Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 801580, 13 pages
doi:10.1155/2010/801580
Research Article
A Note on Symmetric Properties of
the Twisted q-Bernoulli Polynomials and
the Twisted Generalized q-Bernoulli Polynomials
L C. Jang,
1
H. Yi,
2
K. Shivashankara,
3
T. Kim,
4
Y. H. Kim,
4
and B. Lee
5
1
Department of Mathematics and Computer Science, KonKuk University,
Chungju 138-70, Republic of Korea
2
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore 570# 005, India
4
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
5
Department of Wireless Communications Engineering, Kwangwoon University,
p
−v
p
p
p
−1
. When one talks of q-extension, q is variously considered as an indeterminate,
a complex number q ∈ C, or a p-adic number q ∈ C
p
.Ifq ∈ C, one normally assumes
|q| < 1. If q ∈ C
p
, then we assume |q − 1|
p
<p
−1/p−1
, so that q
x
expx log q for |x|
p
≤ 1
cf. 1–32.
2 Advances in Difference Equations
For N, d ∈ N,weset
X X
d
lim
←
N
t
e
t
− 1
e
xt
∞
n0
B
n
x
t
n
n!
1.3
cf. 17, 18, 21, 24, 26. Let UDX be the set of uniformly differentiable functions on X. For
f ∈ UDX,thep-adic invariant integral on Z
p
is defined as
I
f
X
n
xfx n.
We note that
I
f
n
I
f
n−1
i0
f
i
1.5
cf. 1–32.Kim18 studied the symmetric properties of the q-Bernoulli numbers and
polynomials as follows:
t log q
qe
t
− 1
e
xt
p
n
lim
n →∞
C
p
n
be the locally constant space, where C
p
n
{ξ | ξ
p
n
1} is the
cyclic group of order p
n
. For w ∈ C
p
∞
, we denote the locally constant function by
φ
w
: Z
p
−→ C
p
,x−→ w
x
2.1
cf. 2, 3, 21, 24. If we take fxφ
n0
B
q
n,w
t
n
n!
, 2.3
log q t
wqe
t
− 1
e
tx
∞
n0
B
q
n,w
x
t
n
n!
2.4
see 31.From1.5, 2.2, 2.3,and2.4, we can derive
.
2.5
By 1.5, we can see that
1
log q t
Z
p
w
nx
q
nx
e
nxt
dx −
Z
p
w
x
q
x
e
xt
dx
w
n
i0
w
i
q
i
e
it
∞
k0
n−1
i0
i
k
w
i
q
i
t
k
k!
.
2.6
4 Advances in Difference Equations
In 1.4,itiseasytoshowthat
1
q
x
e
xt
dx
Z
p
w
nx
q
nx
e
nxt
dx
. 2.7
For each integer k ≥ 0, let
S
q
k,w
n
0
k
1
k
wq 2
k
w
q
x
e
xt
dx
n
Z
p
w
x
q
x
e
xt
dx
Z
p
w
nx
q
nx
e
nxt
dx
∞
q
k,w
n − 1
,
2.10
for all k, n ∈ N.Letu
1
,u
2
∈ N and w ∈ C
p
∞
; then we have
Z
p
w
u
1
x
1
u
2
x
2
q
u
1
1
u
2
x
e
u
1
u
2
xt
dx
t log q
w
u
1
u
2
q
u
1
u
2
e
u
1
t
− 1
1
x
q
u
1
x
e
u
1
xt
dx
∞
l0
u
1
−1
k0
k
l
w
k
q
k
t
l
2
; x, t
Z
p
w
u
1
x
1
u
2
x
2
q
u
1
x
1
u
2
x
2
e
u
1
x
1
1
u
2
xt
dx
.
2.13
Then we have
T
w
u
1
,u
2
; x, t
t log q
e
u
1
u
2
t
w
u
u
2
q
u
2
e
u
2
t
− 1
. 2.14
Advances in Difference Equations 5
From 2.13, we derive
T
w
u
1
,u
2
; x, t
1
u
1
Z
u
2
x
2
q
u
2
x
2
e
u
2
x
2
t
Z
p
w
u
1
u
2
x
q
u
1
u
2
x
u
1
i,w
u
1
u
2
x
u
i
1
t
i
i!
∞
l0
S
q
u
2
l,w
u
2
u
1
2
x
S
q
u
2
n−i,w
u
2
u
1
− 1
u
i−1
1
u
n−i
2
t
n
n!
.
2.16
By the symmetry of p-adic invariant integral on Z
p
,wealsoseethat
x
2
u
1
xt
dx
2
⎛
⎝
u
2
Z
p
w
u
1
x
1
q
u
1
x
1
e
u
1
x
1
n
i0
n
i
B
q
u
2
i,w
u
2
u
1
x
S
q
u
1
n−i,w
u
1
u
2
− 1
B
q
u
1
i,w
u
1
u
2
x
S
q
u
2
n−i,w
u
2
u
1
− 1
u
i−1
1
u
n−i
2
− 1
u
i−1
2
u
n−i
1
,
2.18
where
n
i
is the binomial coefficient.
