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. Ryoo3
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
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
2
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
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 Zp . From those p-adic q-integrals,
we present a systemic study of some families of extended Carlitz type q-Bernoulli numbers and
polynomials in p-adic number field.
1. Introduction
Let p be a fixed prime number. Throughout this paper, Zp , Qp , C, and Cp will, respectively,
denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex
number field, and the completion of algebraic closure of Qp . Let N be the set of natural
numbers and Z {0} ∪ N.
Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1 . When
one talks of q-extension, q is considered as an indeterminate, a complex number q ∈ C, or
p-adic number q ∈ Cp . If q ∈ C, we normally assume that |q| < 1, and if q ∈ Cp , we normally
assume that |1 − q|p < 1. We use the notation
xq 1 − qx
.
1−q
1.1
The q-factorial is defined as
nq ! nq n − 1q · · · 2q 1q ,
1.2
2
Journal of Inequalities and Applications
and the Gaussian q-binomial coefficient is defined by
nq n − 1q · · · n − k 1q
nq !
n
,
k q n − kq !kq !
kq !
1.3
nn − 1 · · · n − k 1
n
n
.
lim
k
q→1 k
k!
q
1.4
see 1. Note that
From 1.3, we easily see that
n1
n
n
n
n
qk
qn1−k
,
k
k−1 q
k q
k q
k−1 q
q
1.5
see 2, 3. For a fixed positive integer f, f, p 1, let
X Xf lim
←
−
X∗ Z
, X1 Zp ,
fpN Z
N
a fpN Zp x ∈ X | x ≡ a modfpN ,
a fpZp ,
1.6
0<a<fp
a,p1
where a ∈ Z and 0 ≤ a < fpN see 1–14.
We say that f is a uniformly differential function at a point a ∈ Zp and denote this
property by f ∈ UDZp if the difference quotients
fx − f y
Ff x, y x−y
1.7
have a limit l f a as x, y → a, a. For f ∈ UDZp , let us begin with the expression
1
N
p q
N
−1
p
x0
0≤x<pN
fxqx fxμq x pN Zp ,
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 Zp is defined as the limit N → ∞ of the sums if exists. The p-adic q-integral qVolkenborn integral of f ∈ UDZp is defined by
Iq f fxdμq x X
Zp
fxdμq x lim N−→∞
1
p
N
fxqx ,
q 0≤x<pN
1.9
see 12. Carlitz’s q-Bernoull numbers βk,q can be defined recursively by β0,q 1 and by the
rule that
⎧
⎨1,
k
∗
q qβ 1 − βk,q ⎩0,
∗
if k 1,
1.10
if k > 1,
∗
with the usual convention of replacing β∗ i by βi,q
, see 1–13.
It is well known that
∗
βn,q
∗
βn,q
x
Zp
Zp
xnq dμq x
xnq dμq x,
X
n
y x q dμq y X
n ∈ Z ,
n
y x q dμq y ,
1.11
n ∈ Z ,
∗
see 1, where βn,q
x are called the nth Carlitz’s q-Bernoulli polynomials see 1, 12, 13.
Let χ be the Dirichlet’s character with conductor f ∈ N, then the generalized Carlitz’s
q-Bernoulli numbers attached to χ are defined as follows:
∗
βn,χ,q
X
χxxnq dμq x,
1.12
see 13. Recently, many authors have studied in the different several areas related to qtheory 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 Zp . First, we derive some interesting equations of p-adic q-integrals on Zp .
From these equations, we give some interesting formulae for the higher-order Carlitz’s type
q-Bernoulli numbers and polynomials in the p-adic number field.
2. On the Generalized Higher-Order q-Bernoulli Numbers
and Polynomials
In this section, we assume that q ∈ Cp with |1 − q|p < 1. We first consider the q-extension of
Bernoulli polynomials as follows:
∞
∞
tn
βn,q x q−y exyqt dμq y −t exmqt qxm .
n!
