Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 575240, 17 pages
doi:10.1155/2010/575240
Research Article
Some Identities on the Generalized q-Bernoulli
Numbers and Polynomials Associated with
q-Volkenborn Integrals
T. Kim,
1
J. Choi,
1
B. Lee,
2
and C. S. Ryoo
3
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
3
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Correspondence should be addressed to T. Kim, [email protected]
Received 23 August 2010; Accepted 30 September 2010
Academic Editor: Alberto Cabada
Copyright q 2010 T. Kim 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 give some interesting equation of p-adic q-integrals on
Z
−1
. When
one talks of q-extension, q is considered as an indeterminate, a complex number q ∈ C, or
p-adic number q ∈ C
p
. If q ∈ C, we normally assume that |q| < 1, and if q ∈ C
p
, we normally
assume that |1 − q|
p
< 1. We use the notation
x
q
1 − q
x
1 − q
. 1.1
The q-factorial is defined as
n
q
!
n
q
q
!
k
q
!
n
q
n − 1
q
···
n − k 1
q
k
q
!
,
1.3
see 1.Notethat
n
k − 1
q
q
k
n
k
q
q
n1−k
n
k − 1
q
n
k
q
,
1.5
see 2, 3. For a fixed positive integer f, f, p1, let
X X
Z
p
x ∈ X | x ≡ a
modfp
N
,
1.6
where a ∈ Z and 0 ≤ a<fp
N
see 1–14.
We say that f is a uniformly differential function at a point a ∈ Z
p
and denote this
property by f ∈ UDZ
p
if the difference quotients
F
f
x, y
f
x
0≤x<p
N
f
x
μ
q
x p
N
Z
p
,
1.8
representing a q-analogue of the Riemann sums for f, see 1–3, 11–18. The integral of f
Journal of Inequalities and Applications 3
on Z
p
is defined as the limit N →∞ of the sums if exists.Thep-adic q-integral q-
Volkenborn integral of f ∈ UDZ
p
is defined by
I
q
f
q
0≤x<p
N
f
x
q
x
,
1.9
see 12. Carlitz’s q-Bernoull numbers β
k,q
can be defined recursively by β
0,q
1andbythe
rule that
q
qβ
∗
1
k
− β
∗
k,q
x
X
x
n
q
dμ
q
x
,n∈ Z
,
β
∗
n,q
x
Z
p
y x
q-Bernoulli numbers attached to χ are defined as follows:
β
∗
n,χ,q
X
χ
x
x
n
q
dμ
q
x
,
1.12
see 13. Recently, many authors have studied in the different several areas related to q-
theory see 1–13. In this paper, we present a systemic study of some families of multiple
Carlitz’s type q-Bernoulli numbers and polynomials by using the integral equations of p-adic
q-integrals on Z
p
. First, we derive some interesting equations of p-adic q-integrals on Z
p
.
t
dμ
q
y
−t
∞
m0
e
xm
q
t
q
xm
.
2.1
4 Journal of Inequalities and Applications
From 2.1,wenotethat
β
n,q
x
1
1 − q
−q
x
l
l
1 − q
l
n
1 − q
n−1
n−1
l0
n − 1
l
q
l1x
1
1 − q
l1
mx
x m
n−1
q
.
2.2
Note that
lim
q → 1
β
n,q
x
−n
∞
m0
x m
n−1
B
n
x
,
y
−n
∞
m0
q
mx
x m
n−1
q
1
1 − q
n
n
l0
n
l
−q
x
r times
q
−x
1
···x
r
e
xx
1
···x
r
q
t
dμ
q
x
1
···dμ
q
x
r
.
2.5
Journal of Inequalities and Applications 5
q
dμ
q
x
1
···dμ
q
x
r
1
1 − q
n
n
l0
n
l
−1
l
X
r times
q
−x
1
···x
r
x x
1
··· x
r
n
q
dμ
q
x
1
···dμ
q
x
r
l
r
lf
r
q
f
n−r
q
f−1
a
1
, ,a
r
0
β
r
n,q
f
a
1
··· a
r
x
n
l
−1
l
q
la
1
···a
r
x
l
r
lf
r
q
f
n−r
q
f−1
a
n!
X
χ
y
q
−y
e
xy
q
t
dμ
q
y
.
2.9
6 Journal of Inequalities and Applications
From 2.9, one derives
β
n,χ,q
x
N
q
fp
N
−1
y0
a x fy
n
q
1
1 − q
n
f−1
a0
χ
a
n
l0
n−1
q
−n
∞
m0
χ
m
x m
n−1
q
.
