Many mathematicians have studied various relations beween Euler number
E_{n}
, Bernoulli number
B_{n}
and Genocchi number
G_{n}
(see
[1

18]
). They have found numerous important applications in number theory. Howard, T.Agoh, S.H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers
[1
,
5
,
9
,
15]
. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the qextension of Euler and Genocchi numbers and polynomials
[6
,
8
,
10
,
11
,
14
,
17]
. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.
[1

18]
). They have found numerous important applications in number theory. Howard, T.Agoh, S.H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers
[1
,
5
,
9
,
15]
. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the
q
extension of Euler and Genocchi numbers and polynomials
[6
,
8
,
10
,
11
,
14
,
17]
. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.
AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
1. Introduction
The Genocchi number
G_{n}
, the Bernoulli number
B_{n}
(
n
∈ ℕ
_{0}
= {0, 1, 2, ... }) and the Euler number
E_{n}
are defined by the following generating function.
For a real or complex parameter
α
, the generalized Bernoulli polynomials
of order
α
∈ ℤ, and the generalized Euler polynomials
of order
α
∈ ℤ are defined by the following generating functions (see, for details, [4, p.253 et seq.], [14, Section 2.8] and [18, Section 1.6]).
and
The Genocchi polynomials
G_{n}
(
x
) of order
k
∈ ℕ are defined by
The Euler numbers
E_{n}
and Euler polynomials
E_{n}
(
x
) are defined by
where [
x
] is the greatest integer not exceeding
x
(see
[6
,
8
,
9
,
10
,
11
,
13
,
15
,
16]
).
By(1.1), we have
where ℕ is the set of positive integers. The Genocchi number
G_{n}
satisfy the recurrence relation
Therefore, we find out that
G
_{2}
= –1,
G
_{4}
= 1,
G
_{6}
= –3,
G
_{8}
= 17,
G
_{10}
= –155,
G
_{12}
= 2073,
G
_{14}
= –38227, .... That is,
G
_{2n+1}
= 0(
n
≥ 1).
The Stirling number of the first kind
s
(
n, k
) can be defined by means of
or by the generating function
We get (1.9) from (1.7) and (1.8)
with
s
(
n
, 0) = 0(
n
> 0),
s
(
n, n
) = 1(
n
≥ 0),
s
(
n
, 1) = (–1)
^{n−1}
(
n
–1)!(
n
> 0),
s
(
n, k
) = 0(
k
>
n
or
k
< 0). Stirling number of the second kind
S
(
n, k
) can be defined by
or by the generating function
We get (1.12) from (1.10) and (1.11)
with
S
(
n
, 0) = (
n
> 0),
S
(
n, n
) = 1(
n
≥ 0),
S
(
n
, 1) = 1(
n
> 0),
S
(
n, k
) = 0 (
k
>
n
or
k
< 0).
We begin with discussing Euler numbers, Genocchi numbers, Bernoulli numbers, Stirling numbers of the first kind, Stirling numbers of the second kind. In the paper, we organized the entire contents as follows. In Section 2, we define the extension term of generalized Euler polynomials of the second kind and prove them. We also study some interesting relations about
a polynomial of
x
and
α
with integers coefficients. In Section3, the extension term of generalized Euler polynomials of the second kind will be used to induce the main results of this paper. We also obtain some identities involving the Genocchi numbers, Genocchi polynomials, the Euler numbers, Euler polynomials and prove them.
2. Some relations within the an extension terms of the generalized Euler polynomials of the second kind
In this section, we study some relations of the extension term of generalized Euler polynomials of the second kind and research for properties between
. First of all, we define the generalized Euler polynomials
of the second kind as follows. In
[7]
, we introduced the generalized Euler polynomials
of the second kind and investigate their properties. First of all, we introduce the generalized Euler polynomials
of the second kind as follows. This completes with the usual convention of replacing
(see, for details,
[7]
).
Definition 2.1.
Let
x
be a real or complex parameter,
n
≥
k
(
n, k
∈ ℕ). Then we define
We derive that
where
For a real and complex parameter
α
, the generalized Euler polynomials
, each of degree
n
in
x
as well as in
α
, are defined by means of the generating function.
Definition 2.2.
Let
α
be a real or complex parameter. Then we define
By using Definition 2.2, we have the addition theorem of polynomials
and the relation of polynomials
and numbers
Theorem 2.3.
(Addition theorem) Let α, x, y
∈ ℂ
and n be nonnegative integers. Then we get
Proof
. For
n
be nonnegative integers, we have
By comparing the coefficients of both side, we complete the proof of the Theorem 2.3.
By using the Definition 2.2, we have the following Theorem 2.4.
Theorem 2.4.
Let n
≥
k
(
n, k, l
∈ ℕ).