Zp
n0
m0
2.1
4
Journal of Inequalities and Applications
From 2.1, we note that
n n x l l
βn,q x −q
n
lq
1 − q l0 l
1
n l
n x l
−q
n−1
l
1 − ql
1−q
l0
1
n
1−q
n−1
n−1 n−1
l
l0
q
l1x
1
1 − ql1
−1l1
2.2
∞
n−1 −n
n − 1 lxm
mx
q
q
n−1
l
1−q
m0
l0
−n
∞
qmx x mn−1
q .
m0
Note that
lim βn,q x −n
q→1
∞
x mn−1 Bn x,
2.3
m0
where Bn x are called the n th ordinary Bernoulli polynomials. In the special case, x 0,
βn,q 0 βn,q are called the n th q-Bernoulli numbers.
By 2.2, we have the following lemma.
Lemma 2.1. For n ≥ 0, one has
βn,q x Zp
∞
n
q−y x y q dμq y −n qmx x mn−1
q
m0
n 1
n x l l
.
−q
n
lq
1 − q l0 l
2.4
Now, one considers the q-Bernoulli polynomials of order r ∈ N as follows:
∞
n0
r
βn,q x
tn
n!
Zp
···
Zp
q−x1 ···xr exx1 ···xr qt dμq x1 · · · dμq xr .
r times
2.5
Journal of Inequalities and Applications
5
By 2.5, one sees that
r
βn,q x
Zp
···
Zp
q−x1 ···xr x x1 · · · xr nq dμq x1 · · · dμq xr r times
2.6
r
n 1
l
n
l xl
.
−1 q
n
lq
1 − q l0 l
r
r
In the special case, x 0, the sequence βn,q 0 βn,q is refereed to as the q-extension of Bernoulli
numbers of order r. For f ∈ N, one has
r
βn,q x
· · · q−x1 ···xr x x1 · · · xr nq dμq x1 · · · dμq xr X X
r times
1
1−q
n
n n
l0
l
−1l
f−1
2.7
lr
qlxa1 ···ar r
lf q
a1 ,...,ar 0
f−1
n−r a1 · · · ar x
r
.
f q
βn,qf
f
a1 ,...,ar 0
By 2.5 and 2.7, one obtains the following theorem.
Theorem 2.2. For r ∈ Z , f ∈ N, one has
r
1
βn,q x 1−q
n−r
f q
n
f−1 n
lr
n
−1l qla1 ···ar x r
l
lf
l0 a ,...,a 0
1
f−1
q
r
r
βn,qf
a1 ,...,ar 0
a1 · · · ar x
.
f
2.8
Let χ be the primitive Dirichlet’s character with conductor f ∈ N, then the generalized qBernoulli polynomials attached to χ are defined by
∞
tn
βn,χ,q x χ y q−y exyqt dμq y .
n!
X
n0
2.9
6
Journal of Inequalities and Applications
From 2.9, one derives
n
χ y q−y x y q dμq y
βn,χ,q x X
fp
−1
n
1
χa lim N a x fy q
N−→∞ fp
a0
q y0
f−1
N
1
1−q
f−1
χa
a0
−n
f−1
n l
n
χa
−1l qlxa l
lf q
a0
l0
n
∞ m0
∞
2.10
n−1 −n x a mf q
χmx mn−1
q .
m0
By 2.9 and 2.10, one can give the generating function for the generalized q-Bernoulli polynomials
attached to χ as follows:
Fχ,q x, t −t
∞
χmexmqt m0
∞
βn,χ,q x
n0
tn
.
n!
2.11
From 1.3, 2.10, and 2.11, one notes that
f−1
n
1 βn,χ,q x χa
q−fy a x fy q dμqf y
f q a0
Zp
f−1
n−1 f q
χaβn,qf
a0
2.12
ax
.
f
In the special case, x 0, the sequence βn,χ,q 0 βn,χ,q are called the n th generalized qBernoulli numbers attached to χ.
