# On a Class ofq-Bernoulli, q-Euler, and q-Genocchi Polynomials

Authors: N. I. Mahmudov; M. Momenzadeh
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The main purpose of this paper is to introduce and investigate a class ofq-Bernoulli, q-Euler, and q-Genocchi polynomials. The q-analogues of well-known formulas are derived. In addition, the q-analogue of the Srivastava-Pintér theorem is obtained. Some new identities, involving q-polynomials, are proved.


---

## Body

## 1. Introduction

Throughout this paper, we always make use of the classical definition of quantum concepts as follows.Theq-shifted factorial is defined by
(1)
(
a
;
q
)
0
=
1
,
(
a
;
q
)
n
=
∏
j
=
0
n
-
1
(
1
-
q
j
a
)
,
n
∈
ℕ
,
(
a
;
q
)
∞
=
∏
j
=
0
∞
(
1
-
q
j
a
)
,
|
q
|
<
1
,
a
∈
ℂ
.
It is known that
(2)
(
a
;
q
)
n
=
∑
k
=
0
n
[
n
k
]
q
q
(
1
/
2
)
k
(
k
-
1
)
(
-
1
)
k
a
k
.
The q-numbers and q-factorial are defined by
(3)
[
a
]
q
=
1
-
q
a
1
-
q
,
(
q
≠
1
,
a
∈
ℂ
)
;
[
0
]
q
!
=
1
,
[
n
]
q
!
=
[
n
]
q
[
n
-
1
]
q
!
.
The q-polynomial coefficient is defined by
(4)
[
n
k
]
q
=
(
q
;
q
)
n
(
q
;
q
)
n
-
k
(
q
;
q
)
k
,
(
k
⩽
n
,
k
,
n
∈
ℕ
)
.
In the standard approach to the q-calculus two exponential functions are used, these q-exponential functions and improved type q-exponential function (see [1]) are defined as follows:
(5)
e
q
(
z
)
=
∑
n
=
0
∞
z
n
[
n
]
q
!
=
∏
k
=
0
∞
1
(
1
-
(
1
-
q
)
q
k
z
)
,
hhhhhhh
0
<
|
q
|
<
1
,
|
z
|
<
1
|
1
-
q
|
,
E
q
(
z
)
=
e
1
/
q
(
z
)
=
∑
n
=
0
∞
q
(
1
/
2
)
n
(
n
-
1
)
z
n
[
n
]
q
!
=
∏
k
=
0
∞
(
1
+
(
1
-
q
)
q
k
z
)
,
0
<
|
q
|
<
1
,
z
∈
ℂ
,
ℰ
q
(
z
)
=
e
q
(
z
2
)
E
q
(
z
2
)
=
∑
n
=
0
∞
(
-
1
,
q
)
n
2
n
z
n
[
n
]
q
!
=
∏
k
=
0
∞
(
1
+
(
1
-
q
)
q
k
(
z
/
2
)
)
(
1
-
(
1
-
q
)
q
k
(
z
/
2
)
)
,
0
<
q
<
1
,
|
z
|

<
2
1
-
q
.
The form of improved type of q-exponential function ℰ
q
(
z
) motivated us to define a new q-addition and q-subtraction as
(6)
(
x
⊕
q
y
)
n
:
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
k
(
-
1
,
q
)
n
-
k
2
n
x
k
y
n
-
k
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
(
x
⊖
q
y
)
n
:
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
k
(
-
1
,
q
)
n
-
k
2
n
x
k
(
-
y
)
n
-
k
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
.
It follows that
(7)
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
∑
n
=
0
∞
(
x
⊕
q
y
)
n
t
n
[
n
]
q
!
.
The Bernoulli numbers {
B
m
}
m
≥
0 are rational numbers in a sequence defined by the binomial recursion formula:
(8)
∑
k
=
0
m
(
m
k
)
B
k
-
B
m
=
{
1
,
m
=
1
,
0
,
m
>
1
,
or equivalently, the generating function
(9)
∑
k
=
0
∞
B
k
t
k
k
!
=
t
e
t
-
1
.
q-Analogues of the Bernoulli numbers were first studied by Carlitz [2] in the middle of the last century when he introduced a new sequence {
β
m
}
m
⩾
0:
(10)
∑
k
=
0
m
(
m
k
)
β
k
q
k
+
1
-
β
m
=
{
1
,
m
=
1
,
0
,
m
>
1
.
Here and in the remainder of the paper, for the parameter q we make the assumption that |
q
|
<
1. Clearly we recover (8) if we let q
→
1 in (10). The q-binomial formula is known as
(11)
(
1
-
a
)
q
n
=
(
a
;
q
)
n
=
∏
j
=
0
n
-
1
(
1
-
q
j
a
)
=
∑
k
=
0
n
[
n
k
]
q
q
(
1
/
2
)
k
(
k
-
1
)
(
-
1
)
k
a
k
.
The above q-standard notation can be found in [3].Carlitz has introduced theq-Bernoulli numbers and polynomials in [2]. Srivastava and Pintér proved some relations and theorems between the Bernoulli polynomials and Euler polynomials in [4]. They also gave some generalizations of these polynomials. In [4–16], the authors investigated some properties of the q-Euler polynomials and q-Genocchi polynomials. They gave some recurrence relations. In [17], Cenkci et al. gave the q-extension of Genocchi numbers in a different manner. In [18], Kim gave a new concept for the q-Genocchi numbers and polynomials. In [19], Simsek et al. investigated the q-Genocchi zeta function and l-function by using generating functions and Mellin transformation. There are numerous recent studies on this subject by, among many other authors, Cigler [20], Cenkci et al. [17, 21], Choi et al. [22], Cheon [23], Luo and Srivastava [8–10], Srivastava et al. [4, 24], Nalci and Pashaev [25] Gaboury and Kurt, [26], Kim et al. [27], and Kurt [28].We first give the definitions of theq-numbers and q-polynomials. It should be mentioned that the definition of q-Bernoulli numbers in Definition 1 can be found in [25].Definition 1.
