# Identities of Symmetry for Higher-Order Generalizedq-Euler Polynomials

Authors: D. V. Dolgy; D. S. Kim; T. G. Kim; J. J. Seo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate the properties of symmetry in two variables related to multiple Eulerq-l-function which interpolates higher-order q-Euler polynomials at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to generalized higher-order q-Euler polynomials and alternating generalized q-power sums.


---

## Body

## 1. Introduction

Throughout this paper, the notationsℕ, ℤ, ℝ, and ℂ denote the sets of positive integers, integers, real numbers, and complex numbers, respectively, and ℤ
≥
0
:
=
ℕ
∪
{
0
}. Let χ be a Dirichlet character with d
∈
ℕ with conductor d
≡
1
(
mod
⁡
2
). Then the generalized Euler polynomials attached to χ are defined by the following generating function (see [1–3]):
(1)
2
∑
a
=
0
d
-
1
‍
χ
(
a
)
(
-
1
)
a
e
(
a
+
x
)
t
e
d
t
+
1
=
∑
n
=
0
∞
‍
E
n
,
χ
(
x
)
t
n
n
!
.
The generalized Euler polynomials of order r
∈
ℕ attached to χ are also defined by the generating function:
(2)
(
2
∑
a
=
0
d
-
1
‍
χ
(
a
)
(
-
1
)
a
e
(
a
+
x
)
t
e
d
t
+
1
)
r
=
∑
n
=
0
∞
‍
E
n
,
χ
(
r
)
(
x
)
t
n
n
!
.
When x
=
0, E
n
,
χ
(
r
)
=
E
n
χ
(
r
)
(
0
) are called the generalized Euler numbers attached to χ (see [2, 4]).Assume thatq
∈
ℂ with |
q
|
<
1 and define q-numbers by (see [2–15])
(3)
[
x
]
q
=
1
-
q
x
1
-
q
.
Note that lim
⁡
q
→
1
[
x
]
q
=
x.In [4, 8], Kim initiated to consider various q-extensions (or (
h
,
q
)-extensions) of Euler numbers and polynomials and constructed analytic continuations which interpolate his q-numbers and polynomials. Until recently, many authors have studied q-Euler or (
h
,
q
)-Euler polynomials due to him (see [1–21]). In [4], Kim defined the (
h
,
q
)-extension of generalized higher-order Euler polynomials attached to χ which is given by the generating function:
(4)
F
q
,
χ
(
h
,
r
)
(
t
,
x
)
=
[
2
]
q
r
∑
m
1
,
…
,
m
r
=
0
∞
‍
q
∑
j
=
1
r
‍
(
h
-
j
+
1
)
m
j
(
-
1
)
∑
j
=
1
r
‍
m
j
×
(
∏
j
=
1
r
‍
χ
(
m
j
)
)
e
[
x
+
∑
l
=
1
r
‍
m
l
]
q
t
=
∑
n
=
0
∞
‍
E
n
,
χ
,
q
(
h
,
r
)
(
x
)
t
n
n
!
,
where h
∈
ℤ and r
∈
ℕ.Note that(5)
lim
⁡
q
→
1
F
q
(
h
,
r
)
(
t
,
x
)
=
(
2
∑
a
=
0
d
-
1
‍
χ
(
a
)
(
-
1
)
a
e
(
a
+
x
)
t
e
d
t
+
1
)
r
=
∑
n
=
0
∞
‍
E
n
,
χ
(
r
)
(
x
)
t
n
n
!
.
When x
=
0, E
n
,
χ
,
q
(
h
,
r
)
=
E
n
,
χ
,
q
(
h
,
r
)
(
0
) are called the (
h
,
q
)-extension of generalized higher-order Euler numbers attached to χ.We find from (4) that
(6)
E
n
,
χ
,
q
(
h
,
r
)
(
x
)
=
∑
l
=
0
n
‍
(
n
l
)
q
l
x
E
l
,
χ
,
q
(
h
,
r
)
[
x
]
q
n
-
l
=
(
q
x
E
χ
,
q
(
h
,
r
)
+
[
x
]
q
)
n
,
with the usual convention about replacing (
E
χ
,
q
(
h
·
r
)
)
n by E
n
,
χ
,
q
(
h
·
r
).In [4], Dirichlet-type multiple (
h
,
q
)
-
l-function is defined by Kim to be
(7)
l
q
,
r
(
h
)
(
s
,
x
∣
χ
)
=
1
Γ
(
s
)
∫
0
∞
‍
F
q
,
χ
(
h
,
r
)
(
-
t
,
x
)
t
s
-
1
d
t
=
[
2
]
q
r
∑
m
1
,
…
,
m
r
=
0
∞
‍
q
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
(
-
1
)
∑
l
=
1
r
‍
m
l
[
m
1
+
·
·
·
+
m
r
+
x
]
q
s
,
where s
,
h
∈
ℂ and x
∈
ℝ, with x
≠
0
,
-
1
,
-
2
,
….By using Cauchy residue theorem, we get(8)
l
q
,
r
(
h
)
(
-
n
,
x
∣
χ
)
=
E
n
,
χ
,
q
(
h
,
r
)
(
x
)
,
n
∈
ℤ
≥
0
.
In this paper, we investigate certain properties of symmetry in two variables related to Dirichlet-type multiple (
h
,
q
)-function which interpolates the (
h
,
q
)-extension of generalized higher-order Euler polynomials attached to χ at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to (
h
,
q
)-extension of generalized higher-order Euler polynomials and alternating generalized q-power sums.


