# q-Extensions for the Apostol Type Polynomials

Authors: Nazim I. Mahmudov; Marzieh Eini Keleshteri
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The aim of this work is to introduce an extension forq-standard notations. The q-Apostol type polynomials and study some of their properties. Besides, some relations between the mentioned polynomials and some other known polynomials are obtained.


---

## Body

## 1. Introduction, Preliminaries, and Definitions

Throughout this research we always apply the following notations.N indicates the set of natural numbers, N
0 indicates the set of nonnegative integers, R indicates the set of all real numbers, and C denotes the set of complex numbers. We refer the readers to [1] for all the following q-standard notations. The q-shifted factorial is defined as
(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
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
|
q
|
<
1
,
a
∈
C
.
The q-numbers and q-factorials are defined by
(2)
[
a
]
q
=
1
-
q
a
1
-
q
(
q
≠
1
)
;
[
0
]
!
=
1
;
[
n
]
q
!
=
[
1
]
q
[
2
]
q
⋯
[
n
]
q
,
i
i
i
i
i
i
i
i
i
i
i
n
∈
N
,
a
∈
C
,
respectively. The q-polynomial coefficient is defined by
(3)
[
n
k
]
q
=
[
n
]
q
!
[
k
]
q
!
[
n
-
k
]
q
!
.
The q-analogue of the function (
x
+
y
)
n is defined by
(4)
(
x
+
y
)
q
n
:
=
∑
k
=
0
n
[
n
k
]
q
q
(
1
/
2
)
k
(
k
-
1
)
x
n
-
k
y
k
,
n
∈
N
0
.
The q-binomial formula is known as
(5)
(
1
-
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
.
In the standard approach to the q-calculus, two exponential functions are used:
(6)
e
q
(
z
)
=
∑
n
=
0
∞
z
n
[
n
]
q
!
=
∏
j
=
0
∞
1
(
1
-
(
1
-
q
)
q
j
z
)
,
fffffffffffffffff
0
<
|
q
|
<
1
,
|
z
|
<
1
|
1
-
q
|
,
(7)
E
q
(
z
)
=
∑
k
=
0
∞
q
(
1
/
2
)
k
(
k
-
1
)
z
k
[
k
]
q
!
=
∏
j
=
0
∞
(
1
+
(
1
-
q
)
q
j
z
)
,
gggggggggggggggggggggg
0
<
|
q
|
<
1
,
z
∈
C
.
As an immediate result of these two definitions, we have e
q
(
z
)
E
q
(
-
z
)
=
1.Recently, Luo and Srivastava [2] introduced and studied the generalized Apostol-Bernoulli polynomials B
n
α
(
x
;
λ
) and the generalized Apostol-Euler polynomials E
n
α
(
x
;
λ
). Kurt [3] gave the generalization of the Bernoulli polynomials B
n
[
m
-
1
,
α
]
(
x
) of order α and studied their properties. They also studied these polynomials systematically; see [2, 4–9]. There are numerous recent investigations on this subject by many other authors; see [3, 10–20]. More recently, Tremblay et al. [10] further gave the definition of B
n
[
m
-
1
,
α
]
(
x
;
λ