Letq
∈
ℂ
,
0
<
|
q
|
<
1. The q-Bernoulli numbers 𝔟
n
,
q and polynomials 𝔅
n
,
q
(
x
,
y
) are defined by means of the generating functions:
(12)
𝔅
^
(
t
)
:
=
t
e
q
(
-
t
/
2
)
e
q
(
t
/
2
)
-
e
q
(
-
t
/
2
)
=
t
ℰ
q
(
t
)
-
1
=
∑
n
=
0
∞
𝔟
n
,
q
t
n
[
n
]
q
!
,
|
t
|
<
2
π
,
t
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
∑
n
=
0
∞
𝔅
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
,
|
t
|
<
2
π
.Definition 2.
Letq
∈
ℂ
,
0
<
|
q
|
<
1. The q-Euler numbers 𝔢
n
,
q and polynomials 𝔈
n
,
q
(
x
,
y
) are defined by means of the generating functions:
(13)
𝔈
^
(
t
)
:
=
2
e
q
(
-
t
/
2
)
e
q
(
t
/
2
)
+
e
q
(
-
t
/
2
)
=
2
ℰ
q
(
t
)
+
1
=
∑
n
=
0
∞
𝔢
n
,
q
t
n
[
n
]
q
!
,
|
t
|
<
π
,
2
ℰ
q
(
t
)
+
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
∑
n
=
0
∞
𝔈
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
,
|
t
|
<
π
.Definition 3.
Letq
∈
ℂ
,
0
<
|
q
|
<
1. The q-Genocchi numbers 𝔤
n
,
q and polynomials 𝔊
n
,
q
(
x
,
y
) are defined by means of the generating functions:
(14)
𝔊
^
(
t
)
:
=
2
t
e
q
(
-
t
/
2
)
e
q
(
t
/
2
)
+
e
q
(
-
t
/
2
)
=
2
t
ℰ
q
(
t
)
+
1
=
∑
n
=
0
∞
𝔤
n
,
q
t
n
[
n
]
q
!
,
|
t
|
<
π
,
2
t
ℰ
q
(
t
)
+
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
∑
n
=
0
∞
𝔊
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
,
|
t
|
<
π
.Note that Cigler [20] defined q-Genocchi numbers as
(15)
t
e
q
(
t
)
+
e
q
(
-
t
)
e
q
(
t
)
+
e
q
(
-
t
)
=
∑
n
=
0
∞
g
2
n
,
q
(
-
1
)
n
-
1
(
-
q
;
q
)
2
n
-
1
t
2
n
[
2
n
]
q
!
.
Then comparing 𝔤
n
,
q with g
n
,
q, we see that
(16)
(
-
1
)
n
-
1
2
2
n
+
1
𝔤
2
n
+
2
,
q
=
(
-
q
;
q
)
2
n
+
1
g
2
n
+
2
,
q
.Definition 4.
Letq
∈
ℂ
,
0
<
|
q
|
<
1. The q-tangent numbers 𝔗
n
,
q are defined by means of the generating functions:
(17)
tanh
q
t
=
-
i
tan
q
(
i
t
)
=
e
q
(
t
)
-
e
q
(
-
t
)
e
q
(
t
)
+
e
q
(
-
t
)
=
ℰ
q
(
2
t
)
-
1
ℰ
q
(
2
t
)
+
1
=
∑
n
=
1
∞
𝔗
2
n
+
1
,
q
(
-
1
)
k
t
2
n
+
1
[
2
n
+
1
]
q
!
.It is obvious that, by lettingq tend to 1 from the left side, we lead to the classic definition of these polynomials:
(18)
𝔟
n
,
q
:
=
𝔅
n
,
q
(
0
)
,
lim
⁡
q
→
1
-
𝔅
n
,
q
(
x
)
=
B
n
(
x
)
,
lim
⁡
q
→
1
-
𝔅
n
,
q
(
x
,
y
)
=
B
n
(
x
+
y
)
,
lim
⁡
q
→
1
-
𝔟
n
,
q
=
B
n
,
𝔢
n
,
q
:
=
𝔈
n
,
q
(
0
)
,
lim
⁡
q
→
1
-
𝔈
n
,
q
(
x
)
=
E
n
(
x
)
,
lim
⁡
q
→
1
-
𝔈
n
,
q
(
x
,
y
)
=
E
n
(
x
+
y
)
,
lim
⁡
q
→
1
-
𝔢
n
,
q
=
E
n
,
𝔤
n
,
q
:
=
𝔊
n
,
q
(
0
)
,
lim
⁡
q
→
1
-
𝔊
n
,
q
(
x
)
=
G
n
(
x
)
,
lim
⁡
q
→
1
-
𝔊
n
,
q
(
x
,
y
)
=
G
n
(
x
+
y
)
lim
⁡
q
→
1
-
𝔤
n
,
q
=
G
n
.
Here B
n
(
x
)
,
E
n
(
x
), and G
n
(
x
) denote the classical Bernoulli, Euler, and Genocchi polynomials, respectively, which are defined by
(19)
t
e
t
-
1
e
t
x
=
∑
n
=
0
∞
‍
B
n
(
x
)
t
n
n
!