Let one consider the higher-order q-Bernoulli polynomials attached to χ as follows:
r
···
X
X
i1
χxi exx1 ···xr qt q−x1 ···xr dμq x1 · · · dμq xr ∞
n0
r
βn,χ,q x
tn
,
n!
r times
r
where βn,χ,q x are called the n th generalized q-Bernoulli polynomials of order r attaches to χ.
2.13
Journal of Inequalities and Applications
7
By 2.13, one sees that
r
βn,χ,q x
1
1−q
n−r
f q
n
n n l
l0
−q
r
a1 ,...,ar 0
r
f−1
a1 ,...,ar 0
f−1
x l
i1
χai βn,qf
r
χai ql
r
i1
ai
i1
r
lr
r
lf q
x a1 · · · ar
.
f
2.14
r
In the special case, x 0, the sequence βn,χ,q 0 βn,χ,q are called the n th generalized q-Bernoulli
numbers of order r attaches to χ.
By 2.13 and 2.14, one obtains the following theorem.
Theorem 2.3. Let χ be the primitive Dirichlet’s character with conductor f ∈ N. For n ∈ Z , r ∈ N,
one has
r
βn,χ,q x
1
1−q
n
n n l
l0
−q
a1 ,...,ar 0
r
a1 ,...,ar 0
r
f−1
n−r f q
f−1
x l
χai i1
r
βn,qf
χai ql
r
i1
i1
ai
lr
r
lf q
x a1 · · · ar
.
f
2.15
For h ∈ Z, and r ∈ N, one introduces the extended higher-order q-Bernoulli polynomials as
follows:
h,r
βn,q x
Zp
···
Zp
q
r
j1 h−j−1xj
x x1 · · · xr nq dμq x1 · · · dμq xr .
2.16
r times
From 2.16, one notes that
h,r
βn,q x
lh−1 n r!
n
r
l lx
,
−1 q lh−1
n
l
r
1 − q l0
q!
r
q
1
2.17
and
f−1
r
n−r h,r
h,r x a1 · · · ar
βn,q x f q
q j1 h−jaj βn,qf
.
f
a1 ,...,ar 0
h,r
h,r
2.18
In the special case, x 0, βn,q 0 βn,q are called the n th h, q-Bernoulli numbers of order
r.
By 2.17, one obtains the following theorem.
8
Journal of Inequalities and Applications
Theorem 2.4. For h ∈ Z, r ∈ N, one has
h,r
βn,q x
1
1−q
n
n n l0
l
−q
x l
lh−1 r
lh−1
r
q
r!
,
rq !
f−1
r
n−r h,r
h,r x a1 · · · ar
βn,q x f q
.
q j1 h−jaj βn,qf
f
a1 ,...,ar 0
2.19
Let χ be the primitive Dirichlet’s character with conductor f ∈ N, then one considers the
generalized h, q-Bernoulli polynomials attached to χ of order r as follows:
h,r
βn,χ,q x
⎛
⎞
r
· · · q j1 h−j−1xj ⎝ χ xj ⎠x x1 · · · xr nq dμq x1 · · · dμq xr .
2.20
X
j1
X
r
r times
By 2.20, one sees that
h,r
βn,χ,q x
⎞
⎛
f−1
r
r
n−r h,r x a1 · · · ar
j1 h−jaj ⎝
⎠
βn,qf
.
q
χ aj
f q
f
a1 ,...,ar 0
j1
h,r
2.21
h,r
In the special case, x 0, βn,χ,q 0 βn,χ,q are called the n th generalized h, q-Bernoulli numbers
attached to χ of order r.
From 2.20 and 2.21, one notes that
h−1,r
h,r
h−1,r
βn,χ,q q − 1 βn1,χ,q βn,χ,q .
2.22
By 2.16, it is easy to show that
h,r
βn,χ,q
Zp
···
Zp
x1 · · · xr nq q
r
j1 h−j−1xj
dμq x1 · · · dμq xr r times
Zp
···
Zp
x1 · · · xr nq
rj1 h−j−2xj
dμq x1 · · · dμq xr .
x1 · · · xr q q − 1 1 q
2.23
Thus, one has
h−1,r
h,r
h−1,r
βn,q q − 1 βn1,q βn,q .