## 2. Identities for the(
h
,
q
)-Extension of Generalized Higher-Order Euler Polynomials

In this section, we assume thatχ is a Dirichlet character with conductor d
∈
ℕ with d
≡
1
(
mod
⁡
2
).Letw
1
,
w
2
,
r
∈
ℕ with w
1
≡
1
(
mod
⁡
2
) and w
2
≡
1
(
mod
⁡
2
) and h
∈
ℤ. First, we observe that
(9)
1
[
2
]
q
w
1
r
l
q
w
1
,
r
(
h
)
(
s
,
w
2
x
+
w
2
w
1
∑
l
=
1
r
‍
j
l
∣
χ
)
=
∑
m
1
,
…
,
m
r
=
0
∞
‍
(
(
-
1
)
m
1
+
·
·
·
+
m
r
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
×
(
+
w
2
w
1
(
j
1
+
·
·
·
+
j
r
)
]
q
w
1
s
[
w
2
w
1
m
1
+
·
·
·
+
m
r
+
w
2
x
+
w
2
w
1
(
j
1
+
·
·
·
+
j
r
)
]
q
w
1
s
)
-
1
∏
l
=
1
r
‍
χ
(
m
l
)
)
=
∑
m
1
,
…
,
m
r
=
0
∞
‍
(
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
(
-
1
)
∑
l
=
1
r
‍
m
l
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
[
w
1
]
q
s
×
(
+
w
2
w
1
(
j
1
+
·
·
·
+
j
r
)
]
q
w
1
s
[
w
2
(
j
1
+
·
·
·
+
j
r
)
+
w
1
w
2
x
+
w
1
(
m
1
+
·
·
·
+
m
r
)
]
q
s
)
-
1
∏
l
=
1
r
‍
χ
(
m
l
)
)
=
[
w
1
]
q
s
×
∑
n
1
,
…
,
n
r
=
0
∞
‍
∑
i
1
,
…
,
i
r
=
0
d
w
2
-
1
‍
(
(
[
w
1
w
2
(
x
+
d
∑
l
=
1
r
‍
n
l
)
+
w
2
∑
l
=
1
r
‍
j
l
+
w
1
∑
l
=
1
r
‍
i
l
]
q
s
)
-
1
(
-
1
)
∑
l
=
1
r
‍
(
i
l
+
n
l
)
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
(
i
l
+
n
l
d
b
)
×
(
∏
l
=
1
r
‍
χ
(
i
l
)
)
×
(
[
w
1
w
2
(
x
+
d
∑
l
=
1
r
‍
n
l
)
+
w
2
∑
l
=
1
r
‍
j
l
+
w
1
∑
l
=
1
r
‍
i
l
]
q
s
(
∏
l
=
1
r
‍
χ
(
i
l
)
)
)
-
1
=
[
w
1
]
q
s
×
∑
n
1
,
…
,
n
r
=
0
∞
‍
∑
i
1
,
…
,
i
r
=
0
w
2
d
-
1
‍
(
(
[
w
1
w
2
(
x
+
d
∑
l
=
1
r
‍
n
l
)
+
w
2
∑
l
=
1
r
‍
j
l
+
w
1
∑
l
=
1
r
‍
i
l
]
q
s
)
-
1
(
-
1
)
∑
l
=
1
r
‍
(
i
l
+
n
l
)
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
(
i
l
+
n
l
w
2
d
)
×
(
∏
l
=
1
r
‍
χ
(
i
l
)
)
×
(
[
w
1
w
2
(
x
+
d
∑
l
=
1
r
‍
n
l
)
+
w
2
∑
l
=
1
r
‍
j
l
+
w
1
∑
l
=
1
r
‍
i
l
]
q
s
(
∏
l
=
1
r
‍
χ
(
i
l
)
)
)
-
1
.
Thus, by (9), we get
(10)
[
w
2
]
q
s
[
2
]
q
w
1
r
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
q
b
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
l
q
w
1
,
r
(
h
)
(
s
,
w
2
x
+
w
2
w
1
∑
l
=
1
r
‍
j
l
∣
χ
)
=
[
w
1
]
q
s
[
w
2
]
q
s
×
∑
i
1
,
…
,
i
r
=
0
d
w
2
-
1
‍
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
∑
n
1
,
…
,
n
r
=
0
∞
‍
(
+
w
1
∑
l
=
1
r
‍
j
l
+
w
2
∑
l
=
1
r
‍
i
l
]
q
s
)
-
e
8991