,
2
e
t
+
1
e
t
x
=
∑
n
=
0
∞
‍
E
n
(
x
)
t
n
n
!
,
2
t
e
t
+
1
e
t
x
=
∑
n
=
0
∞
‍
G
n
(
x
)
t
n
n
!
.The aim of the present paper is to obtain some results for the above newly definedq-polynomials. It should be mentioned that q-Bernoulli and q-Euler polynomials in our definitions are polynomials of x and y and when y
=
0, they are polynomials of x. First advantage of this approach is that for q
→
1
-
,
𝔅
n
,
q
(
x
,
y
) (𝔈
n
,
q
(
x
,
y
)
,
𝔊
n
,
q
(
x
,
y
)) becomes the classical Bernoulli 𝔅
n
(
x
+
y
) (Euler 𝔈
n
(
x
+
y
)
,Genocchi 𝔊
n
,
q
(
x
,
y
)) polynomial and we may obtain the q-analogues of well-known results, for example, Srivastava and Pintér [11], Cheon [23], and so forth. Second advantage is that, similar to the classical case, odd numbered terms of the Bernoulli numbers 𝔟
k
,
q and the Genocchi numbers 𝔤
k
,
q are zero, and even numbered terms of the Euler numbers 𝔢
n
,
q are zero.


## 2. Preliminary Results

In this section we will provide some basic formulae for theq-Bernoulli, q-Euler, and q-Genocchi numbers and polynomials in order to obtain the main results of this paper in the next section.Lemma 5.
Theq-Bernoulli numbers 𝔟
n
,
q satisfy the following q-binomial recurrence:
(20)
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔟
k
,
q
-
𝔟
n
,
q
=
{
1
,
n
=
1
,
0
,
n
>
1
.Proof.
By a simple multiplication of (8) we see that
(21)
𝔅
^
(
t
)
ℰ
q
(
t
)
=
t
+
𝔅
^
(
t
)
.
So
(22)
∑
n
=
0
∞
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔟
k
,
q
t
n
[
n
]
q
!
=
t
+
∑
n
=
0
∞
𝔟
n
,
q
t
n
[
n
]
q
!
.
The statement follows by comparing t
m coefficients.We use this formula to calculate the first few𝔟
k
,
q:
(23)
𝔟
0
,
q
=
1
,
𝔟
1
,
q
=
-
1
2
,
𝔟
2
,
q
=
1
4
q
(
q
+
1
)
q
2
+
q
+
1
=
q
[
2
]
q
4
[
3
]
q
,
𝔟
3
,
q
=
0
.The similar property can be proved forq-Euler numbers
(24)
∑
k
=
0
m
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔢
k
,
q
+
𝔢
m
,
q
=
{
2
,
m
=
0
,
0
,
m
>
0
.
and q-Genocchi numbers
(25)
∑
k
=
0
m
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔤
k
,
q
+
𝔤
m
,
q
=
{
2
,
m
=
1
,
0
,
m
>
1
.Using the above recurrence formulae we calculate the first few𝔢
n
,
q and 𝔤
n
,
q terms as well:
(26)
𝔢
0
,
q
=
1
,
𝔤
0
,
q
=
0
,
𝔢
1
,
q
=
-
1
2
,
𝔤
1
,
q
=
1
,
𝔢
2
,
q
=
0
,
𝔤
2
,
q
=
-
[
2
]
q
2
=
-
q
+
1
2
,
𝔢
3
,
q
=
[
3
]
q
[
2
]
q
-
[
4
]
q
8
=
q
(
1
+
q
)
8
,
𝔤
3
,
q
=
0
.Remark 6.
The first advantage of the newq-numbers 𝔟
k
,
q
,
𝔢
k
,
q, and 𝔤
k
,
q is that similar to classical case odd numbered terms of the Bernoulli numbers 𝔟
k
,
q and the Genocchi numbers 𝔤
k
,
q are zero, and even numbered terms of the Euler numbers 𝔢
n
,
q are zero.Next lemma gives the relationship betweenq-Genocchi numbers and q-Tangent numbers.Lemma 7.
For anyn
∈
ℕ, we have
(27)
𝔗
2
n
+
1
,
q
=
𝔤
2
n
+
2
,
q
(
-
1
)
k
-
1
2
2
n
+
1
[
2
n
+
2
]
q
.Proof.
First we recall the definition ofq-trigonometric functions:
(28)
cos
⁡
q
t
=
e
q
(
i
t
)
+
e
q
(
-
i
t
)
2
,
sin
q
t
=
e
q
(
i
t
)
-
e
q
(
-
i
t
)
2
i
,
i
tan
q
t
=
e
q
(
i
t
)
-
e
q
(
-
i
t
)
e
q
(
i
t
)
+
e
q
(
-
i
t
)
,
cot
q
t
=
i
e
q
(
i
t
)
+
e
q
(
-
i
t
)
e
q
(
i
t
)
-
e
q
(
-
i
t
)
.
Now by choosing z
=
2
i
t in 𝔅
^
(
z
), we get
(29)
𝔅
^
(
2
i
t
)
=
2
i
t
ℰ
q
(
2
i
t
)
-
1
=
t
e
q
(
-
i
t
)
sin
q
t
=
∑
n
=
0
∞
𝔟
n
,
q
(
2
i
t
)
n
[
n
]
q
!
.
It follows that
(30)
𝔅
^
(
2
i
t
)
=
t
e
q
(
-
i
t
)
sin
q
t
=
t
sin
q
t
(
cos
⁡
q
t
-
i
sin
q
t
)
=
t
cot
q
t
-
i
t
=
𝔟
0
,
q
+
2
i
t
𝔟
1
,
q
+
∑
n
=
2
∞
𝔟
n
,
q
(
2
i
t
)
n
[
n
]
q
!
