Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 392025, 14 pages
doi:10.1155/2011/392025
Research Article
On the q-Bernoulli Numbers and
Polynomials with Weight α
T. Kim and J. Choi
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to J. Choi, jeschoi@kw.ac.kr
Received 13 May 2011; Accepted 26 July 2011
Academic Editor: Elena Litsyn
Copyright q 2011 T. Kim and J. Choi. 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 present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α. From these studies, we derive some interesting identities on the q-Bernoulli
numbers and polynomials with weight α.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp will denote the
ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of
algebraic closure of Qp , respectively. The p-adic norm of Cp is defined as |x|p p−r , where
x pr m/n with p, m p, n 1, r ∈ Q and m, n ∈ Z. Let N and Z be the set of natural
numbers and integers, respectively, Z N ∪ {0}. Let q ∈ Cp with |1 − q|p < p−1/p−1 . The
notation of q-number is defined by xw 1 − wx /1 − w and xq 1 − qx /1 − q, see
1–13.
As the well known definition, the Bernoulli polynomials are defined by
∞
tn
t
x
e
Bn x .
t
n!
e −1
n0
1.1
In the special case, x 0, Bn 0 Bn are called the nth Bernoulli numbers. That is, the recurrence formula for the Bernoulli numbers is given by
2
Abstract and Applied Analysis
n
B0 1,
B 1 − Bn ⎧
⎨1
if n 1,
⎩0
if n > 1,
1.2
with the usual convention about replacing Bi with Bi .
In 1, 2, q-extension of Bernoulli numbers are defined by Carlitz as follows:
⎧
⎨1
n
q qβ 1 − βn,q ⎩0
β0,q 1,
if n 1,
if n > 1,
1.3
with the usual convention about replacing βi with βi,q .
By 1.2 and 1.3, we get limq → 1 βi,q Bi . In this paper, we assume that α ∈ N.
In 7, the q-Bernoulli numbers with weight α are defined by Kim as follows:
⎧ α
⎨
n
α
q qα βα 1 − βn,q αq
⎩
0
α
β0,q 1,
if n 1,
1.4
if n > 1,
α
with the usual convention about replacing βα i with βi,q .
Let UDZp be the space of uniformly differentiable functions on Zp . For f ∈ UDZp ,
the p-adic q-integral on Zp is defined as
Iq f Zp
fxdμq x lim N →∞
1
pN
N
−1
p
fxqx ,
1.5
q x0
see4, 5. From 1.5, we note that
n−1
n−1
q − 1
qn Iq fn Iq f q − 1
ql fl ql f l,
log
q
l0
l0
1.6
where fn x fx n and f l dfx/dx|xl .
By 1.4, 1.5, and 1.6, we set
α
βn,q Zp
xnqα dμq x,
where n ∈ Z ,
1.7
see7. The q-Bernoulli polynomials are also given by
α
βn,q x n
n
n
αlx α
βl,q .
x y qα dμq x xn−l
qα q
l
Zp
l0
1.8
Abstract and Applied Analysis
3
The purpose of this paper is to derive a new concept of higher-order q-Bernoulli
numbers and polynomials with weight α from the fermionic p-adic q-integral on Zp . Finally,
we present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α.
2. Higher Order q-Bernoulli Numbers with Weight α
Let β ∈ Z and α ∈ N in this paper. For k ∈ N and n ∈ Z , we consider the expansion of
higher-order q-Bernoulli polynomials with weight α as follows:
β,k|α
βn,q x Zp
···
Zp
x1 · · · xk xnqα qx1 β−1···xk β−k dμq x1 · · · dμq xk .
2.1
From 2.1, we note that
β,k|α
βn,q x 1−q
k−n
αnq
n
n
1
n
1 − q αnq
−1 q
l
l0
k−1 l αlx
n
n
l0
−1l qαlx l
i0 αl β − i
k−1 αlβ−i
i0 1 − q
αlβ
k
αlβ
k
2.2
q
k!
kq !
,
where αl q 1 − qα 1 − qα−1 · · · 1 − qα−l1 /1 − q1 − q2 · · · 1 − ql and kq ! kq · · · 2q 1q .
Therefore, we obtain the following theorem.
Theorem 2.1. For n ∈ Z and k ∈ N, we have
1
n
1 − q αnq
β,k|α
βn,q x n
n
l0
l
−1l qαlx αlβ
k
αlβ
k
q
k!
kq !
.
