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Abstract and Applied Analysis
Volume 2011, Article ID 867217, 9 pages
doi:10.1155/2011/867217
Research Article
A Note on the Generalized q-Bernoulli Measures
with Weight α
T. Kim,1 S. H. Lee,1 D. V. Dolgy,2 and C. S. Ryoo3
1
Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
Hanrimwon, 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 21 April 2011; Accepted 16 May 2011
Academic Editor: Gabriel Turinici
Copyright q 2011 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 discuss a new concept of the q-extension of Bernoulli measure. From those measures, we derive
some interesting properties on the generalized q-Bernoulli numbers with weight α attached to χ.
1. Introduction
Let p be a fixed prime number. Throughout this paper Zp , Qp , and Cp will, respectively, denote
the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion
of algebraic closure of Qp . Let N be the set of natural numbers and Z N ∪ {0}. Let νp be the
normalized exponential valuation of Cp with |p|p p−νp p 1/p see 1–14.
When we talk of q-extension, q is variously considered as an indeterminate, a complex
number q ∈ C, or a p-adic number q ∈ Cp . Throughout this paper we assume that q ∈ Cp with
|1 − q|p < 1, and we use the notation of q-number as
xq 1 − qx
,
1−q
1.1
see 1–14. Thus, we note that limq → 1 xq x.
In 2, Carlitz defined a set of numbers ξk ξk q inductively by
ξ0 1,
k
qξ 1 − ξk with the usual convention of replacing ξk by ξk .
1,
0,
if k 1,
if k > 1,
1.2
2
Abstract and Applied Analysis
These numbers are q-extension of ordinary Bernoulli numbers Bk . But they do not remain finite when q 1. So he modified 1.2 as follows:
⎧
⎨1,
k
q qβ 1 − βk,q ⎩0,
β0,q 1,
if k 1,
if k > 1,
1.3
with the usual convention of replacing βk by βk,q .
The numbers βk,q are called the k-th Carlitz q-Bernoulli numbers.
In 1, Carlitz also considered the extended Carlitz’s q-Bernoulli numbers as follows:
h
β0,q
h
,
hq
qh
⎧
⎨1,
k
h
qβh 1 − βk,q
⎩0,
if k 1,
if k > 1,
1.4
h
with the usual convention of replacing βh k by βk,q
.
Recently, Kim considered q-Bernoulli numbers, which are different extended Carlitz’s
q-Bernoulli numbers, as follows: for α ∈ N and n ∈ Z ,
α
β
0,q 1,
n
α
q qα β
α 1 − β
n,q k
⎧ α
⎪
,
⎪
⎪
⎪
⎨ αq
if n 1,
1.5
⎪
⎪
⎪
⎪
⎩0,
if n > 1,
with the usual convention of replacing β
α by β
k,q see 3.
α
The numbers β
are called the k-th q-Bernoulli numbers with weight α.
α
k,q
For fixed d ∈ Z with p, d 1, we set
X Xd lim
←
N
X∗ Z
,
dpN Z
X1 Zp ,
a dpZp ,
1.6
0<a<dp
a,p1
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z satisfies the condition 0 ≤ a < dpN .
Let UDZp be the space of uniformly differentiable functions on Zp . For f ∈ UDZp ,
the p-adic q-integral on Zp is defined by Kim as follows:
Iq f Zp
1
fxdμq x lim N N →∞ p
q
N
−1
p
fxqx ,
x0
1.7
Abstract and Applied Analysis
3
see 3, 4, 15, 16. By 1.5 and 1.7, the Witt’s formula for the q-Bernoulli numbers with
weight α is given by
xnqα dμq x β
n,q ,
α
Zp
where n ∈ Z .
1.8
The q-Bernoulli polynomials with weight α are also defined by
α
β
n,q x n
n
l
l0
αlx βl,q .
xn−l
qα q
α
1.9
α
From 1.7, 1.8, and 1.9, we can derive the Witt’s formula for β
n,q x as follows:
Zp
xy
n
qα
α
dμq y β
n,q x,
where n ∈ Z .
1.10
For n ∈ Z and d ∈ N, the distribution relation for the q-Bernoulli polynomials with
weight α are known that
α
β
n,q x d−1
dnqα dq
a0
qa β
n,qd
α
x a
d
,
1.11
see 3. Recently, several authors have studied the p-adic q-Euler and Bernoulli measures
on Zp see 8, 9, 11, 13, 14. The purpose of this paper is to construct p-adic q-Bernoulli
distribution with weight α p-adic q-Bernoulli unbounded measure with weight α on Zp
and to study their integral representations. Finally, we construct the generalized q-Bernoulli
numbers with weight α and investigate their properties related to p-adic q-L-functions.