) and studied their properties. On the other hand, Mahmudov and Keleshteri [21, 22] studied various two dimensional q-polynomials. Motivated by these papers, we define generalized Apostol type q-polynomials as follows.Definition 1.
Letq
,
α
∈
C, m
∈
N, and0
<
|
q
|
<
1. The generalized q-Apostol-Bernoulli numbers B
n
,
q
[
m
-
1
,
α
] and polynomials B
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) in x
,
y of order α are defined, in a suitable neighborhood of t
=
0, by means of the generating functions:
(8)
(
t
m
λ
e
q
(
t
)
-
T
m
-
1
,
q
(
t
)
)
α
=
∑
n
=
0
∞
‍
B
n
,
q
[
m
-
1
,
α
]
(
λ
)
t
n
[
n
]
q
!
,
(
t
m
λ
e
q
(
t
)
-
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
x
)
E
q
(
t
y
)
=
∑
n
=
0
∞
‍
B
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
t
n
[
n
]
q
!
,
where T
m
-
1
,
q
(
t
)
=
∑
k
=
0
m
-
1
(
t
k
/
[
k
]
q
!
).Definition 2.
Letq
,
α
∈
C, 0
<
|
q
|
<
1, and m
∈
N. The generalized q-Apostol-Euler numbers E
n
,
q
[
m
-
1
,
α
] and polynomials E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) in x
,
y of order α are defined, in a suitable neighborhood of t
=
0, by means of the generating functions:
(9)
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
=
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
]
(
λ
)
t
n
[
n
]
q
!
,
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
x
)
E
q
(
t
y
)
=
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
t
n
[
n
]
q
!
.Definition 3.
Letq
,
α
∈
C
,
0
<
|
q
|
<
1, and m
∈
N. The generalized q-Apostol-Genocchi numbers G
n
,
q
[
m
-
1
,
α
] and polynomials G
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) in x
,
y of order α are defined, in a suitable neighborhood of t
=
0, by means of the generating functions:
(10)
(
2
m
t
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
=
∑
n
=
0
∞
‍
G
n
,
q
[
m
-
1
,
α
]
(
λ
)
t
n
[
n
]
q
!
,
(
2
m
t
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
x
)
E
q
(
t
y
)
=
∑
n
=
0
∞
‍
G
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
t
n
[
n
]
q
!
.Clearly, form
=
1, one has
(11)
B
n
,
q
[
0
,
α
]
(
x
,
y
;
λ
)
=
B
n
,
q
(
α
)
(
x
,
y
;
λ
)
,
E
n
,
q
[
0
,
α
]
(
x
,
y
;
λ
)
=
E
n
,
q
(
α
)
(
x
,
y
;
λ
)
,
G
n
,
q
[
0
,
α
]
(
x
,
y
;
λ
)
=
G
n
,
q
(
α
)
(
x
,
y
;
λ
)
.
For m
=
1 and λ
=
1, one has
(12)
B
n
,
q
[
0
,
α
]
(
x
,
y
;
1
)
=
B
n
,
q
(
α
)
(
x
,
y
)
,
E
n
,
q
[
0
,
α
]
(
x
,
y
;
1
)
=
E
n
,
q
(
α
)
(
x
,
y
)
,
G
n
,
q
[
0
,
α
]
(
x
,
y
;
1
)
=
G
n
,
q
(
α
)
(
x
,
y
)
.
For x
=
y
=
0, one has
(13)
B
n
,
q
[
m
-
1
,
α
]
(
0,0
;
λ
)
=
B
n
,
q
[
m
-
1
,
α
]
(
λ
)
,
E
n
,
q
[
m
-
1
,
α
]
(
0,0
;
λ
)
=
E
n
,
q
[
m
-
1
,
α
]
(
λ
)
,
G
n
,
q
[
m
-
1
,
α
]
(
0,0
;
λ
)
=
G
n
,
q
[
m
-
1
,
α
]
(
λ
)
.