2.10
By 2.9 and 2.10, one can give the generating function for the generalized q-Bernoulli polynomials
attached to χ as follows:
F
χ,q
x,t
−t
∞
m0
χ
q
f−1
a0
χ
a
Z
p
q
−fy
a x fy
n
q
dμ
q
f
y
f
n−1
r times
r
i1
χ
x
i
e
xx
1
···x
r
q
t
q
−x
1
···x
r
dμ
q
x
1
n,χ,q
x
1
1 − q
n
n
l0
n
l
−q
x
l
f−1
a
1
, ,a
r
0
a
1
, ,a
r
0
r
i1
χ
a
i
β
r
n,q
f
x a
1
··· a
r
f
.
2.14
In the special case, x 0, the sequence β
−q
x
l
f−1
a
1
, ,a
r
0
r
i1
χ
a
i
q
l
r
i1
a
i
β
r
n,q
f
x a
1
··· a
r
f
.
2.15
For h ∈ Z, and r ∈ N, one introduces the extended higher-order q-Bernoulli polynomials as
follows:
β
h,r
n,q
x
Z
p
···
Z
p
β
h,r
n,q
x
1
1 − q
n
n
l0
n
l
−1
l
q
lx
lh−1
r
r
0
q
r
j1
h−ja
j
β
h,r
n,q
f
x a
1
··· a
r
f
.
2.18
In the special case, x 0, β
h,r
n,q
0β
h,r
n,q
are called the n th h, q-Bernoulli numbers of order
r.
By 2.17, one obtains the following theorem.
lh−1
r
q
r!
r
q
!
,
β
h,r
n,q
x
f
n−r
q
f−1
a
1
, ,a
r
X
···
X
r times
q
r
j1
h−j−1x
j
⎛
⎝
r
j1
χ
x
j
⎞
⎠
x x
1
··· x
r
a
1
, ,a
r
0
q
r
j1
h−ja
j
⎛
⎝
r
j1
χ
a
j
⎞
⎠
β
h,r
n,q
f
x a
By 2.16, it is easy to show that
β
h,r
n,χ,q
Z
p
···
Z
p
r times
x
1
··· x
r
n
q
q
r
j1
h−j−1x
j
dμ
q
··· x
r
q
q − 1
1
q
r
j1
h−j−2x
j
dμ
q
x
1
···dμ
q
x
r
.
2.23
Thus, one has
2
···n−r−1x
r
dμ
q
x
1
···dμ
q
x
r
Z
p
···
Z
p
q
−x
1
···x
r
q
l
q − 1
l
Z
p
···
Z
p
x
1
··· x
r
l
q
q
−x
1
···x
r
q
−x
1
l,q
,
Z
p
···
Z
p
q
n−2x
1
n−3x
2
···n−r−1x
r
dμ
q
x
1
···dμ
q
x
r
p
x
j
q
q
h−2x
dμ
q
x
Z
p
q − 1
x
q
1
n
q
h−2x
,
n
l0
n
l
q − 1
l
β
0,r
l,q
n−1
r
n−1
r
q
r!
r
q
x by
β
0,r
n,q
x
Z
p
···
Z
p
r times
x x
1
··· x
r
n
q
q
−2x
1
−3x
2
l
q
lx
l−1
r
l−1
r
q
r!
r
q
!
.
2.29
By 2.29, one obtains the following theorem.
Theorem 2.6. For r ∈ N and n ∈ Z
, one has
1 − q
n
β
0,r
q
!
.
2.30
By using multivariate p-adic q-integral on Z
p
, one sees that
q
nx
n−1
r
n−1
r
q
r!
r
q
!
Z
p
···
q − 1
x x
1
··· x
r
q
1
n
q
−2x
1
···−r1x
r
dμ
q
x
1
···dμ
q
x
r
1
···−r1x
r
dμ
q
x
1
···dμ
q
x
r
n
l0
n
l
q − 1
l
β
0,r
l,q
n
l
q − 1
l
β
0,r
l,q
x
.
2.32
Journal of Inequalities and Applications 11
It is easy to show that
Z
p
···
Z
p
r times
x x
1
i
1
, ,i
r
0
q
−
r
l1
li
l
×
Z
p
···
Z
p
q
−f
r
l1
l1x
l
x
2.33
From 2.33, one notes that
β
0,r
n,q
x
f
n−r
q
f−1
i
1
, ,i
r
0
q
−i
1
−2i
2
− −ri
r
β
0,r
q
q
−2x
1
−3x
2
−···−r1x
r
dμ
q
x
1
···dμ
q
x
r
Z
p
···
Z
p
x
r
n
l0
n
l
x
n−l
q
q
lx
Z
p
···
Z
p
x
1
··· x
r
p
r times
x y x
1
··· x
r
n
q
q
−2x
1
−3x
2
−···−r1x
r
dμ
q
x
1
···dμ
q
x
r
q
−2x
1
−3x
2
−···−r1x
r
dμ
q
x
1
···dμ
q
x
r
.