From Theorem 2.1, if we take u
2
1, then we have the following corollary.
Corollary 2.2. For m ≥ 0, one we has
B
q
i,w
u
1
x
where
n
i
is the binomial coefficient.
6 Advances in Difference Equations
By 2.17, 2.18,and2.19, we can see that
T
w
u
1
,u
2
; x, t
e
u
1
u
2
xt
u
1
Z
p
q
u
2
x
2
e
u
2
x
2
t
dx
2
Z
p
w
u
1
u
2
x
q
u
1
u
2
x
e
u
u
1
x
1
t
dx
1
u
1
−1
i0
w
u
2
i
q
u
2
i
e
u
2
it
1
u
1
2
/u
1
itu
1
dx
1
∞
n0
u
1
−1
i0
B
q
u
1
n,w
u
1
u
2
x
u
2
u
1
,u
2
; x, t
∞
n0
u
2
−1
i0
B
q
u
2
n,w
u
2
u
1
x
u
1
u
2
i
−1
i0
B
q
u
1
n,w
u
1
u
2
x
u
2
u
1
i
u
n−1
1
w
u
2
i
q
u
2
i
q
u
1
i
.
2.22
We note that by setting u
2
1inTheorem 2.3, we get the following multiplication
theorem for the twisted q-Bernoulli polynomials.
Theorem 2.4. For m ∈ Z
, u
1
∈ N, one has
B
q
n,w
u
1
x
u
n−1
1
u
1
−1
p
.Letχ be Dirichlet’s character with conductor
d ∈ N. Then the generalized Bernoulli numbers B
n,χ
and polynomials B
n,χ
x attached to χ
are defined as
t
d−1
a0
χ
a
e
at
e
dt
− 1
∞
n0
B
n,χ
t
n
n!
cf. 2, 18, 23, 27.
Let C
p
∞
n≥1
C
p
n
lim
n →∞
C
p
n
be the locally constant space, where C
p
n
{w | w
p
n
1} is the cyclic group of order p
n
. For w ∈ C
p
∞
, we denote the locally constant function by
φ
w
x
dx
t log q
d−1
a0
χ
a
w
a
q
a
e
at
w
d
q
d
e
dt
− 1
.
3.4
Now we define the twisted generalized Bernoulli numbers B
q
n,χ,w
n0
B
q
n,χ,w
t
n
n!
, 3.5
t log q
d−1
a0
χ
a
w
a
q
a
e
at
e
xt
w
d
q
w
x
q
x
dx B
q
n,χ,w
,
X
χ
y
x y
n
w
y
q
y
dy B
q
n,χ,w
x
.
3.7
8 Advances in Difference Equations
nd
X
χ
x
e
xt
w
x
q
x
dx
X
e
ndxt
w
ndx
q
ndx
dx
w
nd
q
nd
X
χ
x
e
ndxt
w
nx
q
nx
dx −
X
χ
x
e
xt
w
x
q
x
dx
nd−1
l0
.
3.9
Let us define the p-adic twisted q-function T
q
k,w
χ, n as follows:
T
q
k,w
χ, n
n
l0
χ
l
l
k
w
l
q
l
.
3.10
By 3.9 and 3.10,weseethat
1
∞
k0
T
q
k,w
χ, nd − 1
t
k
k!
.
3.11
Thus,
X
χ
x
nd x
k
w
nx
q
nx
dx −
nd
− B
q
n,χ,w
t log q
T
q
k,w
χ, nd − 1
,
3.13
Advances in Difference Equations 9
for all k, n,d ∈ N. For all u
1
,u
2
,d ∈ N, we have
d
X
X
χ
1
x
1
u
2
x
2
dx
1
dx
2
X
e
du
1
u
2
xt
w
du
1
u
2
x
q
du
1
u
2
du
1
q
du
1
− 1
e
du
2
t
w
du
2
q
du
2
− 1
×
d−1
a0
χ
a
e
u
.
3.14
The twisted generalized Bernoulli numbers B
k,q
n,χ,w
and polynomials B
k,q
n,χ,w
x attached
to χ of order k are defined as
t log q
d−1
a0
χ
a
w
a
q
a
e
at
w
d
w
a
q
a
e
at
w
d
q
d
e
dt
− 1
k
e
xt
∞
n0
B
k,q
n,χ,w
x
t
n
n!
e
m
i1
x
i
u
2
xu
1
t
w
m
i1
x
i
u
2
xu
1
q
m
i1
x
×
X
m
m
i1
χ
x
i
e
m
i1
x
i
u
1
yu
2
t
w
m
i1
x
dx
1
···dx
m
X
···
X
fx
1
, ,x
m
dx
1
···dx
m
.In3.17,wenotethat
K
q
w
m, χ; u
1
,u
2
is symmetric in u
1
,u
2
2
t
w
m
i1
x
i
u
2
q
m
i1
x
i
u
2
dx
1
···dx
m
× e
u
1
u
2
xt
x
m
q
u
2
x
m
dx
m
X
e
du
1
u
2
x
q
du
1
u
2
x
dx
×
X
m−1
m−1
i1
x
i
u
2
dx
1
···dx
m−1
× e
u
1
u
2
yt
w
u
1
u
2
y
q
u
1
u
2
y
.