## 2. Properties of the Apostol Typeq-Polynomials

In this section, we show some basic properties of the generalizedq-polynomials. We only prove the facts for one of them. Obviously, by applying the similar technique, other ones can be proved.Proposition 4.
The generalizedq-polynomials B
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
), E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
), and G
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) satisfy the following relations:
(14)
B
n
,
q
[
m
-
1
,
α
+
β
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
B
k
,
q
[
m
-
1
,
α
]
(
x
,
0
;
λ
)
×
B
n
-
k
,
q
[
m
-
1
,
β
]
(
0
,
y
;
λ
)
,
E
n
,
q
[
m
-
1
,
α
+
β
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
x
,
0
;
λ
)
×
E
n
-
k
,
q
[
m
-
1
,
β
]
(
0
,
y
;
λ
)
,
G
n
,
q
[
m
-
1
,
α
+
β
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
G
k
,
q
[
m
-
1
,
α
]
(
x
,
0
;
λ
)
×
G
n
-
k
,
q
[
m
-
1
,
β
]
(
0
,
y
;
λ
)
.Proof.
We only prove the second identity. By using Definition2, we have
(15)
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
+
β
]
(
x
,
y
;
λ
)
t
n
[
n
]
q
!
=
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
+
β
e
q
(
t
x
)
E
q
(
t
y
)
=
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
x
)
×
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
β
E
q
(
t
y
)
=
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
]
(
x
,
0
;
λ
)
t
n
[
n
]
q
!
×
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
β
]
(
0
,
y
;
λ
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
‍
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
x
,
0
;
λ
)
E
n
-
k
,
q
[
m
-
1
,
β
]
(
0
,
y
;
λ
)
t
n
[
n
]
q
!
.
Comparing the coefficients of the term t
n
/
[
n
]
q
! in both sides gives the result.Corollary 5.
The generalizedq-polynomials B
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
), E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
), and G
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) satisfy the following relations:
(16)
B
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
B
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
x
n
-
k
,
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
x
n
-
k
,
G
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
G
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
x
n
-
k
.Proposition 6.
The generalizedq-polynomials B
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
), E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
), and G
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) satisfy the following relations:
(17)
λ
B
n
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
-
B
n
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
[
k
]
q
B
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
B
n
-
k
,
q
[
0
,
-
1
]
(
λ
)
,
f
o
r
n
≥
1
,
(18)
λ
E
n
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
+
E
n
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
2
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
E
n
-
k
,
q
[
0
,
-
1
]
(
λ
)
,
(19)
λ
G
n
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
+
G
n
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
2
∑
k
=
0
n
‍
[
n
k
]
q
[
k
]
q
G
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
G
n
-
k
,
q
[
0
,
-
1
]
(
λ
)
,
f
o
r
n
≥
1
.Proof.
We only prove (18). By using Definition 2 and starting from the left hand side of the relation (18), we have
(20)
∑
n
=
0
∞
‍
(
λ
E
n
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
+
E
n
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
)
t
n
[
n
]
q
!
=
λ
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
)
E
q
(
t
y
)
+
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
E
q
(
t
y
)
=
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
E
q
(
t
y
)
(
λ
e
q
(
t
)
+
1
)
=
2
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
E
q
(
t
y
)
(
2
λ
e
q
(
t
)
+
1
)
-
1
=
2
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
t
n
[
n
]
q
!
∑
n
=
0
∞
‍
E
n
,
q
[
0
,
-
1
]
(
λ
)
t
n
[
n
]
q
!
=
2
∑
n
=
0
∞
‍
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
E
n
-
k
,
q
[
0
,
-
1
]
(
λ
)
t
n
[
n
]
q
!
.
Comparing the coefficients of the term t
n
/
[
n
]
q
! in both sides gives the result.