(
(
-
1
)
∑
l
=
1
r
‍
(
i
l
+
j
l
+
n
l
)
×
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
(
∏
l
=
1
r
‍
χ
(
i
l
)
)
×
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
+
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
i
l
)
×
(
[
w
1
w
2
(
x
+
d
∑
l
=
1
r
‍
n
l
)
+
w
2
∑
l
=
1
r
‍
j
l
+
w
1
∑
l
=
1
r
‍
i
l
]
q
s
)
-
1
)
×
q
w
1
w
2
d
∑
l
=
1
r
‍
(
h
-
l
+
1
)
n
l
.
By using the same method as (10), we get
(11)
[
w
1
]
q
s
[
2
]
q
w
2
s
∑
j
1
,
…
,
j
r
=
0
d
w
2
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
l
q
w
2
,
r
(
h
)
(
s
,
w
1
x
+
w
1
w
2
∑
l
=
1
r
‍
j
l
∣
χ
)
=
[
w
1
]
q
s
[
w
2
]
q
s
×
∑
j
1
,
…
,
j
r
=
0
d
w
2
-
1
‍
∑
i
1
,
…
,
i
r
=
0
d
w
1
-
1
‍
∑
n
1
,
…
,
n
r
=
0
∞
‍
(
+
w
1
∑
l
=
1
r
‍
j
l
+
w
2
∑
l
=
1
r
‍
i
l
]
q
s
)
-
1