2.36
By 2.35 and 2.36, one obtains the following corollary.
12 Journal of Inequalities and Applications
Corollary 2.8. For r ∈ N and n ∈ Z
, one has
β
0,r
n,q
l0
n
l
y
n−l
q
q
ly
β
0,r
l,q
x
.
2.37
Now, one also considers the polynomial of β
h,1
n,q
x. From the integral equation on Z
p
, one
notes that
β
h,1
n,q
l0
n
l
−1
l
q
lx
l h − 1
l h − 1
q
.
2.38
By 2.38, one easily gets
β
h,1
n,q
x
1
1 − q
−1
l
q
lx
1 − q
lh−1
−n
1 − q
n−1
n−1
l0
n−1
l
−1
l
q
x
q
lx
1 − q
lh
q
h − 1
1 − q
∞
m0
q
h−1m
x m
n
q
.
2.39
Thus, one obtains the following theorem.
Theorem 2.9. For h ∈ Z and n ∈ Z
, one has
β
h,1
n,q
x
From the definition of p-adic q-integral on Z
p
, one notes that
Z
p
q
h−2x
1
x x
1
n
q
dμ
q
x
1
1
f
q
f−1
i0
2.41
Journal of Inequalities and Applications 13
Thus, one has
β
h,1
n,q
x
1
f
q
f−1
i0
q
h−1i
i
n
q
β
h,1
n,q
f
x x
1
n
q
x x
1
q
q − 1
1
q
x
1
h−3
dμ
q
x
1
.
2.43
From 2.43, one has
h,1
n,q
x
q − 1
β
h−1,1
n1,q
x
β
h−1,1
n,q
x
.
2.45
By 2.38 and 2.43, one easily sees that
Z
p
q
h−2x
1
q
n
dμ
q
x
1
n
l0
n
l
x
n−l
q
q
lx
Z
p
q
h−2x
1
dμ
q
x
1
−
Z
p
q
h−2x
1
x x
1
n
q
dμ
q
x
1
q
x
n
x
h−2x
1
x
1
1
n
q
dμ
q
x
1
−
Z
p
q
h−2x
1
x
1
n
q
dμ
q
h − 1
q
.
2.49
14 Journal of Inequalities and Applications
From 2.46 and 2.48, one can derive the recurrence relation for β
h,1
n,q
as follows:
q
h−1
β
h,1
n,q
1
− β
h,1
n,q
δ
n,1
,
2.50
where δ
n,1
is kronecker symbol.
By 2.46, 2.48,and2.50, one obtains the following theorem.
β
h,1
n,q
x 1
− β
h,1
n,q
q
x
n
x
n−1
q
h
q − 1
x
n
q
−
q − 1
δ
n,1
,
2.52
where δ
n,1
is kronecker symbol.
From the definition of p-adic q-integral on Z
p
, one notes that
Z
p
q
−h−2x
1
1 − x x
1
n
q
−1
dμ
q
−1
x
1
n,q
−1
1 − x
−1
n
q
nh−2
β
h,1
n,q
x
.
2.54
Note that
B
n
1 − x
lim
q−→ 1
β
h,1
2.55
where B
n
x are t he n th ordinary Bernoulli polynomials.
In the special case, x 1, one gets
β
h,1
n,q
−1
−1
n
q
nh−2
β
h,1
n,q
1
−1
n
q
n−1
β
fh−2x
1
dμ
q
f
x
1
Z
p
fx x
1
n
q
q
h−2x
1
dμ
q
x
1
,f∈ N.
2.57
,w
2
, ,w
r
∈ Z
p
, and
δ
1
,δ
2
, ,δ
r
∈ Z, one defines Barnes’ type multiple q-Bernoulli polynomials as follows:
β
r
n,q
x | w
1
, ,w
r
: δ
1
, ,δ
r
Z
x
1
···dμ
q
x
r
.
2.59
From 2.59, one can easily derive the following equation:
β
r
n,q
x | w
1
, ,w
r
: δ
1
, ,δ
r
1
1 − q
δ
r
lw
1
δ
1
q
lw
2
δ
2
q
···
lw
r
δ
r
q
.
2.60
Let δ
r
δ
1 − q
n
n
l0
n
l
−1
l
q
lx
lw
1
δ
1
r−1
r
lw
1
δ
1
1
···w
1
r times
: δ
1
,δ
1
1 ,δ
1
r − 1
⎞
⎟
⎟
⎟
⎠
1
1 − q
n
n
l0
n
l
2.62
Acknowledgments
The authors express their gratitude to The referees for their valuable suggestions and
comments. This paper was supported by the research grant of Kwangwoon University in
2010.
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