=
1
-
i
t
+
∑
n
=
2
∞
𝔟
n
,
q
(
2
i
t
)
n
[
n
]
q
!
.
Since the function t
cot
q
t is even in the above sum odd coefficients 𝔟
2
k
+
1
,
q, k
=
1,2
,
…, are zero, and we get
(31)
t
cot
q
t
=
1
+
∑
n
=
2
∞
𝔟
n
,
q
(
2
i
t
)
n
[
n
]
q
!
=
1
+
∑
n
=
1
∞
𝔟
n
,
q
(
2
i
t
)
2
n
[
2
n
]
q
!
.
By choosing z
=
2
i
t in 𝔊
^
(
z
), we get
(32)
𝔊
^
(
2
i
t
)
=
4
i
t
ℰ
q
(
2
i
t
)
+
1
=
2
i
t
e
q
(
-
i
t
)
cos
⁡
q
t
=
∑
n
=
0
∞
𝔤
n
,
q
(
2
i
t
)
n
[
n
]
q
!
,
𝔊
^
(
2
i
t
)
=
4
i
t
ℰ
q
(
2
i
t
)
+
1
=
2
i
t
e
q
(
-
i
t
)
cos
⁡
q
t
=
2
i
t
cos
⁡
q
t
(
cos
⁡
q
t
-
i
sin
q
t
)
=
2
i
t
+
2
t
tan
q
t
=
𝔤
0
,
q
+
2
i
t
𝔤
1
,
q
+
∑
n
=
2
∞
𝔤
n
,
q
(
2
i
t
)
n
[
n
]
q
!
=
2
i
t
+
∑
n
=
2
∞
𝔤
n
,
q
(
2
i
t
)
n
[
n
]
q
!
.
It follows that
(33)
2
t
tan
q
t
=
∑
n
=
1
∞
𝔤
2
n
,
q
(
2
i
t
)
2
n
[
2
n
]
q
!
,
tan
q
t
=
∑
n
=
1
∞
𝔤
2
n
,
q
(
-
1
)
n
(
2
t
)
2
n
-
1
[
2
n
]
q
!
,
tanh
q
t
=
-
i
tan
q
(
i
t
)
=
-
i
∑
n
=
1
∞
𝔤
2
n
,
q
(
-
1
)
n
(
2
i
t
)
2
n
-
1
[
2
n
]
q
!
=
-
∑
n
=
1
∞
𝔤
2
n
,
q
(
2
t
)
2
n
-
1
[
2
n
]
q
!
=
-
∑
n
=
1
∞
𝔤
2
n
+
2
,
q
(
2
t
)
2
n
+
1
[
2
n
+
2
]
q
!
,
Thus
(34)
tanh
q
t
=
-
i
tan
q
(
i
t
)
=
e
q
(
t
)
-
e
q
(
-
t
)
e
q
(
t
)
+
e
q
(
-
t
)
=
ℰ
q
(
2
t
)
-
1
ℰ
q
(
2
t
)
+
1
=
∑
n
=
1
∞
𝔗
2
n
+
1
,
q
(
-
1
)
k
t
2
n
+
1
[
2
n
+
1
]
q
!
,
𝔗
2
n
+
1
,
q
=
𝔤
2
n
+
2
,
q
(
-
1
)
k
-
1
2
2
n
+
1
[
2
n
+
2
]
q
.The following result is aq-analogue of the addition theorem, for the classical Bernoulli, Euler, and Genocchi polynomials.Lemma 8 (addition theorems).
For allx
,
y
∈
ℂ we have
(35)
𝔅
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
[
n
k
]
q
𝔟
k
,
q
(
x
⊕
q
y
)
n
-
k
,
𝔅
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔅
k
,
q
(
x
)
y
n
-
k
,
𝔈
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
[
n
k
]
q
𝔢
k
,
q
(
x
⊕
q
y
)
n
-
k
,
𝔈
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔈
k
,
q
(
x
)
y
n
-
k
,
𝔊
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
[
n
k
]
q
𝔤
k
,
q
(
x
⊕
q
y
)
n
-
k
,
𝔊
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔊
k
,
q
(
x
)
y
n
-
k
.Proof.
We prove only the first formula. It is a consequence of the following identity:(36)
∑
n
=
0
∞
𝔅
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
=
t
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
∑
n
=
0
∞
𝔟
n
,
q
t
n
[
n
]
q
!
∑
n
=
0
∞
(
x
⊕
q
y
)
n
t
n
[
n
]
q
!
=
∑
n
=
0
∞
‍
∑
k
=
0
n
‍
[
n
k
]
q
𝔟
k
,
q
(
x
⊕
q
y
)
n
-
k
t
n
[
n
]
q
!
.In particular, settingy
=
0 in (35), we get the following formulae for q-Bernoulli, q-Euler and q-Genocchi polynomials, respectively:
(37)
𝔅
n
,
q
(
x
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔟
k
,
q
x
n
-
k
,
𝔈
n
,
q
(
x
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔢
k
,
q
x
n
-
k
,
(38)
𝔊
n
,
q
(
x
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔤
k
,
q
x
n
-
k
.
Setting y
=
1 in (35), we get
(39)
𝔅
n
,
q
(
x
,
1
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔅
k
,
q
(
x
)
,
𝔈
n
,
q
(
x
,
1
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔈
k
,
q
(
x
)
,
𝔊
n
,
q
(
x
,
1
)
=
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔊
k
,
q
(
x
)
.