(
(
-
1
)
∑
l
=
1
r
‍
(
i
l
+
j
l
+
n
l
)
×
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
(
∏
l
=
1
r
‍
χ
(
i
l
)
)
×
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
+
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
i
l
)
×
(
[
w
1
w
2
(
x
+
d
∑
l
=
1
r
‍
n
l
)
+
w
1
∑
l
=
1
r
‍
j
l
+
w
2
∑
l
=
1
r
‍
i
l
]
q
s
)
-
1
)
×
q
w
1
w
2
d
∑
l
=
1
r
‍
(
h
-
l
+
1
)
n
l
.Therefore, by (10) and (11), we obtain the following theorem.Theorem 1.
Forw
1
,
w
2
∈
ℕ with w
1
≡
1
(
mod
⁡
2
) and w
2
≡
1
(
mod
⁡
2
), one has
(12)
[
2
]
q
w
2
r
[
w
2
]
q
s
∑
j
1
,
…
,
j
r
=
0
w
1
d
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
l
q
w
1
,
r
(
h
)
×
(
s
,
w
2
x
+
w
2
w
1
∑
l
=
1
r
‍
j
l
∣
χ
)
=
[
2
]
q
w
1
r
[
w
1
]
q
s
∑
j
1
,
…
,
j
r
=
0
w
2
d
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
l
q
w
2
,
r
(
h
)
×
(
s
,
w
1
x
+
w
1
w
2
∑
l
=
1
r
‍
j
l
∣
χ
)
.By (8) and Theorem 1, we obtain the following theorem.Theorem 2.
Forn
∈
ℤ
≥
0 and w
1
,
w
2
∈
ℕ with w
1
≡
1
(
mod
⁡
2
) and w
2
≡
1
(
mod
⁡
2
), one has
(13)
[
2
]
q
w
2
r
[
w
1
]
q
n
∑
j
1
,
…
,
j
r
=
0
w
1
d
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
E
n
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
+
w
2
w
1
∑
l
=
1
r
‍
j
l
)
=
[
2
]
q
w
1
r
[
w
2
]
q
n
×
∑
j
1
,
…
,
j
r
=
0
w
2
d
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
E
n
,
χ
,
q
w
2
(
h
,
r
)
(
w
1
x
+
w
1
w
2
∑
l
=
1
r
‍
j
l
)
.From (6), we note that
(14)
E
n
,
χ
,
q
(
h
,
r
)
(
x
+
y
)
=
(
q
x
+
y
E
χ
,
q
(
h
,
r
)
+
[
x
+
y
]
q
)
n
=
(
q
x
+
y
E
χ
,
q
(
h
,
r
)
+
q
x
[
y
]
q
+
[
x
]
q
)
n
=
∑
i
=
0
n
‍
(
n
i
)
q
x
i
(
q
y
E
χ
,
q
(
h
,
r
)
+
[
y
]
q
)
i
[
x
]
q
n
-
i
=
∑
i
=
0
n
‍
(
n
i
)
q
x
i
E
i
,
χ
,
q
(
h
,
r
)
(
y
)
[
x
]
q
n
-
i
.
By (14), we get
(15)
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
E
n
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
+
w
2
w
1
∑
l
=
1
r
‍
j
l
)
=
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
∑
i
=
0
n
‍
(
n
i
)
q
i
w
2
(
j
1
+
·
·
·
+
j
r
)
E
i
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
)
×
[
w
2
(
j
1
+
·
·
·
+
j
r
)
w
1
]
q
w
1
n
-
i
=
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
×
∑
i
=
0
n
‍
(
n
i
)
q
(
n
-
i
)
w
2
∑
l
=
1
r
‍
j
l
E
n
-
i
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
)
[
w
2
w
1
∑
l
=
1
r
‍
j
l
]
q
w
1
i
=
∑
i
=
0
n
‍
(
n
i
)
(
[
w
2
]
q
[
w
1
]
q
)
i
E
n
-
i
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
)
×
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
n
-
i
+
1
)
j
l
×
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
[
j
1
+
·
·
·
+
j
r
]
q
w
2
i
=
∑
i
=
0
n
‍
(
n
i
)
(
[
w
2
]
q
[
w
1
]
q
)
i
E
n
-
i
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
)
S
n
,
i
,
q
w
2
(
h
,
r
)
(
w
1
d
∣
χ
)
,
where
(16)
S
n
,
i
,
q
(
h
,
r
)
(
w
∣
χ
)
=
∑
j
1
,
…
,
j
r
=
0
w
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
∑
l
=
1
r
‍
(
h
-
l
+
n
-
i
+
1
)
j
l
×
[
j
1
+
·
·
·
+
j
r
]
q
i
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
.
From (15), we have
(17)
[
2
]
q
w
2
r
[
w
1
]
q
n
×
∑
j
1
,
…
,
j
r
=
0
d
w
1
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
w
2
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
×
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