2. p-Adic q-Bernoulli Distribution with Weight α
Let X be any compact-open subset of Qp , such as Zp or Z∗p . A p-adic distribution μ on X is
defined to be an additive map from the collection of compact open set in X to Qp :
μ
n
Uk
k1
n
μUk additivity ,
2.1
k1
where {U1 , U2 , . . . , Un } is any collection of disjoint compact opensets in X.
The set Zp has a topological basis of compact open sets of the form a pn Zp .
Consequently, if U is any compact open subset of Zp , it can be written as a finite disjoint
union of sets
U
k
aj pn Zp ,
j1
where n ∈ N and a1 , a2 , . . . , ak ∈ Z with 0 ≤ ai < pn for i 1, 2, . . . k.
2.2
4
Abstract and Applied Analysis
Indeed, the p-adic ball a pn Zp can be represented as the union of smaller balls
a pn Zp p−1
a bpn pn1 Zp .
2.3
b0
Lemma 2.1. Every map μ from the collection of compact-open sets in X to Qp for which
p−1
μ a pN Zp a bpN dpN1 Zp
2.4
b0
holds whenever a pN Zp ⊂ X, extends to a p-adic distribution ( p-adic unbounded measure) on X.
α
Now we define a map μk,q on the balls in Zp as follows:
α μk,q a
pn Zp
n k
p qα
{a}n
a α
n q fk,qpn
,
pn
p q
2.5
where {a}n is the unique number in the set {0, 1, 2, . . . , pn − 1} such that {a}n ≡ amod pn .
If a ∈ {0, 1, 2, . . . , pn − 1}, then
p−1
b0
α
μk,q
a bpn pn1 Zp
k
p−1 pn1
a bpn
qα abpn α
f pn1
n1 q
k,q
pn1
b0 p
q
n k k
p−1
p q α p q p n α a/pn b
a
bpn α
q n .
q f pn p
k,q p
p q p qpn b0
2.6
α
From 2.6, we note that μk,q is p-adic distribution on Zp if and only if
k
p−1
p qpn α a/pn b
a
α
bpn α
fk,qpn
q f pn p
.
k,q p
pn
p qpn b0
2.7
α
Theorem 2.2. Let α ∈ N and k ∈ Z . Then we see that μk,q a pn Zp is p-adic distribution on Zp if
and only if
k
p−1
p qpn α a/pn b
a
α
bpn α
fk,qpn
q f pn p
.
n
k,
q
p
p
p qpn b0
2.8
fk,qpn x β
k,qpn x.
2.9
One sets
α
α
Abstract and Applied Analysis
5
From 2.5 and 2.9, one gets
α μk,q a
p Zp
n
k
p n qα
a
α
.
n qa β
k,qpn
pn
p q
2.10
By 1.11, 2.10, and Theorem 2.2, we obtain the following theorem.
α
Theorem 2.3. Let μk,q be given by
α
μk,q
a dp Zp
N
k
dpN qα
a
a α
N q β dpN
.
k,q
dpN
dp q
2.11
α
Then μk,q extends to a Qq-valued distribution on the compact open sets U ⊂ X.
From 2.11, one notes that
X
α
dμk,q x
α
μk,q x dpN Zp
N
dp
−1
lim
N →∞
x0
N k N
dp qα dp−1
a
α
lim N .
qa β
dpN
k,q
N → ∞ dp
dpN
q a0
2.12
By 1.11 and 2.12, one gets
X
α
α
dμk,q x β
k,q .
2.13
Therefore, we obtain the following theorem.
Theorem 2.4. For α ∈ N and k ∈ Z , one has
dμk,q x β
k,q .