Then we derive that
where
Proof
. By (2.1) and (2.2), we easily have
By Definition 2.2, (1.4) and (1.8) we have
which readily yields
Therefore, we complete the proof of Theorem 2.4
Remark 2.1.
From (2.3) and Theorem 2.4, we find out that
ρ
^{(α)}
(0, 0) = 1,
ρ
^{(α)}
(1, 0) = 0,
ρ
^{(α)}
(1, 1) = 1,
ρ
^{(α)}
(2, 0) = –
α
,
ρ
^{(α)}
(2, 1) = 0,
ρ
^{(α)}
(2, 2) = 1,
ρ
^{(α)}
(3, 0) = 0,
ρ
^{(α)}
(3, 1) = –3
α
,
ρ
^{(α)}
(3, 2) = 0,
ρ
^{(α)}
(3, 3) = 1,
ρ
^{(α)}
(4, 0) = 2
α
+3
α
^{2}
,
ρ
^{(α)}
(4, 1) = 0,
ρ
^{(α)}
(4, 2) = –6
α
,
ρ
^{(α)}
(4, 3) = 0,
ρ
^{(α)}
(4, 4) = 1, . . .
Thus, we know that
is a polynomial of
x
. Setting
n
= 1, 2, 3, 4, 5 in Theorem 2.4, we get to
We also find out an extension terms of the generalized Euler polynomials that can be represented by
c
(
n, k
) with Stirling numbers of the first kind, Stirling numbers of the second kind.
Since
we have the following theorem.
Theorem 2.5.
Let n, k
∈ ℕ,
then by (1.1) and Definition 2.2, we have
3. Some relations between an extension of the generalized Euler polynomials and Euler numbers, Genocchi numbers and themselves
In this section, we access some relations between an extension terms of generalized Euler polynomials and Euler numbers, Euler polynomials, Genocchi numbers, Genocchi polynomials of order k. We construct relations among an extension terms of generalized Euler polynomials themselves.
Theorem 3.1.
Let n
≥
k
(
n, k
∈ ℕ).
Relation between
and Genocchi numbers G_{n}, we have
where
Proof
. By (1.1), we have
Let us that
Therefore, we have
And we expand to degree of
α
, we obtain
We deduce that by the generalized Euler polynomials
By (2.4), we may immediately obtain Theorem 3.1. This completes the proof of Theorem 3.1.
We find out that
G
^{∗}
(0) = 1,
G
^{∗}
(1) = 0,
G
^{∗}
(2) = –1,
G
^{∗}
(3) = 0,
G
^{∗}
(4) = 5,
G
^{∗}
(5) = 6,
G
^{∗}
(6) = –61, ....
Remark 3.1.
Let
n
≥
k
(
n, k
∈ ℕ). Then we have
where
Theorem 3.2.
Let n
≥
k
(
n, k
∈ ℕ).
Then we obtain
Proof
. By applying Theorem 2.4, we have
It follows from Definition 2.2 that
On the other hand, we have from (2.1)
Substituting (3.9) in (3.8) we get
By (3.10), we may immediately obtain Theorem 3.2. This completes the proof of Theorem 3.2.
Theorem 3.3.
Let n
≥
k
(
n, k
∈ ℕ).
Relation between
and Euler numbers E_{n}, we have
where
Proof
. By definition (1.1)
Therefore, according to (3.1), (3.2), we have
By Theorem 3.1, we may immediately obtain Theorem 3.3.
We find out that
E
^{∗}
(0) = 1,
E
^{∗}
(1) = 0,
E
^{∗}
(2) = –1,
E
^{∗}
(3) = 0,
E
^{∗}
(4) = 5,
E
^{∗}
(5) = 0,
E
^{∗}
(6) = –61, .... Thus, we easily see that
G
^{∗}
=
E
^{∗}
.
By Definition 2.2, we have
Therefore, we have the following theorem.
Theorem 3.4.
Let α be a real or complex parameter. Relation between
and Euler polynomials
,
we have
For
α
= 1 in (3.13), we have the following corollary.
Corollary 3.5.
For α
= 1,
we have
Theorem 3.6.
Let n, α
∈ ℕ.
Relation between
and Genocchi polynomials
we have
Proof
. By Definition 2.2,
Therefore, we may immediately obtain Theorem 3.6. This completes the proof of Theorem 3.6.
BIO
Y. H. Kim is a graduate student in Hannam University. Her research interests focus on functional analysis and special functions.
Department of Mathematics, Hannam University, Daejeon, 306791, Korea
email: sunbidori@hanmail.net
H. Y. Jung is a graduate student in Hannam University. His research interests focus on the scientific computing and functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306791, Korea
email: tiger8049@korea.ac.kr
C. S. Ryoo received Ph.D. degree from Kyushu University. His research interests focus on the numerical verification method, scientific computing and padic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306791, Korea
email: ryoocs@hnu.kr
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