2.24
Journal of Inequalities and Applications
9
From 2.16 and 2.23, one can also derive
Zp
···
Zp
qn−2x1 n−3x2 ···n−r−1xr dμq x1 · · · dμq xr r times
· · · Zp q−x1 ···xr qnx1 ···xr q−x1 −2x2 −···−rxr dμq x1 · · · dμq xr n
n l q − 1 Zp · · · Zp x1 · · · xr lq q−x1 ···xr q−x1 −2x2 −···−rxr dμq x1 · · · dμq xr l
l0
n
n l 0,r
q − 1 βl,q ,
l
l0
n−1 r!
r
.
···
qn−2x1 n−3x2 ···n−r−1xr dμq x1 · · · dμq xr n−1 rq !
Zp
Zp
Zp
r
q
2.25
It is easy to see that
n n j0
j
j
q−1
Zp
j
xq qh−2x dμq x
Zp
n
q − 1 xq 1 qh−2x dμq x
nh−1
.
n h − 1q
2.26
By 2.23, 2.25, and 2.26, one obtains the following theorem.
Theorem 2.5. For h ∈ Z, r ∈ N, and n ∈ Z , one has
h−1,r
h,r
h−1,r
βn,q q − 1 βn1,q βn,q ,
n−1 n
n l 0,r
r!
r
.
q − 1 βl,q n−1 r
l
q!
l0
r
2.27
q
Furthermore, one gets
n n l0
l
l h,1
q − 1 βl,q nh−1
.
n h − 1q
2.28
10
Journal of Inequalities and Applications
0,r
Now, one considers the polynomials of βn,q x by
0,r
βn,q x Zp
···
Zp
x x1 · · · xr nq q−2x1 −3x2 −···−r−1xr dμq x1 · · · dμq xr r times
1
1−q
n
n n
l
l0
l−1 −1l qlx l−1r r
q
2.29
r!
.
rq !
By 2.29, one obtains the following theorem.
Theorem 2.6. For r ∈ N and n ∈ Z , one has
l−1 n n 0,r
r!
n
.
1 − q βn,q x −1l qlx l−1r l
r
q!
r q
l0
2.30
By using multivariate p-adic q-integral on Zp , one sees that
q
nx
n−1 r
n−1
r
q
Zp
r!
rq !
···
Zp
qnxn−2x1 ···n−r−1xr dμq x1 · · · dμq xr r times
Zp
Zp
n n l0
···
n
q − 1 x x1 · · · xr q 1 q−2x1 ···−r1xr dμq x1 · · · dμq xr l
l
q−1
Zp
···
Zp
2.31
x x1 · · · xr lq q−2x1 ···−r1xr dμq x1 · · · dμq xr n n l0
l
l 0,r
q − 1 βl,q x.
Therefore, one obtains the following corollary.
Corollary 2.7. For r ∈ N and n ∈ Z , one has
q
nx
n−1 r
n−1
r
q
n l 0,r
r!
n q − 1 βl,q x.
rq ! l0 l
2.32
Journal of Inequalities and Applications
11
It is easy to show that
···
Zp
Zp
x x1 · · · xr nq q−2x1 ···−r1xr dμq x1 · · · dμq xr r times
f−1
r
n−r f q
q− l1 lil
2.33
i1 ,...,ir 0
×
Zp
···
Zp
q
−f
r
l1 l1xl
x
r
l1 il
f
r
l1
n
dμqf x1 · · · dμqf xr .
xl
qf
From 2.33, one notes that
f−1
n−r 0,r
0,r x i1 · · · ir
.