E
n
,
χ
,
q
w
1
(
h
,
r
)
×
(
w
2
x
+
w
2
w
1
(
j
1
+
·
·
·
+
j
r
)
)
=
[
2
]
q
w
2
r
∑
i
=
0
n
‍
(
n
i
)
[
w
1
]
q
n
-
i
[
w
2
]
q
i
E
n
-
i
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
)
S
n
,
i
,
q
w
2
(
h
,
r
)
×
(
w
1
d
∣
χ
)
.
By using the same method as in (17), we get
(18)
[
2
]
q
w
1
r
[
w
2
]
q
n
∑
j
1
,
…
,
j
r
=
0
d
w
2
-
1
‍
(
-
1
)
∑
l
=
1
r
‍
j
l
q
w
1
∑
l
=
1
r
‍
(
h
-
l
+
1
)
j
l
×
(
∏
l
=
1
r
‍
χ
(
j
l
)
)
E
n
,
χ
,
q
w
2
(
h
,
r
)
(
w
1
x
+
w
1
w
2
∑
l
=
1
r
‍
j
l
)
=
[
2
]
q
w
1
r
∑
i
=
0
n
‍
(
n
i
)
[
w
2
]
q
n
-
i
[
w
1
]
q
i
E
n
-
i
,
χ
,
q
w
2
(
h
,
r
)
×
(
w
1
x
)
S
n
,
i
,
q
w
1
(
h
,
r
)
(
w
2
d
∣
χ
)
.
Therefore, by (17) and (18), we obtain the following theorem.Theorem 3.
Forn
∈
ℤ
≥
0 and w
1
,
w
2
∈
ℕ, with w
1
≡
1
(
mod
⁡
2
) and w
2
≡
1
(
mod
⁡
2
), one has
(19)
[
2
]
q
w
2
r
∑
i
=
0
n
‍
(
n
i
)
[
w
1
]
q
n
-
i
[
w
2
]
q
i
E
n
-
i
,
χ
,
q
w
1
(
h
,
r
)
(
w
2
x
)
S
n
,
i
,
q
w
2
(
h
,
r
)
(
w
1
d
∣
χ
)
=
[
2
]
q
w
1
r
∑
i
=
0
n
‍
(
n
i
)
[
w
2
]
q
n
-
i
[
w
1
]
q
i
E
n
-
i
,
χ
,
q
w
2
(
h
,
r
)
(
w
1
x
)
S
n
,
i
,
q
w
1
(
h
,
r
)
×
(
w
2
d
∣
χ
)
.Now, we observe that(20)
e
[
x
]
q
u
∑
m
1
,
…
,
m
r
=
0
∞
‍
q
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
(
-
1
)
∑
l
=
1
r
‍
m
l
×
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
e
[
y
+
∑
l
=
1
r
‍
m
l
]
q
q
x
(
u
+
v
)
=
e
-
[
x
]
q
u
∑
m
1
,
…
,
m
r
=
0
∞
‍
q
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
(
-
1
)
∑
l
=
1
r
‍
m
l
×
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
e
[
x
+
y
+
∑
l
=
1
r
‍
m
l
]
q
(
u
+
v
)
.
The left hand side of (20) multiplied by [
2
]
q
r is given by
(21)
[
2
]
q
r
e
[
x
]
q
u
×
∑
m
1
,
…
,
m
r
=
0
∞
‍
q
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
(
-
1
)
∑
l
=
1
r
‍
m
l
e
[
y
+
∑
l
=
1
r
‍
m
l
]
q
q
x
(
u
+
v
)
×
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
=
e
[
x
]
q
u
∑
n
=
0
∞
‍
q
n
x
E
n
,
χ
,
q
(
h
,
r
)
(
y
)
(
u
+
v
)
n
n
!
=
(
∑
l
=
0
∞
‍
[
x
]
q
l
u
l
l
!
)
×
(
∑
k
=
0
∞
‍
∑
n
=
0
∞
‍
q
(
k
+
n
)
x
E
k
+
n
,
χ
,
q
(
h
,
r
)
(
y
)
u
k
k
!
v
n
n
!
)
=
∑
m
=
0
∞
‍
∑
n
=
0
∞
‍
(
∑
k
=
0
m
‍
(
m
k
)
q
(
k
+
n
)
x
E
k
+
n
,
χ
,
q
(
h
,
r
)
(
y
)
[
x
]
q
m
-
k
)
×
u
m
m
!
v
n
n
!
.
The right hand side of (20) multiplied by [
2
]
q
r is given by
(22)
[
2
]
q
r
e
-
[
x
]
q
v
∑
m
1
,
…
,
m
r
=
0
∞
‍
(
-
1
)
∑
l
=
1
r
‍
m
l
q
∑
l
=
1
r
‍
(
h
-
l
+
1
)
m
l
×
(
∏
l
=
1
r
‍
χ
(
m
l
)
)
e
[
x
+
∑
l
=
1
r
‍
m
l
]
q
(
u
+
v
)
=
e
-
[
x
]
q
v
∑
n
=
0
∞
‍
E
n
,
χ
,
q
(
h
,
r
)
(
x
+
y
)
(
u
+
v
)
n
n
!
=
(
∑
l
=
0
∞
‍
(
-
[
x
]
q
)
l
l
!
v
l
)
×
(
∑
m
=
0
∞
‍
∑
k
=
0
∞
‍
E
m
+
k
,
χ
,
q
(
h
,
r
)
(
x
+
y
)
u
m
m
!
v
k
k
!
)
=
∑
n
=
0
∞
‍
∑
m
=
0
∞
‍
(
∑
k
=
0
n
‍
(
n
k
)
E
m
+
k
,
χ
,
q
(
h
,
r
)
(
x
+
y
)
(
-
[
x
]
q
)
n
-
k
)
×
u
m
m
!
v
n
n
!
=
∑
n
=
0
∞
‍
∑
m
=
0
∞
‍
(
∑
k
=
0
n
‍
(
n
k
)
E
m
+
k
,
χ
,
q
(
h
,
r
)
(
x
+
y
)
q
(
n
-
k
)
x
[
-
x
]
q
n
-
k
)
×
u
m
m
!
v
n
n
!
.
Therefore, by (21) and (22), we obtain the following theorem.Theorem 4.
Form, n
∈
ℤ
≥
0, one has
(23)
∑
k
=
0
m
‍
(
m
k
)
q
k
x
E
n
+
k
,
χ
,
q
(
h
,
r
)
(
y
)
[
x
]
q
m
-
k
=
∑
k
=
0
n
‍
(
n
k
)
q
-
k
x
E
m
+
k
,
χ
,
q
(
h
,
r
)
(
x
+
y
)
[
-
x
]
q
n
-
k
.



---

*Source: 286239-2014-02-25.xml*