (cf., [1, 6, 25, 27, 28, 33–41]). Choi et al. [6] studied some
-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order
, and multiple Hurwitz zeta function. In [24], Kim et al. defined Apostol's type
-Euler numbers and polynomials using the fermionic
-adic
-integral and obtained the generating functions of these numbers and polynomials, respectively. They also had the distribution relation for Apostol's type
-Euler polynomials and obtained
-zeta function associated with Apostol's type
-Euler numbers and Hurwitz type
-zeta function associated with Apostol's type
-Euler polynomials for negative integers.
In this paper, we will present new -extensions of Apostol's type Euler polynomials using the fermionic
-adic integral on
, and then we give interpolation functions and the Hurwitz type zeta functions of these polynomials. We also give
-extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic
-adic integral on
.
Let be a fixed odd prime number. Throughout this paper
,
, and
will, respectively, denote the ring of
-adic rational integers, the field of
-adic rational numbers, the complex number field, and the completion of algebraic closure of
. Let
be the set of natural numbers and
. 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 one assumes
Now we recall some -notations. The
-basic natural numbers are defined by
and the
-factorial by
. The
-binomial coefficients are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ1_HTML.gif)
Note that , which is the binomial coefficient. The
-shift factorial is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ2_HTML.gif)
Note that . It is well known that the
-binomial formulae are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ3_HTML.gif)
Since , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ4_HTML.gif)
Hence it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ5_HTML.gif)
which converges to as
For a fixed odd positive integer with
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ6_HTML.gif)
where lies in
. The distribution is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ7_HTML.gif)
Let be the set of uniformly differentiable functions on
. For
, the
-adic invariant
-integral is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ8_HTML.gif)
The fermionic -adic invariant
-integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ9_HTML.gif)
where . The fermionic
-adic integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ10_HTML.gif)
It follows that where
. For
, let
. we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ11_HTML.gif)
The classical Euler numbers and the classical Euler polynomials
are defined, respectively, as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ12_HTML.gif)
It is known that the classical Euler numbers and polynomials are interpolated by the Euler zeta function and Hurwitz type zeta function, respectively, as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ13_HTML.gif)
In Section 2, we define new -extensions of Apostol's type Euler polynomials using the fermionic
-adic integral on
which will be called the
-
-Euler polynomials . Then we obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. In Section 3, we define
-extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic
-adic integral on
. We have the interpolation functions of these higher-order
-
-Euler polynomials. In Section 4, we also give
-extensions of Apostol's type Euler polynomials of higher order and have the multiple Euler zeta functions of these
-
-Euler polynomials.
2.
-Extensions of Apostol's Type Euler Polynomials
First, we assume that with
. In
, the
-Euler polynomials are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ14_HTML.gif)
and are called the
-Euler numbers. Then it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ15_HTML.gif)
The generating functions of are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ16_HTML.gif)
By (2.3), the interpolation functions of the -Euler polynomials
are obtained as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ17_HTML.gif)
Thus, we have the following theorem.
Theorem 2.1.
Assume with
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ18_HTML.gif)
Differentiating at
shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ19_HTML.gif)
In , we assume that
with
. The
-Euler polynomials
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ20_HTML.gif)
By (2.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ21_HTML.gif)
For , the Hurwitz type zeta functions for the
-Euler polynomials
are given as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ22_HTML.gif)
For , we have from (2.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ23_HTML.gif)
Now we give new -extensions of Apostol's type Euler polynomials. For
, let
be the cyclic group of order
. Let
be the p-adic locally constant space defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ24_HTML.gif)
First, we assume that with
. For
, we define
-Euler polynomials of Apostol's type using the fermionic
-adic integral as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ25_HTML.gif)
and we will call them the -
-Euler polynomials. Then
are defined as the
-
-Euler numbers. From (2.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ26_HTML.gif)
Let . From (2.12), we easily derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ27_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ28_HTML.gif)
From (2.14) and (2.15), we obtain the following theorem.
Theorem 2.2.
Assume that with
. For
, let
Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ29_HTML.gif)
In , we assume that
with
. Let
with
. We define the
-
-Euler polynomials
to be satisfied the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ30_HTML.gif)
When we differentiate both sides of (2.17) at , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ31_HTML.gif)
Hence we have the interpolation functions of the -
-Euler polynomials as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ32_HTML.gif)
For , we define the Hurwitz type zeta function of the
-
-Euler polynomials as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ33_HTML.gif)
where For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ34_HTML.gif)
3.
-Extensions of Apostol's Type Euler Polynomials of Higher Order
In this section, we give the -extension of Apostol's type Euler polynomials of higher order using the multivariate fermionic
-adic integral.
First, we assume that with
. Let
. We define the
-
-Euler polynomials of order
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ35_HTML.gif)
Note that are called the
-
-Euler number of order
. Using the multivariate fermionic
-adic integral, we obtain from (3.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ36_HTML.gif)
Let be the generating functions of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ37_HTML.gif)
By (2.12) and (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ38_HTML.gif)
Thus we have the following theorem.
Theorem 3.1.
Assume that with
. For
and
, let
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ39_HTML.gif)
In , we assume that
with
and
with
for
. We define the
-
-Euler polynomial
of order
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ40_HTML.gif)
From (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ41_HTML.gif)
For , we define the multiple Hurwitz type zeta functions for
-
-Euler polynomials as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ42_HTML.gif)
where In the special case
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ43_HTML.gif)
4.
-Extension of Apostol's Type Euler Polynomials of Higher Order
In this section, we give the -extension of
-
-Euler polynomials of higher order using the multivariate fermionic
-adic integral.
Assume that with
. For
, we define
-
-Euler polynomials of order
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ44_HTML.gif)
Note that are called the
-
-Euler numbers.
When , the
-
-Euler polynomials are
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ45_HTML.gif)
where is the Gaussian binomial coefficient. From (4.2), we obtain the following theorem.
Theorem 4.1.
Assume that with
. For
and
, let
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ46_HTML.gif)
In , assume that
with
and
with
. Then we can define
-
-Euler polynomials
for
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ47_HTML.gif)
Differentiating both sides of (4.4) at , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ48_HTML.gif)
From (4.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ49_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ50_HTML.gif)
For , we define the Hurwitz type zeta function of
-
-Euler polynomials of order
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ51_HTML.gif)
where
From (4.4) and (4.8), we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F451217/MediaObjects/13660_2009_Article_1955_Equ52_HTML.gif)
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Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University in 2009.
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Hwang, KW., Kim, YH. & Kim, T. Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials.
J Inequal Appl 2009, 451217 (2009). https://doi.org/10.1155/2009/451217
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DOI: https://doi.org/10.1155/2009/451217