βn,q x f q
q−i1 −2i2 −...−rir βn,qf
f
i1 ,...,ir 0
2.34
From the multivariate p-adic q-integral on Zp , one has
Zp
···
Zp
x x1 · · · xr nq q−2x1 −3x2 −···−r1xr dμq x1 · · · dμq xr r times
Zp
Zp
xq qx x1 · · · xr q
n
q−2x1 −3x2 −···−r1xr dμq x1 · · · dμq xr n n
lx
q
·
·
·
x1 · · · xr lq q−2x1 −3x2 −···−r1xr dμq x1 · · · dμq xr ,
xn−l
q
l
Z
Z
p
p
l0
Zp
···
···
Zp
2.35
n
x y x1 · · · xr q q−2x1 −3x2 −···−r1xr dμq x1 · · · dμq xr r times
n n n−l ly
···
y q q
x x1 · · · xr lq q−2x1 −3x2 −···−r1xr dμq x1 · · · dμq xr .
l
Zp
Zp
l0
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
0,r
βn,q x n n
l0
0,r βn,q x
l
0,r
lx
xn−l
q q βl,q ,
n n n−l ly 0,r
y q q βl,q x.
y l
l0
2.37
h,1
Now, one also considers the polynomial of βn,q x. From the integral equation on Zp , one
notes that
h,1
βn,q x Zp
x x1 nq qx1 h−2 dμq x1 1
1−q
n
n n
l0
lh−1
.
−1 q
l
l h − 1q
2.38
l lx
By 2.38, one easily gets
1
h,1
βn,q x 1−q
n−1
n
nl −1l qlx l
n−1
−n
n−1
1−q
l0
−n
∞
1 − qlh−1
l0
n
nl −1l qlx
h−1 n−1
lh−1
1−q
l0 1 − q
n−1 l
−1l qx qlx
1 − qlh
n
nl −1l qlx
h−1 n−1
lh−1
1−q
l0 1 − q
2.39
∞
qhmx x mn−1
h − 1 1 − q
qh−1m x mnq .
q
m0
m0
Thus, one obtains the following theorem.
Theorem 2.9. For h ∈ Z and n ∈ Z , one has
h,1
βn,q x −n
∞
∞
qhmx x mn−1
h − 1 1 − q
qh−1m x mnq .
q
m0
2.40
m0
From the definition of p-adic q-integral on Zp , one notes that
Zp
qh−2x1 x x1 nq dμq x1 1 h−1i n
q
iq
f q i0
f−1
!
Zp
xi
x1
f
"n
qf
2.41
qfh−2x1 dμqf x1 .
Journal of Inequalities and Applications
13
Thus, one has
f−1
1 h−1i n h,1 x i
h,1
.
q
βn,q x iq βn,qf
f
f q i0
2.42
By 2.38, one easily gets
Zp
x x1 nq qx1 h−2 dμq x1 q
−x
Zp
x x1 nq x x1 q q − 1 1 qx1 h−3 dμq x1 .
2.43
From 2.43, one has
h,1
βn,q x q−x
h−1,1
h−1,1
q − 1 βn1,q x βn,q x .
2.44
That is,
h−1,1
h,1
h−1,1
qx βn,q x q − 1 βn1,q x βn,q x.
2.45
By 2.38 and 2.43, one easily sees that
Zp
qh−2x1 x x1 nq dμq x1 Zp
n
qh−2x1 xq qx x1 q dμq x1 n n
l0
l
2.46
lx
xn−l
q q
Zp
q
h−2x1
x1 lq dμq x1 ,
and
qh−1
Zp
qh−2x1 x1 1 xnq dμq x1 −
Zp
qh−2x1 x x1 nq dμq x1 qx nxn−1
h q − 1 xnq − q − 1 xnq .
q
2.47
For x 0, this gives
qh−1
Zp
qh−2x1 x1 1nq dμq x1 −
Zp
qh−2x1 x1 nq dμq x1 ⎧
⎨1,
if n 1,
⎩0,
if n > 1,
2.48
and
h,1
β0,q Zp
qh−2x1 dμq x1 h−1
.
h − 1q
2.49
14
Journal of Inequalities and Applications
h,1
From 2.46 and 2.48, one can derive the recurrence relation for βn,q as follows:
h,1
h,1
2.50
qh−1 βn,q 1 − βn,q δn,1 ,
where δn,1 is kronecker symbol.