Clearly (39) is q-analogues of
(40)
B
n
(
x
+
1
)
=
∑
k
=
0
n
(
n
k
)
B
k
(
x
)
,
E
n
(
x
+
1
)
=
∑
k
=
0
n
(
n
k
)
E
k
(
x
)
,
G
n
(
x
+
1
)
=
∑
k
=
0
n
(
n
k
)
G
k
(
x
)
,
respectively.Lemma 9.
The odd coefficients of theq-Bernoulli numbers, except the first one, are zero. That means 𝔟
n
,
q
=
0 where n
=
2
r
+
1
(
r
∈
ℕ
).Proof.
It follows from the fact that the function(41)
f
(
t
)
=
∑
n
=
0
∞
𝔟
n
,
q
t
n
[
n
]
q
!
-
𝔟
1
,
q
t
=
t
ℰ
q
(
t
)
-
1
+
t
2
=
t
2
(
ℰ
q
(
t
)
+
1
ℰ
q
(
t
)
-
1
)
,By usingq-derivative we obtain the next lemma.Lemma 10.
One has(42)
D
q
,
x
𝔅
n
,
q
(
x
)
=
[
n
]
q
𝔅
n
-
1
,
q
(
x
)
+
𝔅
n
-
1
,
q
(
q
x
)
2
,
D
q
,
x
𝔈
n
,
q
(
x
)
=
[
n
]
q
𝔈
n
-
1
,
q
(
x
)
+
𝔈
n
-
1
,
q
(
q
x
)
2
,
D
q
,
x
𝔊
n
,
q
(
x
)
=
[
n
]
q
𝔊
n
-
1
,
q
(
x
)
+
𝔊
n
-
1
,
q
(
q
x
)
2
.Lemma 11 (difference equations).
One has(43)
𝔅
n
,
q
(
x
,
1
)
-
𝔅
n
,
q
(
x
)
=
(
-
1
;
q
)
n
-
1
2
n
-
1
[
n
]
q
x
n
-
1
,
n
≥
1
,
(44)
𝔈
n
,
q
(
x
,
1
)
+
𝔈
n
,
q
(
x
)
=
2
(
-
1
;
q
)
n
2
n
x
n
,
n
≥
0
,
(45)
𝔊
n
,
q
(
x
,
1
)
+
𝔊
n
,
q
(
x
)
=
2
(
-
1
;
q
)
n
-
1
2
n
-
1
[
n
]
q
x
n
-
1
,
n
≥
1
.Proof.
We prove the identity for theq-Bernoulli polynomials. From the identity
(46)
t
ℰ
q
(
t
)
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
=
t
ℰ
q
(
t
x
)
+
t
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
,
it follows that
(47)
∑
n
=
0
∞
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔅
k
,
q
(
x
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
(
-
1
,
q
)
n
2
n
x
n
t
n
+
1
[
n
]
q
!
+
∑
n
=
0
∞
𝔅
n
,
q
(
x
)
t
n
[
n
]
q
!
.From (43) and (37), (44) and (38), we obtain the following formulae.Lemma 12.
One has(48)
x
n
=
2
n
(
-
1
;
q
)
n
[
n
]
q
∑
k
=
0
n
[
n
+
1
k
]
q
(
-
1
;
q
)
n
+
1
-
k
2
n
+
1
-
k
𝔅
k
,
q
(
x
)
,
x
n
=
2
n
-
1
(
-
1
;
q
)
n
(
∑
k
=
0
n
[
n
k
]
q
(
-
1
;
q
)
n
-
k
2
n
-
k
𝔈
k
,
q
(
x
)
+
𝔈
n
,
q
(
x
)
)
,
x
n
=
2
n
-
1
(
-
1
;
q
)
n
[
n
+
1
]
q
×
(
∑
k
=
0
n
+
1
[
n
+
1
k
]
q
(
-
1
;
q
)
n
+
1
-
k
2
n
+
1
-
k
𝔊
k
,
q
(
x
)
+
𝔊
n
+
1
,
q
(
x
)
)
.The above formulae areq-analogues of the following familiar expansions:
(49)
x
n
=
1
n
+
1
∑
k
=
0
n
(
n
+
1
k
)
B
k
(
x
)
,
x
n
=
1
2
[
∑
k
=
0
n
(
n
k
)
E
k
(
x
)
+
E
n
(
x
)
]
,
x
n
=
1
2
(
n
+
1
)
[
∑
k
=
0
n
+
1
(
n
+
1
k
)
E
k
(
x
)
+
E
n
+
1
(
x
)
]
,
respectively.Lemma 13.
The following identities hold true:(50)
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔅
k
,
q
(
x
,
y
)
-
𝔅
n
,
q
(
x
,
y
)
=
[
n
]
q
(
x
⊕
q
y
)
n
-
1
,
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔈
k
,
q
(
x
,
y
)
+
𝔈
n
,
q
(
x
,
y
)
=
2
(
x
⊕
q
y
)
n
,
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔊
k
,
q
(
x
,
y
)
+
𝔊
n
,
q
(
x
,
y
)
=
2
[
n
]
q
(
x
⊕
q
y
)
n
-
1
.Proof.
We prove the identity for theq-Bernoulli polynomials. From the identity
(51)
t
ℰ
q
(
t
)
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
t
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
+
t
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
,
it follows that
(52)
∑
n
=
0
∞
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔅
k
,
q
(
x
,
y
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
‍
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
k
(
-
1
,
q
)
n
-
k
2
n
x
k
y
n
-
k
t
n
+
1
[
n
]
q
!
+
∑
n
=
0
∞
𝔅
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
.


