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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 579509, 6 pages
doi:10.1155/2010/579509
Research Article
Multivariate Twisted p-Adic q-Integral on
Z
p
Associated with Twisted q-Bernoulli Polynomials
and Numbers
Seog-Hoon Rim, Eun-Jung Moon, Sun-Jung Lee,
and Jeong-Hee Jin
Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
Correspondence should be addressed to Seog-Hoon Rim,
Received 19 June 2010; Accepted 2 October 2010
Academic Editor: Ulrich Abel
Copyright q 2010 Seog-Hoon Rim 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.
Recently, many authors have studied twisted q-Bernoulli polynomials by using the p-adic invariant
q-integral on
Z
p
. In this paper, we define the twisted p-adic q-integral on Z
p
and extend our result
to the twisted q-Bernoulli polynomials and numbers. Finally, we derive some various identities
related to the twisted q-Bernoulli polynomials.
1. Introduction
Let p be a fixed prime number. Throughout this paper, the symbols Z, Z
p
, Q
p
, C, and C
p
will
denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational
numbers, the complex number field, and the completion of the algebraic closure of Q
p
,
respectively. Let N be the set of natural numbers and Z
N ∪{0}. Let v
p
be the normalized
exponential valuation of C
p
with |p|
p
p
−v
p
p
1/p.
When one talks of q-extension, q is variously considered as an indeterminate, a
complex q ∈ C, or p-adic number q ∈ C
p
. If q ∈ C, one normally assumes that |q| < 1. If
q ∈ C
p
, then we assume that |q − 1|
p
< 1.
For n ∈ N, let T
p
be the p-adic locally constant space defined by
T
p
n≥1
C
p
n
lim
n →∞
C
p
n
C
p
∞
, 1.1
where C
p
n
{ζ ∈ C
p
| ζ
p
n
1 for some n ≥ 0} is the cyclic group of order p
n
.
2 Journal of Inequalities and Applications
Let UDZ
p
be the space of uniformly differentiable function on Z
p
.
For f ∈ UDZ
p
, the p-adic invariant q-integral on Z
p
is defined as
I
q
f
Z
p
f
x
dμ
q
x
lim
N →∞
1
p
N
q
p
N
−1
x0
f
x
q
x
, 1.2
compare with 1–3.
It is well known that the twisted q-Bernoulli polynomials of order k are defined as
e
xt
t
e
t
ζq − 1
k
∞
n0
β
k
n,ζ,q
x
t
n
n!
,ζ∈ T
p
, 1.3
and β
k
n,ζ,q
β
k
n,ζ,q
0 are called the twisted q-Bernoulli numbers of order k. When k 1,
the polynomials and numbers are called the twisted q-Bernoulli polynomials and numbers,
respectively. When k 1andq 1, the polynomials and numbers are called the twisted
Bernoulli polynomials and numbers, respectively. When k 1,q 1, and ζ 1, the
polynomials and numbers are called the ordinary Bernoulli polynomials and numbers,
respectively.
Many authors have studied the twisted q-Bernoulli polynomials by using the
properties of the p-adic invariant q-integral on Z
p
cf. 4. In this paper, we define the
twisted p-adic q-integral on Z
p
and extend our result to the twisted q-Bernoulli polynomials
and numbers. Finally, we derive some various identities related to the twisted q-Bernoulli
polynomials.
2. Multivariate Twisted p-Adic q-Integral on Z
p
Associated with
Tw is t ed q-Bernoulli Polynomials
In this section, we assume that q ∈ C
p
with |q−1|
p
< 1. For ζ ∈ T
p
, we define the q, ζ-numbers
as
k
q,ζ
1 − q
k
ζ
1 − q
, for k ∈ Z
p
. 2.1
Note that k
q
k
q,1
1 − q
k
/1 − q.
Let us define
n
k
q,ζ
n
q,ζ
!
k
q,ζ
!
n − k
q,ζ
!
, 2.2
where k
q,ζ
! k
q,ζ
k − 1
q,ζ
···1
q,ζ
. Note that
n
k
n
k
1,1
n!/k!n − k!.
Journal of Inequalities and Applications 3
Now we construct the twisted p-adic q-integral on Z
p
as follows:
I
q,ζ
f
Z
p
f
x
dμ
q,ζ
x
lim
N →∞
p
N
−1
x0
f
x
μ
q,ζ
x p
N
Z
p
lim
N →∞
1
p
N
q
p
N
−1
x0
f
x
q
x
ζ
x
,
2.3
where μ
q,ζ
x p
N
Z
p
q
x
ζ
x
/p
N
q
. From the definition of the twisted p-adic q-integral on
Z
p
, we can consider the twisted q-Bernoulli polynomials and numbers of order k as follows:
β
k
n,q,ζ
x
Z
k
p
x
1
x
2
··· x
k
x
n
q
dμ
q,ζ
x
1
dμ
q,ζ
x
2
···dμ
q,ζ
x
k
lim
N →∞
1
p
N
k
q
p
N
−1
x
1
, ,x
k
0
x
1
x
2
··· x
k
x
n
q
q
x
1
x
2
···x
k
ζ
x
1
x
2
···x
k
1
1 − q
n
n
l0
n
l
−1
l
q
lx
lim
N →∞
1
p
N
k
q
p
N
−1
x
1
, ,x
k
0
q
l1x
1
···l1x
k
ζ
x
1
···x
k
1
1 − q
n
n
l0
n
l
−1
l
q
lx
l 1
k
l 1
k
q,ζ
.