α
X
α
2.14
Let χ be Dirichlet character with conductor d ∈ N. Then one defines the generalized q-Bernoulli
numbers attached to χ as follows:
α
β
n,χ,q X
χxxnqα dμq x
d−1
a
dnqα α
.
qa χaβ
n,qd
d
dq a0
2.15
6
Abstract and Applied Analysis
From 2.11 and 2.15, one can derive the following equation;
X
α
χxdμk,q x
lim
α
χxμk,q x dpN Zp
N
dp
−1
N →∞
x0
N k N
dp qα dp−1
x
α
lim N χxqx β
dpN
k,q
N → ∞ dp
dpN
q x0
⎧
⎫
N k N
⎬
−1
d−1
⎨
p qαd p
dkqα x
a/d
qa χa lim N qdx β
k,qdpN
⎩N → ∞ p
⎭
dq a0
pN
d
q x0
d−1
a
dkqα α
α
β
k,χ,q ,
qa χaβ
k,qd
d
dq a0
pX
α
χxdμk,q x
N1 k N
dp −1
dp
px
qα α
χ px qpx β
dpN1
lim N1 k,q
N → ∞ dp
dpN1
q x0
k
N k N
−1
d−1
p qα dkqpα p qdpα p
pa
pxd a
pdx α
χ pa q lim N q β pdpN
k,q
N →∞ p
pdpN
p q dqp a0
qdp x0
k
k
d−1
a
p qα dkqαp p qα α
pa α
χ p β
k,χ,qp .
χ p χaq βk,qpd
d
p q dqp a0
p q
2.16
For β /
1 ∈ X ∗ , one has
pX
α χxdμk,q1/β βx
α
X
χxdμk,q1/β
pkα/β
p
q
α
,
χ
β
β p q1/β k,χ,qp/β
1 α
.
βx χ
β
β k,χ,q1/β
Therefore, we obtain the following theorem.
Theorem 2.5. For β / 1 ∈ X ∗ , one has
X
pX
α
α
χxdμk,q x β
k,χ,q ,
α
χxdμk,q x
k
p qα α
χ p β
k,χ,qp ,
p q
2.17
Abstract and Applied Analysis
pX
X
7
α χxdμk,q1/β βx
α χxdμk,q1/β βx
pkα/β
p
q
α
,
χ
β
β p q1/β k,χ,qp/β
1 α
χ
β
.
β k,χ,q1/β
2.18
Define
−1 k
β qα α α
α
μk,β,q U μk,q U − β−1 −1 μk,q1/β βU .
β q
2.19
By a simple calculation, one gets
X∗
α
χxdμk,β,q x
X
k
β−1 qα
−1 k
β q
α
χxdμk,q x
α
β
k,χ,q
− β−1
k
β−1 qα α
χxμk,q1/β x
−1 β q pX
k
p qα α
− χ p β
k,χ,qp ,
p q
k
1/β qα 1 α
α β
χxdμk,q1/β βx χ
β k,χ,q1/β
1/β q
X∗
p/βkα
p
q
α
−χ
β
p/β .
β
p/β q k,χ,q
2.20
By 2.19 and 2.20, one gets
X∗
α χxdμk,β,q βx
X
α
χxdμk,q x
α
β
k,χ,q
−1 k β qα
α χxμk,q1/β βx
− β −1 β q pX
−1
k
k
1/β qα 1 α
p qα α
1
β
− χ p β
k,χ,qp − χ
β 1/β q
β k,χ,q1/β
p q
2.21
p/βkα
p
q
α
χ
β
p/β .
β
p/β q k,χ,q
Now one defines the operator χy χy,k,α:q on fq by
k
y qα χy f q χy,k,α:q f q χ y f qy .
y q
2.22
8
Abstract and Applied Analysis
Thus, by 2.22, one gets
x,k,α:q
χ
◦χ
y,k,α:q
k
y qα x,k,α:q
f q χ
χ y f qy
y q
k
k
y qα y qαy χ y χx χ y f qxy
y q
y qy
2.23
k
xy qα χ xy f qxy
xy q
χxy,k,α:q f q
χxy f q .
Let us define χx χy χx,k,α:q ◦ χy,k,α:q . Then one has
χx χy χxy .
2.24
From the definition of χx , one can easily derive the following equation;
1
1
1 1/β
1−χ
1 − x1/β − χp xp/β .
1− x
β
β
β
p
2.25
α
Let fq β
k,χ,q . Then one gets
1−χ
p
k 1 1/β qα
1 α
1 1/β α
α
β
βk,χ,q βk,χ,q − −
1− x
χ
β
β 1/β q
β k,χ,q
k
p qα α
χ p β
k,χ,qp
p q
k p α
1 p/β qα
.
β
χ
β p/β q
β k,χ,qp/β
2.26
By 2.21 and 2.26, one obtains the following equation:
1
α α
χxdμk,β,q βx 1 − χp 1 − x1/β β
k,χ,q ,
β
X∗
2.27
where β / 1 ∈ X ∗ .
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Abstract and Applied Analysis
9
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