## 3. Some New Formulae

The classical Cayley transformationz
→Cay(
z
,
a
)
:
=
(
1
+
a
z
)
/
(
1
-
a
z
) motivated us to derive the formula for ℰ
q
(
q
t
). In addition, by substituting Cay(
z
,
(
q
-
1
)
/
2
) in the generating formula we have
(53)
𝔅
^
q
(
q
t
)
𝔅
^
q
(
t
)
=
(
𝔅
^
q
(
q
t
)
-
q
𝔅
^
q
(
t
)
(
1
+
(
1
-
q
)
t
2
)
)
×
1
1
-
q
×
2
ℰ
q
(
t
)
+
1
.
The right hand side can be presented by q-Euler numbers. Now equating coefficients of t
n we get the following identity. In the case that n
=
0, we find the first improved q-Euler number which is exactly 1.Proposition 14.
For alln
≥
1,
(54)
∑
k
=
0
n
[
n
k
]
q
𝔅
k
,
q
𝔅
n
-
k
,
q
q
k
=
-
q
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
2
n
-
k
𝔅
k
,
q
𝔈
n
-
k
,
q
[
k
-
1
]
q
-
q
2
∑
k
=
0
n
-
1
[
n
-
1
k
]
q
(
-
1
,
q
)
n
-
k
-
1
2
n
-
k
-
1
𝔅
k
,
q
𝔈
n
-
k
-
1
,
q
[
n
]
q
.Let us take aq-derivative from the generating function, after simplifying the equation, by knowing the quotient rule for quantum derivative, and also using(55)
ℰ
q
(
q
t
)
=
1
-
(
1
-
q
)
(
t
/
2
)
1
+
(
1
-
q
)
(
t
/
2
)
ℰ
q
(
t
)
,
D
q
(
ℰ
q
(
t
)
)
=
ℰ
q
(
q
t
)
+
ℰ
q
(
t
)
2
,
one has
(56)
𝔅
^
q
(
q
t
)
𝔅
^
q
(
t
)
=
2
+
(
1
-
q
)
t
2
ℰ
q
(
t
)
(
q
-
1
)
(
q
𝔅
^
q
(
t
)
-
𝔅
^
q
(
q
t
)
)
.It is clear thatℰ
q
-
1
(
t
)
=
ℰ
q
(
-
t
). Now, by equating coefficients of t
n we obtain the following identity.Proposition 15.
For alln
≥
1,
(57)
∑
k
=
0
2
n
[
2
n
k
]
q
𝔅
k
,
q
𝔅
2
n
-
k
,
q
q
k
=
-
q
∑
k
=
0
2
n
[
2
n
k
]
q
(
-
1
,
q
)
2
n
-
k
2
2
n
-
k
𝔅
k
,
q
[
k
-
1
]
q
(
-
1
)
k
+
q
(
1
-
q
)
2
∑
k
=
0
2
n
-
1
[
2
n
-
1
k
]
q
(
-
1
,
q
)
2
n
-
1
-
k
2
2
n
-
1
-
k
𝔅
k
,
q
+
q
(
1
-
q
)
2
Q
S
R
A
R
×
[
k
-
1
]
q
(
-
1
)
k
,
∑
k
=
0
2
n
+
1
[
2
n
+
1
k
]
q
𝔅
k
,
q
𝔅
2
n
-
k
+
1
,
q
q
k
=
q
∑
k
=
0
2
n
+
1
[
2
n
+
1
k
]
q
(
-
1
,
q
)
2
n
+
1
-
k
2
2
n
+
1
-
k
𝔅
k
,
q
[
k
-
1
]
q
(
-
1
)
k
-
q
(
1
-
q
)
2
∑
k
=
0
2
n
[
2
n
k
]
q
(
-
1
,
q
)
2
n
-
k
2
2
n
-
k
𝔅
k
,
q
[
k
-
1
]
q
(
-
1
)
k
.We may also derive a differential equation for𝔅
^
q
(
t
). If we differentiate both sides of the generating function with respect to t, after a little calculation we find that
(58)
∂
∂
t
𝔅
^
q
(
t
)
=
𝔅
^
q
(
t
)
(
1
t
-
(
1
-
q
)
ℰ
q
(
t
)
ℰ
q
(
t
)
-
1
(
∑
k
=
0
∞
4
q
k
4
-
(
1
-
q
)
2
q
2
k
)
)
.If we differentiate𝔅
^
q
(
t
) with respect to q, we obtain, instead,
(59)
∂
∂
q
𝔅
^
q
(
t
)
=
-
𝔅
^
q
2
(
t
)
ℰ
q
(
t
)
∑
k
=
0
∞
4
t
(
k
q
k
-
1
-
(
k
+
1
)
q
k
)
4
-
(
1
-
q
)
2
q
2
k
.Again, using the generating function and combining this with the derivative we get the partial differential equation.Proposition 16.
Consider the following:(60)
∂
∂
t
𝔅
^
q
(
t
)
-
∂
∂
q
𝔅
^
q
(
t
)
=
𝔅
^
q
(
t
)
t
+
𝔅
^
q
2
(
t
)
ℰ
q
(
t
)
t
×
∑
k
=
0
∞
4
t
(
k
q
k
-
1
-
(
k
+
1
)
q
k
)
-
q
k
(
1
-
q
)
4
-
(
1
-
q
)
2
q
2
k
.