3.18
10 Advances in Difference Equations
x
dx
∞
k0
u
1
d−1
i0
χ
i
i
k
w
i
q
i
t
k
k!
∞
k0
X
m
m
i1
χ
x
i
e
m
i1
x
i
u
1
t
w
m
i1
x
i
u
1
x
u
1
e
du
1
t
w
du
1
q
du
1
− 1
d−1
a0
χ
a
e
u
1
at
w
u
1
a
m, χ; u
1
,u
2
∞
l0
B
m,q
l,χ,w
u
1
x
u
l
1
t
l
l!
∞
k0
T
q
k,w
∞
n0
n
j0
n
j
u
j
2
u
n−j−1
1
B
m,q
n−j,χ,w
u
2
x
×
j
k0
T
q
in u
1
and u
2
, we can see that
K
q
w
m, χ; u
1
,u
2
∞
n0
n
j0
n
j
u
j
1
u
n−j−1
2
y
t
n
n!
.
3.21
By comparing the coefficients on both sides of 3.20 and 3.21, we see the following theorem.
Theorem 3.1. For d, u
1
,u
2
,m∈ N, n ∈ Z, one has
n
j0
n
j
u
j
2
u
n−j−1
1
B
m,q
n−j,χ,w
j0
n
j
u
j
1
u
n−j−1
2
B
m,q
n−j,χ,w
u
1
x
j
k0
T
q
k,w
χ, u
2
d − 1
B
q
n−j,χ,w
u
2
x
j
k0
T
q
k,w
χ, u
1
d − 1
j
k
n
j0
n
j
.
3.23
Now we can also calculate
K
q
w
m, χ; u
1
,u
2
∞
n0
n
k0
n
k
u
k−1
1
u
n−k
2
.
3.24
From the symmetric property of K
q
w
m, χ; u
1
,u
2
in u
1
and u
2
, we derive
K
q
w
m, χ; u
1
,u
2
∞
n0
n
x
u
1
u
2
i
t
n
n!
.
3.25
By comparing the coefficients on both sides of 3.24 and 3.26, we obtain the following
theorem.
Theorem 3.3. For d, u
1
,u
2
,m∈ N, n ∈ Z, we have
n
k0
n
k
u
k−1
1
n
k0
n
k
u
k−1
2
u
n−k
1
B
m−1,q
n−k,χ,w
u
2
y
du
2
−1
i0
B
m,q
k,χ,w
1
i
u
n−1
2
du
2
−1
i0
B
q
n,χ,w
u
1
x
u
1
u
2
i
.
3.27
Remark 3.5. In our results for q 1, we can also derive similar results, which were treated
in 27. In this paper, we used the p-adic integrals to derive the symmetric properties of
the q-Bernoulli polynomials. By using the symmetric properties of p-adic integral on X,we
can easily derive many interesting symmetric properties related to Bernoulli numbers and
Contemporary Mathematics, vol. 15, pp. 133–138, 2007.
12 T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of
Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598–608, 2008.
13 T. Kim, “On the analogs of Euler numbers and polynomials associated with p-adic q-integral on
Z
p
at
q −1,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779–792, 2007.
14 T. Kim, “A note on p-adic q-integral on
Z
p
associated with q-Euler numbers,” Advanced Studies in
Contemporary Mathematics, vol. 15, no. 2, pp. 133–137, 2007.
15 T. Kim, “q-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear
Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.
16 T. Kim, “A note on some formulae for the q-Euler numbers and polynomials,” Proc. Jangjeon Math.
Soc., vol. 9, no. 2, pp. 227–232, 2006.
17 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
18 T. Kim, “On the symmetries of the q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008,
Article ID 914367, 7 pages, 2008.
19 T. Kim, “Note on Dedekind t ype DC sums,” Advanced Studies in Contemporary Mathematics, vol. 18,
no. 2, pp. 249–260, 2009.
20 T. Kim, L C. Jang, and H. K. Pak, “A note on q-Euler and Genocchi numbers,” Proceedings of the Japan
Academy, Series A, vol. 77, no. 8, pp. 139–141, 2001.
21 T. Kim, “Note on the q-Euler numbers of higher order,” Advanced Studies in Contemporary Mathematics,
vol. 19, no. 1, pp. 25–29, 2009.
22 T. Kim, M S. Kim, L C. Jang, and S H. Rim, “New q-Euler numbers and polynomials associated with
p-adic q-integrals,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 243–252, 2007.
23 W. Kim, Y H. Kim, and L C. Jang, “On the q-extension of apostol-euler numbers and polynomials,”
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