Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 875098, 11 pages
doi:10.1155/2010/875098
Research Article
On the Twisted q-Analogs of the Generalized Euler
Numbers and Polynomials of H igher Order
Lee Chae Jang,
1
Byungje Lee,
2
and Taekyun Kim
3
1
Department of Mathematics and Computer Science, KonKuk University,
Chungju 138-701, Republic of Korea
2
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
3
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to Lee Chae Jang, [email protected]
Received 12 April 2010; Accepted 28 June 2010
Academic Editor: Istvan Gyori
Copyright q 2010 Lee Chae J ang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the twisted q-extensions of the generalized Euler numbers and polynomials attached
to χ.
1. Introduction and Preliminaries
Let p be an odd prime number. For n ∈ Z
considered as an indeterminate, a complex number q ∈ C,orp-adic number q ∈ C
p
.Ifq ∈ C,
one normally assumes that |q| < 1. If q ∈ C
p
, one normally assumes that |1 − q|
p
< 1. In this
paper, we use the notation
x
q
1 − q
x
1 − q
,
x
−q
1 −
−q
x
1 q
.
n
Z
p
x ∈ X | x ≡ a
mod dp
n
,
1.2
where a ∈ Z lies in 0 ≤ a<dp
n
; compared to 1–16.
Let χ be the Dirichlet’s character with an odd conductor d ∈ N. Then the generalized
ζ-Euler polynomials attached to χ, E
n,χ,ζ
x, are defined as
F
χ,ζ
x, t
2
d−1
l0
, for ζ ∈ T
p
.
1.3
In the special case x 0, E
n,χ,ζ
E
n,χ,ζ
0 are called the nth ζ-Euler numbers attached to χ.
For f ∈ UDZ
p
,thep-adic fermionic integral on Z
p
is defined by
I
−q
f
Z
p
f
x
dμ
−q
x
−1
x
q
x
p
N
−q
, see 7–17.
1.4
Let I
−1
lim
q → 1
I
−q
f. Then, we see that
Z
p
f
x
dμ
−1
x
−1
n
Z
p
f
x
dμ
−1
x
2
n−1
l0
−1
n−1−l
f
l
l
, see 7–17.
1.7
Advances in Difference Equations 3
By 1.7,weseethat
X
χ
y
ζ
y
e
xyt
dμ
−1
y
2
d−1
l0
−1
l
n,χ,ζ
x as follows:
X
χ
x
x
n
ζ
x
dμ
−1
x
E
n,χ,ζ
,
X
χ
y
y x
n
ζ
χle
lt
ζ
d
e
dt
1
e
xt
k
∞
n0
E
k
n,χ,ζ
x
t
n
n!
.
1.10
In the special case x 0, E
k
n,χ,ζ
E
k
dμ
−1
x
1
···dμ
−1
x
k
2
d−1
l0
−1
l
χle
lt
ζ
d
e
dt
1
i1
χ
x
i
x
1
··· x
k
x
n
ζ
x
1
···x
k
dμ
−1
x
1
···dμ
−1
E
n,χ,ζ,q
x
t
n
n!
X
e
xy
q
t
ζ
y
χ
y
dμ
−1
y
2
∞
m0
−1
y
2
∞
m0
χ
m
−ζ
m
e
mx
q
t
2
d−1
a0
χ
a
−1
a
, we can derive the twisted
q-extension of the generalized Euler polynomials of order k attached to χ as follows:
∞
n0
E
k
n,χ,ζ,q
x
t
n
n!
X
···
X
k
i1
χ
x
i
n,χ,ζ,q
x
X
···
X
k
i1
χ
x
i
x
1
··· x
k
x
n
q
ζ
a
i
−ζ
k
j1
a
j
2
k
1 − q
n
n
l0
n
l
−1
l
q
lx
−ζ
k
j1
a
j
∞
m0
m k − 1
m
×
−ζ
d
m
x a
1
··· a
k
md
n
a
1
, ,a
k
0
k
i1
χ
a
i
−ζ
k
j1
a
j
∞
m0
m k − 1
m
k
i1
χ
n
i
e
n
1
···n
k
x
q
t
.
2.5
Therefore, we obtain the following theorem.
Theorem 2.1. For k ∈ N,n ≥ 0, one has
E
k
n,χ,ζ,q
x
2
k
n
q
d−1
a
1
, ,a
r
0
k
i1
χ
a
i
−ζ
k
j1
a
j
2
.
2.6
Let h ∈ Z,r ∈ N. Then we define the extension of E
r
n,χ,ζ,q
x as follows:
∞
n0
E
h,r
n,χ,ζ,q
x
t
n
n!
X
···
X
q
r
j1
h−jx
j
···dμ
−1
x
r
.