## 4. Explicit Relationship between theq-Bernoulli and q-Euler Polynomials

In this section, we give some explicit relationship between theq-Bernoulli and q-Euler polynomials. We also obtain new formulae and some special cases for them. These formulae are extensions of the formulae of Srivastava and Pintér, Cheon, and others.We present naturalq-extensions of the main results in the papers [9, 11]; see Theorems 17 and 19.Theorem 17.
Forn
∈
ℕ
0, the following relationships hold true:
(61)
𝔅
n
,
q
(
x
,
y
)
=
1
2
∑
k
=
0
n
[
n
k
]
q
m
k
-
n
[
𝔅
k
,
q
(
x
)
+
∑
j
=
0
k
[
n
j
]
q
(
-
1
,
q
)
k
-
j
𝔅
j
,
q
(
x
)
2
k
-
j
m
k
-
j
]
h
h
h
h
h
h
h
×
𝔈
n
-
k
,
q
(
m
y
)
=
1
2
∑
k
=
0
n
[
n
k
]
q
m
k
-
n
[
𝔅
k
,
q
(
x
)
+
𝔅
k
,
q
(
x
,
1
m
)
]
𝔈
n
-
k
,
q
(
m
y
)
.Proof.
Using the following identity(62)
t
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
t
ℰ
q
(
t
)
-
1
ℰ
q
(
t
x
)
·
ℰ
q
(
t
/
m
)
+
1
2
·
2
ℰ
q
(
t
/
m
)
+
1
ℰ
q
(
t
m
m
y
)
we have
(63)
∑
n
=
0
∞
𝔅
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
=
1
2
∑
n
=
0
∞
𝔈
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
∑
n
=
0
∞
‍
(
-
1
,
q
)
n
m
n
2
n
t
n
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
x
)
t
n
[
n
]
q
!
+
1
2
∑
n
=
0
∞
𝔈
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
x
)
t
n
[
n
]
q
!
=
:
I
1
+
I
2
.
It is clear that
(64)
I
2
=
1
2
∑
n
=
0
∞
𝔈
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
x
)
t
n
[
n
]
q
!
=
1
2
∑
n
=
0
∞
‍
∑
k
=
0
n
[
n
j
]
q
m
k
-
n
𝔅
k
,
q
(
x
)
𝔈
n
-
k
,
q
(
m
y
)
t
n
[
n
]
q
!
.
On the other hand
(65)
I
1
=
1
2
∑
n
=
0
∞
‍
𝔈
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
×
∑
n
=
0
∞
‍
∑
j
=
0
n
[
n
j
]
q
𝔅
j
,
q
(
x
)
(
-
1
,
q
)
n
-
j
m
n
-
j
2
n
-
j
t
n
[
n
]
q
!
=
1
2
∑
n
=
0
∞
‍
∑
k
=
0
n
[
n
k
]
q
𝔈
n
-
k
,
q
(
m
y
)
×
∑
j
=
0
k
[
n
j
]
q
𝔅
j
,
q
(
x
)
(
-
1
,
q
)
k
-
j
m
n
-
k
m
k
-
j
2
k
-
j
t
n
[
n
]
q
!
.
Therefore
(66)
∑
n
=
0
∞
𝔅
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
=
1
2
∑
n
=
0
∞
∑
k
=
0
n
[
n
k
]
q
m
k
-
n
h
h
h
h
h
h
h
h
h
h
×
[
𝔅
k
,
q
(
x
)
+
∑
j
=
0
k
[
n
j
]
q
(
-
1
,
q
)
k
-
j
𝔅
j
,
q
(
x
)
2
k
-
j
m
k
-
j
]
h
h
h
h
h
h
h
h
h
h
×
𝔈
n
-
k
,
q
(
m
y
)
t
n
[
n
]
q
!
.
It remains to equate the coefficients of t
n.Next we discuss some special cases of Theorem17.Corollary 18.
Forn
∈
ℕ
0 the following relationship holds true:
(67)
𝔅
n
,
q
(
x
,
y
)
=
∑
k
=
0
n
[
n
k
]
q
(
𝔅
k
,
q
(
x
)
+
(
-
1
;
q
)
k
-
1
2
k
[
k
]
q
x
k
-
1
)
𝔈
n
-
k
,
q
(
y
)
.The formula (67) is a q-extension of the Cheon's main result [23].Theorem 19.
Forn
∈
ℕ
0, the following relationships
(68)
𝔈
n
,
q
(
x
,
y
)
=
1
[
n
+
1
]
q
×
∑
k
=
0
n
+
1
1
m
n
+
1
-
k
[
n
+
1
k
]
q
h
h
h
h
h
h
×
(
∑
j
=
0
k
[
k
j
]
q
(
-
1
,
q
)
k
-
j
m
k
-
j
2
k
-
j
𝔈
j
,
q
(
y
)
-
𝔈
k
,
q
(
y
)
)
h
h
h
h
h
h
×
𝔅
n
+
1
-
k
,
q
(
m
x
)
hold true between the q-Bernoulli polynomials and q-Euler polynomials.Proof.
The proof is based on the following identity:(69)
2
ℰ
q
(
t
)
+
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
2
ℰ
q
(
t
)
+
1
ℰ
q
(
t
y
)
·
ℰ
q
(
t
/
m
)
-
1
t
·
t
ℰ
q
(
t
/
m
)
-
1
ℰ
q
(
t
m
m
x
)
.
Indeed
(70)
∑
n
=
0
∞
𝔈
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
𝔈
n
,
q
(
y
)
t
n
[
n
]
q
!
∑
n
=
0
∞
(
-
1
,
q
)
n
m
n
2
n
t
n
-
1
[
n
]
q
!
×
∑
n
=
0
∞
𝔅
n
,
q
(
m
x
)
t
n
m
n
[
n
]
q
!