## 3.q-Analogue of the Luo-Srivastava Addition Theorem

In this section, we state and prove aq-generalization of the Luo-Srivastava addition theorem.Theorem 7.
The following relation holds between generalizedq-Apostol-Euler and q-Apostol-Bernoulli polynomials:
(21)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
j
=
0
n
‍
1
[
n
+
1
]
q
[
n
+
1
j
]
q
×
(
λ
∑
k
=
0
n
-
j
+
1
‍
1
[
n
+
1
]
q
[
n
-
j
+
1
k
]
q
E
k
,
q
[
m
-
1
,
α
-
1
]
(
0
,
y
;
λ
)
-
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
λ
∑
k
=
0
n
-
j
+
1
‍
1
[
n
+
1
]
q
[
n
-
j
+
1
k
]
q
∑
k
=
0
n
-
j
+
1
‍
)
×
B
j
,
q
(
x
,
0
;
λ
)
+
λ
-
1
[
n
+
1
]
q
×
(
2
m
λ
+
1
)
α
B
n
+
1
,
q
(
x
,
0
;
λ
)
.Proof.
We take aid of the following identity to prove (21):
(22)
λ
t
λ
e
q
(
t
)
-
1
e
q
(
t
x
)
e
q
(
t
)
-
t
λ
e
q
(
t
)
-
1
e
q
(
t
x
)
=
t
e
q
(
t
x
)
λ
e
q
(
t
)
-
1
(
λ
e
q
(
t
)
-
1
)
=
t
e
q
(
t
x
)
.
Therefore, we can write
(23)
λ
∑
n
=
0
∞
‍
∑
k
=
0
n
‍
[
n
k
]
q
B
k
,
q
(
x
,
0
;
λ
)
t
n
[
n
]
q
!
-
∑
n
=
0
∞
‍
B
n
,
q
(
x
,
0
;
λ
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
‍
x
n
t
n
+
1
[
n
+
1
]
q
!
[
n
+
1
]
q
=
∑
n
=
0
∞
‍
[
n
]
q
x
n
-
1
t
n
[
n
]
q
!
.
From that we can conclude the following:
(24)
λ
∑
k
=
0
n
‍
[
n
k
]
q
B
k
,
q
(
x
,
0
;
λ
)
-
B
n
,
q
(
x
,
0
;
λ
)
=
[
n
]
q
x
n
-
1
.
That is,
(25)
x
n
=
1
[
n
+
1
]
q
(
λ
∑
k
=
0
n
+
1
‍
[
n
+
1
k
]
q
B
k
,
q
(
x
,
0
;
λ
)
-
B
n
+
1
,
q
(
x
,
0
;
λ
)
)
.
Substituting (25) into the right hand side of (16), we obtain
(26)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
1
[
n
-
k
+
1
]
q
×
(
λ
∑
j
=
0
n
-
k
+
1
‍
[
n
-
k
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
g
g
g
g
g
g
g
i
g
-
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
λ
∑
j
=
0
n
-
k
+
1
‍
[
n
-
k
+
1
j
]
q
∑
j
=
0
n
-
k
+
1
‍
)
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
1
[
n
-
k
+
1
]
q
×
(
λ
∑
j
=
0
n
-
k
‍
[
n
-
k
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
g
g
g
g
g
g
g
g
+
(
λ
-
1
)
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
λ
∑
j
=
0
n
-
k
‍
[
n
-
k
+
1
j
]
q
∑
j
=
0
n
-
k
‍
)
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
λ
[
n
-
k
+
1
]
q
×
∑
j
=
0
n
-
k
‍
[
n
-
k
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
+
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
λ
-
1
[
n
-
k
+
1
]
q
×
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
:
=
I
1
+
I
2
.
Thus, from one hand, we can write
(27)
I
1
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
λ
[
n
-
k
+
1
]
q
×
∑
j
=
0
n
-
k
‍
[
n
-
k
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
=
∑
j
=
0
n
‍
∑
k
=
0
n
-
j
‍
λ
[
n
+
1
]
q
[
n
+
1
n
-
k
+
1
]
q
[
n
-
k
+
1
j
]
q
×
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
B
j
,
q
(
x
,
0
;
λ
)
.
As we know that
(28)
[
m
l
]
q
[
l
n
]
q
=
[
m
n
]
q
[
m
-
n
m
-
l
]
q
,
for
m
≥
l
≥
n
,
we can continue as
(29)
I
1
=
∑
j
=
0
n
‍
∑
k
=
0
n
-
j
‍
λ
[
n
+
1
]
q
[
n
+
1
j
]
q
[
n
-
j
+
1
k
]
q
×
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
B
j
,
q
(
x
,
0
;
λ
)
=
∑
j
=
0
n
‍
λ
[
n
+
1
]
q
[
n
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
×
∑
k
=
0
n
-
j
‍
[
n
-
j
+
1
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
j
=
0
n
‍
λ
[
n
+
1
]
q
[
n
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
×
(
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
-
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
)
.
On the other hand, for I
2, we can write
(30)
I
2