2.7
Then, E
r
n,χ,ζ,q
x are called the nth generalized h, q-Euler polynomials of order r attached
to χ. In the special case x 0, E
r
n,χ,ζ,q
E
r
n,χ,ζ,q
0 are called the nth generalized h, r-Euler
numbers of order r.By1.7, we obtain the Witt’s formula f or E
r
n,χ,ζ,q
x as follows:
E
h,r
n,χ,ζ,q
x
⎤
⎦
n
q
ζ
x
1
···x
r
dμ
−1
x
1
···dμ
−1
x
r
d−1
a
1
, ,a
r
0
n
n
l0
n
l
−1
l
q
lx
r
j1
a
j
−q
dh−rl
ζ
d
; q
d
r
,
2.8
q
k − 1
q
···2
q
1
q
.From2.8,wenotethat
E
h,r
n,χ,ζ,q
x
2
r
1 − q
n
d−1
a
1
, ,a
r
0
r
−1
l
q
lx
r
j1
a
j
∞
m0
m r − 1
m
q
d
−ζ
d
m
q
dh−rm
q
ldm
j1
h−ja
j
×
∞
m0
m r − 1
m
q
d
−ζ
d
m
q
dh−rm
1
1 − q
n
1 − q
dmx
k
a
1
, ,a
r
0
r−1
i1
χ
a
i
×
−ζ
r
j1
a
j
q
r
j1
h−ja
/n! be the generating function for E
h,r
n,χ,ζ,q
x.From
2.8, we can easily derive
F
h,r
q,χ,ζ
t, x
2
r
∞
n
1
, ,n
r
0
q
r
j1
h−jn
j
−ζ
m r − 1
m
q
−ζ
d
m
q
dh−rm
d−1
a
1
, ,a
r
0
r−1
i1
χ
a
i
×
n,χ,ζ,q
x
2
r
∞
n
1
, ,n
r
0
q
r
j1
h−jn
j
−ζ
r
j1
n
j
⎛
⎝
r
m
q
−ζ
d
m
q
dh−rm
d−1
a
1
, ,a
r
0
r−1
i1
χ
a
i
×
−ζ
···a
r
0
⎛
⎝
r
j1
χ
n
j
⎞
⎠
−ζ
r
j1
a
j
q
r
j1
h−ja
j
×
d
r
.
2.11
Let h r. Then we see that
E
r,r
n,χ,ζ,q
x
2
r
1 − q
n
d−1
a
1
, ,a
r
0
r−1
i1
l
q
l
r
j1
a
j
x
−q
ld
ζ
d
; q
d
r
2
r
d
n
q
∞
m0
m r − 1
j1
a
j
q
r
j1
r−ja
j
m
x
r
j1
a
j
d
n
q
d
.
2.12
It is easy to see that
X
···
X
x
r
d−1
a
1
, ,a
r
0
r
i1
χ
a
i
q
mx
r
j1
h−ja
j
2
r
q
mx
d−1
a
1
, ,a
r
0
r
j1
χ
a
j
q
r
j1
m−ja
j
0
r
j1
χ
a
j
q
r
j1
m−ja
j
−ζ
r
j1
a
j
−q
dm−r
ζ
d
j
ζ
x
1
···x
r
×
⎛
⎝
r
j1
χ
x
j
⎞
⎠
dμ
−1
x
1
···dμ
−1
x
r
×
x x
1
··· x
r
l
q
q
−
r
j1
jx
j
ζ
x
1
···x
r
dμ
−1
x
1
···dμ
−1
d−1
a
1
, ,a
r
0
r
j1
χ
a
j
q
r
j1
m−ja
j
−ζ
r
j1
a
j
X
f
x d
dμ
−1
x
X
f
x
dμ
−1
x
2
d−1
l0
−1
r
j1
x
j
×
⎛
⎝
r
j1
χ
x
j
⎞
⎠
dμ
−1
x
1
···dμ
−1
x
r
j1
χ
x
j
⎞
⎠
dμ
−1
x
1
···dμ
−1
x
r
2
d−1
l0
χ
l
−ζ
j1
x
j1
h−1−j
ζ
x
2
x
3
···x
r
dμ
−1
x
2
···dμ
−1
x
r
.
2.17
By 2.17, we obtain the following theorem.
Advances in Difference Equations 9
Theorem 2.4. For h ∈ Z,d ∈ N with 2 d, one has
q
dh−1
2.18
It is easy to see that
q
x
E
h1,r
n,χ,ζ,q
x
q − 1
E
h,r
n1,χ,ζ,q
E
h,r
n,χ,ζ,q
x
.
2.19
Let F
h,1
q,χ,ζ
t, x
q
t
.
2.20
From 2.20, we can derive
E
h,1
n,χ,ζ,q
x
2
∞
m0
χ
m
q
h−1m
−ζ
m
m x
n
q
ld
ζ
d
.
2.21
3. Further Remark
In this section, we assume that q ∈ C with |q| < 1. Let χ be the Dirichlet’s character with an
odd conductor d ∈ N. From the Mellin transformation of F
h,r
q,χ,ζ
t, x in 2.10,wenotethat
1
Γ
s
F
h,r
q,χ,ζ
−t, x
t
s−1
dt 2
r
∞
1
··· m
r
x
s
q
,
3.1
where h, s ∈ C, x
/
0, −1, −2, ,and r ∈ N, ζ e
2πi/d
.By3.1, we can define the Dirichlet’s
type multiple h, q-l-function as follows.
Definition 3.1. For s ∈ C, x ∈ R with x
/
0, −1, −2, , one defines the Dirichlet’s type multiple
h, q-l-function related to higher order h, q-Euler polynomials as
l
h,r
q
s, x | χ
2
r
∞
m
··· m
r
x
s
q
, 3.2
where s, h ∈ C, x
/
0, −1, −2, ···, r ∈ N,andζ e
2πi/d
.
10 Advances in Difference Equations
Note that l
h,r
q
s, x | χ is analytic continuation in whole complex s-plane. In 2.10,we
note that
F
h,r
q,χ,ζ
t, x
2
r
∞
n
1
1
···n
r
x
q
t
∞
n0
E
h,r
n,χ,ζ,q
x
t
n
n!
.
3.3
By Laurent series and Cauchy residue theorem in 3.1 and 3.3,weobtainthe
following theorem.
Theorem 3.2. Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ e
2πi/d
. For
h, s ∈ C, x
/
0, −1, −2, ,r ∈ N, and n ∈ Z
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