2.3
β,k|α
β,k|α
In the special case, x 0, βn,q 0 βn,q are called the nth higher order q-Bernoulli
numbers with weight α.
4
Abstract and Applied Analysis
From 2.1 and 2.2, we can derive
β−α,k|α
β,k|α
β−α,k|α
βn,q
qα − 1 βn1,q
βn,q
.
2.4
By Theorem 2.1 and 2.4, we get
mα,k|α
β0,q
···
Zp
k
j1 αm−jxj
m
m l
qα − 1
l
l0
Zp
q
dμq x1 · · · dμq xk Zp
···
Zp
x1 · · · xk lqα q−
k
j1
jxj
dμq x1 · · · dμq xk 2.5
m
m l 0,k|α
qα − 1 βl,q
l
l0
1−q
k−1 k
1 − qαm−k1i
i0
1−q
k
kl−1
k l
l0
qαm−k1l .
q
From 2.1, we have
i
i j0
j
j
q −1
α
Zp
···
Zp
Zp
···
n−ij β−α−1x1 ···β−α−kxk
Zp
x1 · · · xk qα
q
dμq x1 · · · dμq xk β−1x1 ···β−kxk αx1 ···xk i−1
q
dμq x1 · · · dμq xk x1 · · · xk n−i
qα q
2.6
i−1 i − 1
j0
j β,k|α
qα − 1 βn−ij,q .
j
Thus, we obtain the following theorem.
Theorem 2.2. For i ∈ N, we have
i
i j0
j
i−1 i − 1
j β−α,k|α j β,k|α
α
q − 1 βn−ij,q q − 1 βn−ij,q .
j
j0
α
It is easy to show that
m k αm−k1l
m
j 0,k|α
α
kl−1
1 − qk
q
j q − 1 βj,q
l
q
j0
l0
k−1
i0
1 − qk
1 − qαm−k1i .
2.7
Abstract and Applied Analysis
3. Polynomials βn,q
0,k|α
5
x
0,k|α
In this section, we consider the polynomials βn,q x as follows:
0,k|α
βn,q x Zp
···
Zp
x1 · · · xk xnqα q−
k
j1
jxj
dμq x1 · · · dμq xk .
3.1
From 3.1, we can easily derive the following equation:
k n k
n
1−q αl − i
0,k|α
l αlx
βn,q x −1 q k−1i0
n
α
αl−i
1−q
l
l0
i0 1 − q
1
n
1 − q αnq
n
n
l0
l
αl −1l qαlx αl k
k
q
k!
kq !
3.2
.
By 3.1 and 3.2, we get
Zp
···
Zp
q
k
j1 αn−jxj αnx
dμq x1 · · · dμq xk n
n
l0
Zp
···
Zp
q
k
l1 αn−lxl αnx
l
l 0,k|α
αlq q − 1 βl,q x,
3.3
dμq x1 · · · dμq xk k k−1 qαnx 1 − q
j0 αn − j
qαnx αn k!
.
αn k
k−1 αn−j
k q kq !
j0 1 − q
3.4
Therefore, by 3.3 and 3.4, we obtain the following theorem.
Theorem 3.1. For n ∈ Z , we have
1−q
n
0,k|α
βn,q x
αl n
n
k!
1 l αlx
.
−1 q αl k
n
αq l0 l
k q kq !
3.5
Moreover,
n
n
l0
l
l 0,k|α
qαnx αn k!
.
αlq q − 1 βl,q x αn k
k q kq !
3.6
6
Abstract and Applied Analysis
Let d ∈ N. Then, we have
Zp
···
Zp
x x1 · · · xk nqα q−
dnqα
d−1
q−
k
j1
jxj
dμq x1 · · · dμq xk k
j2 j−1aj
3.7
dkq a1 ,··· ,ak 0
×
⎡
Zp
···
⎣
x
Zp
k
j1 aj
d
⎤n
k
k
xi ⎦ q−d j1 jxj dμqd x1 · · · dμqd xk .
i1
qαd
Thus, by 3.1 and 3.7, we obtain the following theorem.
Theorem 3.2. For d, k ∈ N, and n ∈ Z , we have
0,k|α
βn,q x dnqα
d−1
dkq
a1 ,...,ak 0
q−
k
j2 j−1aj
0,k|α x a1 · · · ak
.