2.4
In the special case x 0in2.4, β
k
n,q,ζ
0β
k
n,q,ζ
are called the twisted q-Bernoulli
numbers of order k.
If we take k 1andζ 1in2.4, we can easily see that
β
n,q
x
1
1 − q
n
n
l0
n
l
−1
l
q
lx
l 1
l 1
q
. 2.5
compare with 4.
Theorem 2.1. For k ∈ Z
and ζ ∈ T
p
, we have
β
k
n,q,ζ
x
1
1 − q
n
n
l0
n
l
−1
l
q
lx
l 1
k
l 1
k
q,ζ
. 2.6
4 Journal of Inequalities and Applications
Moreover, if we take x 0 in Theorem 2.1, then we have the following identity for the
twisted q-Bernoull numbers
β
k
n,q,ζ
1
1 − q
n
n
l0
n
l
−1
l
l 1
k
l 1
k
q,ζ
. 2.7
From the definition of multivariate twisted p-adic q-integral, we also see that
β
k
n,q,ζ
x
Z
k
p
x
1
x
2
··· x
k
x
n
q
dμ
q,ζ
x
1
dμ
q,ζ
x
2
···dμ
q,ζ
x
k
n
l0
n
l
q
lx
x
n−l
q
Z
k
p
x
1
x
2
··· x
k
l
q
dμ
q,ζ
x
1
dμ
q,ζ
x
2
···dμ
q,ζ
x
k
n
l0
n
l
q
lx
x
n−l
q
β
k
l,q,ζ
.
2.8
Corollary 2.2. For k ∈ Z
and ζ ∈ T
p
, one obtains
β
k
n,q,ζ
x
n
l0
n
l
q
lx
x
n−l
q
β
k
l,q,ζ
. 2.9
Note that
q
nx
1
···x
k
n
l0
n
l
q − 1
l
x
1
··· x
k
l
q
. 2.10
We have
Z
k
p
q
nx
1
···x
k
dμ
q,ζ
x
1
dμ
q,ζ
x
2
···dμ
q,ζ
x
k
n
l0
n
l
q − 1
l
β
k
l,q,ζ
. 2.11
It is easy to see that
Z
k
p
q
nx
1
···x
k
dμ
q,ζ
x
1
dμ
q,ζ
x
2
···dμ
q,ζ
x
k
lim
N →∞
1
p
N
k
q
p
N
−1
x
1
, ,x
k
0
q
nx
1
···x
k
q
x
1
···x
k
ζ
x
1
···x
k
n 1
k
n 1
k
q,ζ
.
2.12
By 2.11 and 2.12, we obtain the following theorem.
Journal of Inequalities and Applications 5
Theorem 2.3. For n ∈ Z
,k∈ N and ζ ∈ T
p
, one has
n
l0
n
l
q − 1
l
β
k
l,q,ζ
n 1
k
n 1
k
q,ζ
. 2.13
Now we consider the modified extension of the twisted q-Bernoulli polynomials of
order k as follows:
B
k
n,q,ζ
x
1
1 − q
n
n
i0
−1
i
n
i
q
ix
Z
k
p
q
k
l1
k−lix
i
dμ
q,ζ
x
1
···dμ
q,ζ
x
k
. 2.14
In the special case x 0, we write B
k
n,q,ζ
B
k
n,q,ζ
0, which are called the modified
extension of the twisted q-Bernoulli numbers of order k.
From 2.14, we derive that
B
k
n,q,ζ
x
1
1 − q
n
n
i0
−1
i
n
i
i k
···
i 1
i k
q,ζ
···
i 1
q,ζ
q
ix
1
1 − q
n
n
i0
−1
i
n
i
ik
k
k!
ik
k
q,ζ
k
q,ζ
!
q
ix
.
2.15
Therefore, we obtain the following theorem.
Theorem 2.4. For n ∈ Z
,k∈ N and ζ ∈ T
p
, one has
B
k
n,q,ζ
x
1
1 − q
n
n
i0
−1
i
n
i
ik
k
k!
ik
k
q,ζ
k
q,ζ
!
q
ix
. 2.16
Now, we define B
−k
n,q,ζ
x as follows:
B
−k
n,q,ζ
x
1
1 − q
n
n
i0
−1
i
n
i
q
ix
Z
k
p
q
k
l1
k−lix
i
dμ
q,ζ
x
1
···dμ
q,ζ
x
k
. 2.17
By 2.17, we can see that
B
−k
n,q,ζ
x
1
1 − q
n
n
i0
−1
i
n
i
ik
k
q,ζ
k
q,ζ
!
ik
k
k!
q
ix
. 2.18
Therefore, we obtain the following theorem.
6 Journal of Inequalities and Applications
Theorem 2.5. For n ∈ Z
,k∈ N and ζ ∈ T
p
, one has
B
−k
n,q,ζ
x
1
1 − q
n
n
i0
−1
i
i k
k
q,ζ
nk
n−i
k
q,ζ
!
nk
k
k!
q
ix
. 2.19
In 2.19, we can see the relations between the binomial coefficients and the modified
extension of the twisted q-Bernoulli polynomials of order k.
Acknowledgments
The authors would like to thank the anonymous referee for his/her excellent detail comments
and suggestions. This Research was supported by Kyungpook National University Research
Fund, 2010.
References
1 L C. Jang, “Multiple twisted q-Euler numbers and polynomials associated with p-adic q-integrals,”
Advances in Difference Equations, Article ID 738603, 11 pages, 2008.
2 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
3 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.
4 T. Kim, “Sums of products of q-Bernoulli numbers,” Archiv der Mathematik, vol. 76, no. 3, pp. 190–195,
2001.