-
∑
n
=
0
∞
𝔈
n
,
q
(
y
)
t
n
-
1
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
m
x
)
t
n
m
n
[
n
]
q
!
=
:
I
1
-
I
2
.
It follows that
(71)
I
2
=
1
t
∑
n
=
0
∞
𝔈
n
,
q
(
y
)
t
n
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
m
x
)
t
n
m
n
[
n
]
q
!
=
1
t
∑
n
=
0
∞
∑
k
=
0
n
[
n
k
]
q
1
m
n
-
k
𝔈
k
,
q
(
y
)
𝔅
n
-
k
,
q
(
m
x
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
1
[
n
+
1
]
q
×
∑
k
=
0
n
+
1
[
n
+
1
k
]
q
1
m
n
+
1
-
k
𝔈
k
,
q
(
y
)
𝔅
n
+
1
-
k
,
q
(
m
x
)
t
n
[
n
]
q
!
,
I
1
=
1
t
∑
n
=
0
∞
𝔅
n
,
q
(
m
x
)
t
n
m
n
[
n
]
q
!
×
∑
n
=
0
∞
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
m
n
-
k
2
n
-
k
𝔈
k
,
q
(
y
)
t
n
[
n
]
q
!
=
1
t
∑
n
=
0
∞
‍
∑
k
=
0
n
[
n
k
]
q
1
m
n
-
k
𝔅
n
-
k
,
q
(
m
x
)
×
∑
j
=
0
k
[
k
j
]
q
(
-
1
,
q
)
k
-
j
m
k
-
j
2
k
-
j
𝔈
j
,
q
(
y
)
t
n
-
1
[
n
]
q
!
.Next we give an interesting relationship between theq-Genocchi polynomials and the q-Bernoulli polynomials.Theorem 20.
Forn
∈
ℕ
0, the following relationship
(72)
𝔊
n
,
q
(
x
,
y
)
=
1
[
n
+
1
]
q
×
∑
k
=
0
n
+
1
1
m
n
-
k
[
n
+
1
k
]
q
h
h
h
h
h
h
×
(
∑
j
=
0
k
[
k
j
]
q
(
-
1
,
q
)
k
-
j
m
k
-
j
2
k
-
j
𝔊
j
,
q
(
x
)
-
𝔊
k
,
q
(
x
)
)
h
h
h
h
h
h
×
𝔅
n
+
1
-
k
,
q
(
m
y
)
,
𝔅
n
,
q
(
x
,
y
)
=
1
2
[
n
+
1
]
q
×
∑
k
=
0
n
+
1
1
m
n
-
k
[
n
+
1
k
]
q
h
h
h
h
h
h
×
(
∑
j
=
0
k
[
k
j
]
q
(
-
1
,
q
)
k
-
j
m
k
-
j
2
k
-
j
𝔅
j
,
q
(
x
)
+
𝔅
k
,
q
(
x
)
)
h
h
h
h
h
h
×
𝔊
n
+
1
-
k
,
q
(
m
y
)
holds true between the q-Genocchi and the q-Bernoulli polynomials.Proof.
Using the following identity(73)
2
t
ℰ
q
(
t
)
+
1
ℰ
q
(
t
x
)
ℰ
q
(
t
y
)
=
2
t
ℰ
q
(
t
)
+
1
ℰ
q
(
t
x
)
·
(
ℰ
q
(
t
m
)
-
1
)
m
t
·
t
/
m
ℰ
q
(
t
/
m
)
-
1
·
ℰ
q
(
t
m
m
y
)
we have
(74)
∑
n
=
0
∞
𝔊
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
=
m
t
∑
n
=
0
∞
𝔊
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
×
∑
n
=
0
∞
‍
(
-
1
,
q
)
n
m
n
2
n
t
n
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
-
m
t
∑
n
=
0
∞
𝔊
n
,
q
(
x
,
y
)
t
n
[
n
]
q
!
∑
n
=
0
∞
𝔅
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
=
m
t
∑
n
=
0
∞
(
∑
k
=
0
n
[
n
k
]
q
(
-
1
,
q
)
n
-
k
m
n
-
k
2
n
-
k
𝔊
k
,
q
(
x
)
-
𝔊
n
,
q
(
x
)
)
t
n
[
n
]
q
!
×
∑
n
=
0
∞
𝔅
n
,
q
(
m
y
)
t
n
m
n
[
n
]
q
!
=
m
t
∑
n
=
0
∞
‍
∑
k
=
0
n
1
m
n
-
k
[
n
k
]
q
h
h
h
h
h
h
h
h
h
×
(
∑
j
=
0
k
[
k
j
]
q
(
-
1
,
q
)
k
-
j
m
k
-
j
2
k
-
j
𝔊
j
,
q
(
x
)
-
𝔊
k
,
q
(
x
)
)
h
h
h
h
h
h
h
h
h
×
𝔅
n
-
k
,
q
(
m
y
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
1
[
n
+
1
]
q
∑
k
=
0
n
+
1
1
m
n
-
k
[
n
+
1
k
]
q
h
h
h
h
h
h
h
h
h
×
(
∑
j
=
0
k
‍
[
k
j
]
q
(
-
1
,
q
)
k
-
j
m
k
-
j
2
k
-
j
𝔊
j
,
q
(
x
)
-
𝔊
k
,
q
(
x
)
)
h
h
h
h
h
h
h
h
h
×
𝔅
n
+
1
-
k
,
q
(
m
y
)
t
n
[
n
]
q
!
.
The second identity can be proved in a like manner.



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*Source: 696454-2014-04-28.xml*