βn,qα
d
3.8
From 3.1, we note that
0,k|α
βn,q x n
n
αlx βl,q
xn−l
qα q
0,k|α
,
l
n
n n−l αlx 0,k|α
0,k|α
βn,q
xy y qα q βl,q x.
l
l0
l0
4. Polynomials βn,q
h,1|α
3.9
x
h,1|α
For h ∈ Z, let us define weighted h, q-Bernoulli polynomials βn,q x as follows:
h,1|α
βn,q x Zp
x x1 nqα qx1 h−1 dμq x1 .
4.1
By 4.1, we easily see that
h,1|α
βn,q x 1
αnq 1
−q
n
n
n
l0
l
Therefore, by 4.2, we obtain the following theorem.
−1l qαlx
αl h
.
αl hq
4.2
Abstract and Applied Analysis
7
Theorem 4.1. For h ∈ Z and n ∈ Z , we have
h,1|α
βn,q x 1
αnq 1
−q
n
n
n
l0
−1l qαlx
l
αl h
.
αl hq
4.3
From 4.1, we can derive the following equation:
qαx
Zp
x x1 nqα qx1 h−1 dμq x1 q −1
α
Zp
x x1 h−α−1
dμq x1 x1 n1
qα q
4.4
Zp
x x1 nqα qx1 h−α−1 dμq x1 .
By 4.4, we easily get
qαx βn,q
h,1|α
h−α−1,1|α
h−α−1,1|α
x qα − 1 βn1,q
x βn,q
x.
4.5
From 4.1, we have
h,1|α
βn,q x
Zp
x x1 nqα qx1 h−1 dμq x1 n
n
l0
l
αlx βl,q
xn−l
qα q
h,1|α
,
4.6
h,1|α
h,1|α
where βl,q 0 βl,q .
By 4.6, we get the following recurrence formula:
n
h,1|α
h,1|α
xqα ,
βn,q x qαx βq
with the usual convention about replacing βq
From 1.6, we note that
for n ≥ 1,
with βn,q
h,1|α n
h,1|α
4.7
.
q−1 f 0.
qIq f1 Iq f q − 1 f0 log q
4.8
For h ∈ Z , by 4.8, we have
qh
Zp
fx 1qh−1x dμq x q−1 f 0.
fxdμq x q − 1 hf0 log
q
Zp
4.9
If h ∈ {−1, −2, −3, . . .}, then we get
qh
Zp
fx 1qh−1x dμq x q−1 f 0.
fxdμq x 1 − q hf0 log q
Zp
4.10
8
Abstract and Applied Analysis
Let h ∈ Z . By 4.9, we get
qh
Zp
x x1 1nqα qh−1x1 dμq x1 −
x x1 nqα qh−1x1 dμq x1 Zp
4.11
α
αx
q − 1 hxnqα xn−1
qα q .
αq
From 4.6 and 4.11, we note that
qh βn,q
h,1|α
x 1 − βn,q
h,1|α
α
αx
x q − 1 hxnqα n
xn−1
qα q .
αq
4.12
If we take x 0 in 4.12, then we have
h,1|α
β0,q
h
,
hq
h,1|α
qh βn,q 1
−
h,1|α
βn,q
⎧ α
⎨
αq
⎩
0
if n 1,
4.13
if n > 1.
Therefore, by 4.12 and 4.13, we obtain the following theorem.
Theorem 4.2. For h ∈ Z , we have
h,1|α
β0,q
h
,
hq
qh βn,q
h,1|α
h,1|α
1 − βn,q
⎧ α
⎨
αq
⎩
0
if n 1,
4.14
if n > 1.
By 4.7 and Theorem 4.2, we obtain the following corollary.
Corollary 4.3. For h ∈ Z , we have
h,1|α
β0,q
h
,
hq
qh
⎧ α
⎨
n
h,1|α
h,1|α
qα βq
1 − βn,q αq
⎩
0
n
h,1|α
h,1|α
with the usual convention about replacing βq
with βn,q .
if n 1,
if n > 1,
4.15
Abstract and Applied Analysis
9
From 4.1, we have
h,1|α
β0,q
qx1 h−1 dμq x1 Zp
h
,
hq
if h ∈ Z .
4.16
It is not difficult to show that
h,1|α
βn,q−1 1 − x Zp
1 − x x1 nq−α q−x1 h−1 dμq−1 x1 −1n qαnh−1
Zp
−1n qαnh−1 βn,q
x x1 qα qx1 h−1 dμq x1 h,1|α
4.17
x.
Therefore, by 4.17, we obtain the following theorem.
Theorem 4.4. For h, n ∈ Z , we have
h,1|α
h,1|α
βn,q−1 1 − x −1n qαnh−1 βn,q x.