By 2.46, 2.48, and 2.50, one obtains the following theorem.
Theorem 2.10. For h ∈ Z and n ∈ Z , one has
h,1
βn,q x
n
n
l0
h,1
h,1
qh−1 βn,q x 1 − βn,q
l
h,1
lx
xn−l
q q βl,q ,
qx nxn−1
h q − 1 xnq − q − 1 xnq .
q
2.51
Furthermore,
h−1,1
h−1,1
h,1
qh−2 q − 1 βn1,q 1 qh−2 βn,q 1 − βn,q δn,1 ,
2.52
where δn,1 is kronecker symbol.
From the definition of p-adic q-integral on Zp , one notes that
Zp
q−h−2x1 1 − x x1 nq−1 dμq−1 x1 2.53
n nh−2
−1 q
Zp
q
h−2x1
x x1 nq dμq x1 .
By 2.53, one sees that
h,1
h,1
βn,q−1 1 − x −1n qnh−2 βn,q x.
2.54
Note that
h,1
h,1
Bn 1 − x lim βn,q−1 1 − x lim −1n qnh−2 βn,q x −1n Bn x,
q−→1
q−→1
2.55
where Bn x are the n th ordinary Bernoulli polynomials.
In the special case, x 1, one gets
h,1
h,1
h,1
βn,q−1 −1n qnh−2 βn,q 1 −1n qn−1 βn,q ,
if n > 1.
2.56
Journal of Inequalities and Applications
15
It is not difficult to show that
f−1
n−1 f q
qlh−1
l0
Zp
!
"n
l
x x1 qfh−2x1 dμqf x1 f
Zp
qf
n
fx x1 q qh−2x1 dμq x1 ,
2.57
f ∈ N.
That is,
f−1
n−1 l
h,1 lh−1 h,1
q
βn,qf x f q
βn,q fx .
f
l0
2.58
Let one consider Barnes’ type multiple q-Bernoulli polynomials. For w1 , w2 , . . . , wr ∈ Zp , and
δ1 , δ2 , . . . , δr ∈ Z, one defines Barnes’ type multiple q-Bernoulli polynomials as follows:
r
βn,q x | w1 , . . . , wr : δ1 , . . . , δr r
···
w1 x1 · · · wr xr xnq q j1 δj −1xj dμq x1 · · · dμq xr .
Zp
Zp
2.59
r times
From 2.59, one can easily derive the following equation:
r
βn,q x | w1 , . . . , wr : δ1 , . . . , δr n 1
lw1 δ1 lw2 δ2 · · · lwr δr n
.
−1l qlx
n
l
lw
1 − q l0
1 δ1 q lw2 δ2 q · · · lwr δr q
2.60
Let δr δ1 r − 1, then one has
⎛
⎞
lw δ r−1 n ⎜
⎟
1
1
r!
1
n
⎟
r ⎜
r
l lx
.
q
βn,q ⎜x | w1 · · · w1 : δ1 , δ1 1 . . . , δ1 r − 1⎟ −1
n
lw1 δ1 r−1
⎝ ⎠
l
r
1 − q l0
q!
r
q
r times
2.61
Therefore, one obtains the following theorem.
16
Journal of Inequalities and Applications
Theorem 2.11. For w1 ∈ Zp , r ∈ N, and δ1 ∈ Z, one has
⎛
⎞
⎜
⎟
⎟
r ⎜
βn,q ⎜x | w1 · · · w1 : δ1 , δ1 1 . . . , δ1 r − 1⎟
⎝ ⎠
r times
2.62
lw δ r−1 n 1
1
r!
n
l lx
.
−1 q lw δrr−1 n
1
1
l
r
1 − q l0
q!
r
q
1
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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