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
λ
-
1
[
n
-
k
+
1
]
q
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
=
∑
k
=
0
n
‍
[
n
+
1
k
]
q
λ
-
1
[
n
+
1
]
q
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
k
=
0
n
+
1
‍
[
n
+
1
k
]
q
λ
-
1
[
n
+
1
]
q
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
-
λ
-
1
[
n
+
1
]
q
B
0
,
q
(
x
,
0
;
λ
)
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
,
and, as B
0
,
q
(
x
,
0
;
λ
)
=
0, we have
(31)
I
2
=
∑
k
=
0
n
+
1
‍
[
n
+
1
k
]
q
λ
-
1
[
n
+
1
]
q
B
n
-
k
+
1
,
q
(
x
,
0
;
λ
)
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
j
=
0
n
+
1
‍
[
n
+
1
j
]
q
λ
-
1
[
n
+
1
]
q
B
j
,
q
(
x
,
0
;
λ
)
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
j
=
0
n
‍
[
n
+
1
j
]
q
λ
-
1
[
n
+
1
]
q
B
j
,
q
(
x
,
0
;
λ
)
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
+
λ
-
1
[
n
+
1
]
q
B
n
+
1
,
q
(
x
,
0
;
λ
)
E
0
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
.
Adding I
2 to I
1 we obtain
(32)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
I
1
+
I
2
=
∑
j
=
0
n
‍
λ
[
n
+
1
]
q
[
n
+
1
j
]
q
B
j
,
q
(
x
,
0
;
λ
)
×
(
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
-
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
)
+
∑
j
=
0
n
‍
[
n
+
1
j
]
q
λ
-
1
[
n
+
1
]
q
B
j
,
q
(
x
,
0
;
λ
)
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
+
λ
-
1
[
n
+
1
]
q
B
n
+
1
,
q
(
x
,
0
;
λ
)
E
0
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
.
Consequently,
(33)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
j
=
0
n
‍
1
[
n
+
1
]
q
[
n
+
1
j
]
q
×
(
λ
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
-
λ
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
g
g
g
g
i
g
+
(
λ
-
1
)
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
)
×
B
j
,
q
(
x
,
0
;
λ
)
+
λ
-
1
[
n
+
1
]
q
×
B
n
+
1
,
q
(
x
,
0
;
λ
)
E
0
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
j
=
0
n
‍
1
[
n
+
1
]
q
[
n
+
1
j
]
q
×
(
λ
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
1
,
y
;
λ
)
-
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
)
×
B
j
,
q
(
x
,
0
;
λ
)
+
(
λ
-
1
)
[
n
+
1
]
q
B
n
+
1
,
q
(
x
,
0
;
λ
)
×
E
0
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
=
∑
j
=
0
n
‍
1
[
n
+
1
]
q
[
n
+
1
j
]
q
h
h
h
h
×
(
λ
∑
k
=
0
n
-
j
+
1
‍
[
n
-
j
+
1
k
]
q
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
g
g
g
g
g
g
g
g
-
E
n
-
j
+
1
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
λ
∑
k
=
0
n
-
j
+
1
‍
[
n
-
j
+
1
k
]
q
∑
k
=
0
n
-
j
+
1
‍
)
×
B
j
,
q
(
x
,
0
;
λ
)
+
(
λ
-
1
)
[
n
+
1
]
q
(
2
m
λ
+
1
)
α
B
n
+
1
,
q
(
x
,
0
;
λ
)
.Takingm
=
1 in Theorem 7, we get a q-generalization of the Luo-Srivastava addition theorem [2].Corollary 8.
The following relation holds between generalizedq-Apostol-Euler and q-Apostol-Bernoulli polynomials:
(34)
E
n
,
q
(
α
)
(
x
,
y
;
λ
)
=
∑
j
=
0
n
‍
2
[
j
+
1
]
q
[
n
j
]
q
×
(
E
j
+
1
,
q
(
α
)
(
0
,
y
;
λ
)
-
E
j
+
1
,
q
(
α
)
(
0
,
y
;
λ
)
)
×
B
n
-
j
,
q
(
x
,
0
;
λ
)
+
λ
-
1
[
n
+
1
]
q
(
2
λ
+
1
)
α
×
B
n
+
1
,
q
(
x
,
0
;
λ
)
.Lettingq
↑
1, we get the Luo-Srivastava addition theorem (see [12]):
(35)
E
n
(
α
)
(
x
+
y
;
λ
)
=
∑
j
=
0
n
‍
2
j
+
1
(
n
j
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
(
E
j
+
1
(
α
)
(
y
;
λ
)
-
E
j
+
1
(
α
)
(
y
;
λ
)
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
B
n
-
j
,
q
(
x
;
λ
)
+
λ
-
1
n
+
1
(
2
λ
+
1
)
α
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
B
n
+
1
(
x
;
λ
)
.Next theorem gives relationship betweenE
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) and G
n
,
q
(
x
,
0
).Theorem 9.
The following relation holds between generalizedq-Apostol-Euler and q-Apostol-Genocchi polynomials:
(36)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
1
2
∑
k
=
0
n
‍
1
[
k
+
1
]
q
iiiiiiiiiiiiiiiiiiiiiiiiiiii
×
(
λ
∑
j
=
k
n
‍
[
n
j
]
q
[
j
k
]
q
E
n
-
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
∑
k
=
0
n
‍
[
n
k
]
q
E
n
-
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
)
iiiiiiiiiiiiiiiiiiiiiiiiiiii
×
G
k
+
1
,
q
(
x
,
0
)
.Proof.
The proof follows from the following identity:(37)
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
x
)
E
q
(
t
y
)
=
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
E
q
(
t
y
)
2
t
e
q
(
t
)
+
1
×
e
q
(
t
x
)
e
q
(
t
)
+
1
2
t
.Theorem 10.
The following relation holds between generalizedq-Apostol-Euler and q-Stirling polynomials S
q
(
i
,
j
) of the second kind:
(38)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
∑
j
=
k
n
‍
[
n
n
-
j
]
q
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
E
n
-
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
S
q
(
j
,
k
)
x
k
(
x
)
.Proof.
Theq-Stirling polynomials S
q
(
n
,
k
) of the second kind are defined by means of the following generating function:
(39)
x
n
=
∑
k
=
0
n
‍
S
q
(
n
,
k
)
x
k
(
x
)
,
where x
k
(
x
)
=
x
(
x
-
[
1
]
q
)
(
x
-
[
2
]
q
)
⋯
(
x
-
[
k
-
1
]
q
); see [23]. Replacing identity (39) in the right hand side of (16), we have
(40)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
n
‍
[
n
k
]
q
E
k
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
i
i
i
i
×
∑
k
=
0
n
-
k
‍
S
q
(
n
-
k
,
k
)
x
k
(
x
)
=
∑
k
=
0
n
‍
∑
j
=
k
n
‍
[
n
n
-
j
]
q
i
i
i
i
×
E
n
-
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
S
q
(
j
,
k
)
x
k
(
x
)
.Theorem 11.
The relationship(41)
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
=
∑
k
=
0
[
n
/
2
]
‍
∑
j
=
0
n
-
2
k
‍
[
n
k
]
q
[
n
-
2
k
j
]
q
[
k
]
q
!
[
2
]
q
n
[
k
]
q
2
!
i
i
i
i
×
E
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
H
n
-
2
k
-
j
,
q
(
x
)
holds between the polynomials E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
) and the q-Hermite polynomials defined by (see [24])
(42)
e
q
(
t
x
)
E
q
2
(
-
t
2
[
2
]
q
)
=
∑
n
=
0
∞
‍
H
n
,
q
(
x
)
t
n
[
n
]
q
!
.Proof.
Indeed,(43)
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
]
(
x
,
y
;
λ
)
t
n
[
n
]
q
!
=
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
e
q
(
t
x
)
E
q
(
t
y
)
=
(
2
m
λ
e
q
(
t
)
+
T
m
-
1
,
q
(
t
)
)
α
E
q
(
t
y
)
e
q
(
t
x
)
×
E
q
2
(
-
t
2
[
2
]
q
)
e
q
2
(
t
2
[
2
]
q
)
=
∑
n
=
0
∞
‍
E
n
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
t
n
[
n
]
q
!
×
∑
n
=
0
∞
‍
H
n
,
q
(
x
)
t
n
[
n
]
q
!
∑
n
=
0
∞
‍
t
2
n
[
2
]
q
n
[
n
]
q
2
!
=
∑
n
=
0
∞
‍
∑
j
=
0
n
‍
[
n
j
]
q
E
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
H
n
-
j
,
q
(
x
)
t
n
[
n
]
q
!
×
∑
n
=
0
∞
‍
t
2
n
[
2
]
q
n
[
n
]
q
2
!
=
∑
n
=
0
∞
‍
∑
k
=
0
[
n
/
2
]
‍
[
n
]
q
!
[
2
]
q
n
[
k
]
q
2
!
[
n
-
2
k
]
q
!
×
∑
j
=
0
n
-
2
k
‍
[
n
-
2
k
j
]
q
E
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
H
n
-
2
k
-
j
,
q
(
x
)
t
n
[
n
]
q
!
=
∑
n
=
0
∞
‍
∑
k
=
0
[
n
/
2
]
‍
∑
j
=
0
n
-
2
k
‍
[
n
k
]
q
[
n
-
2
k
j
]
q
[
k
]
q
!
[
2
]
q
n
[
k
]
q
2
!
i
i
i
i
i
i
i
×
E
j
,
q
[
m
-
1
,
α
]
(
0
,
y
;
λ
)
H
n
-
2
k
-
j
,
q
(
x
)
t
n
[
n
]
q
!
.



---

*Source: 868167-2014-06-02.xml*