4.18
For x 1 in Theorem 4.4, we get
h,1|α
h,1|α
βn,q−1 −1n qαnh−1 βn,q 1
h,1|α
−1n qαn−1 βn,q
4.19
if n > 1.
Therefore, by 4.19, we obtain the following corollary.
Corollary 4.5. For h ∈ Z and n ∈ N with n > 1, we have
h,1|α
h,1|α
βn,q−1 −1n qαn−1 βn,q .
4.20
Let d ∈ N. By 4.1, we see that
Zp
qh−1x1 x x1 nqα dμq x1 d−1
n
dnqα xa
x1
qha
qx1 h−1d dμqα x1 .
d
dq a0
qαd
Zp
4.21
10
Abstract and Applied Analysis
By 4.1 and 4.21, we obtain the following equation:
h,1|α
βn,q x d−1
dnqα dq
qha βn,q
h,1|α
x a
a0
d
,
4.22
where d ∈ N and h ∈ Z .
h,k|α
5. Polynomials βn,q x and h k
From 2.1, we note that
h,k|α
βn,q x Zp
···
Zp
x x1 · · · xk nqα qh−1x1 ···h−kxk dμq x1 · · · dμq xk n
n
1
αl h · · · αl h − k 1
,
−1l qαlx
n n
hq · · · αl h − k 1q
αl
1 − q αq l0 l
qh
Zp
···
Zp
x 1 x1 · · · xk nqα qh−1x1 ···h−kxk dμq x1 · · · dμq xk Zp
···
Zp
x x1 · · · xk nqα qh−1x1 ···h−kxk dμq x1 · · · dμq xk q−1 h
n
5.1
α αx
q
αq
Zp
···
Zp
x x2 · · · xk nqα qh−1x2 ···h−kxk dμq x2 · · · dμq xk Zp
···
Zp
x x2 · · · xk n−1
qα
× qαh−1x2 ···h−k1xk dμq x2 · · · dμq xk .
5.2
From 5.1, we have
qh βn,q
h,k|α
h−1,k−1|α
nα h,k−1|α
h,k|α
βn,q
x 1 βn,q x q − 1 hβn,q
x qαx
x.
αq
5.3
Therefore, by 5.3, we obtain the following theorem.
Theorem 5.1. For h, n ∈ Z and k ∈ N, we have
qh βn,q
h,k|α
h−1,k−1|α
α h,k−1|α
h,k|α
βn,q
x 1 − βn,q x q − 1 hβn,q
x nqαx
x.
αq
5.4
Abstract and Applied Analysis
11
It is easy to show that
qαx
Zp
···
Zp
x x1 · · · xk nqα qhx1 h−1x2 ···h1−kxk dμq x1 · · · dμq xk qα − 1
Zp
···
Zp
Zp
···
Zp
h−αx1 ···h−α1−kxk
dμq x1 · · · dμq xk x x1 · · · xk n1
qα q
5.5
x x1 · · · xk nqα qh−αx1 ···h−α1−kxk dμq x1 · · · dμq xk h1−α,k|α
h1−α,k|α
qα − 1 βn1,q
x βn,q
x.
Thus, by 5.5, we obtain the following proposition.
Proposition 5.2. For h, n ∈ Z , we have
qαx βn,q
h1,k|α
h1−α,k|α
h1−α,k|α
x qα − 1 βn1,q
x βn,q
x.
5.6
For d ∈ N, we get
Zp
⎡
···
Zp
⎣x k
⎤n
xj ⎦ q
j1
dnqα
d−1
k
j1 h−jxj
dμq x1 · · · dμq xk qα
qh
k
j1
aj −
k
j2 j−1aj
5.7
dkq a1 ,...,ak 0
×
Zp
···
⎡
⎣
x
Zp
k
j1
aj
d
⎤n
k
k
xj ⎦ qd j1 h−jxj dμqd x1 · · · dμqd xk .
j1
qαd
Thus, we obtain
h,k|α
βn,q x dnqα
d−1
dkq
a1 ,...,ak 0
qh
k
j1
aj −
k
j2 j−1aj
h,k|α x a1 · · · ak
.
βn,qd
d
5.8
Equation 5.8 is multiplication formula for the q-Bernoulli polynomials of order h, k with
weight α.
12
Abstract and Applied Analysis
k,k|α
k|α
Let us define βn,q x βn,q x. Then we see that
n
n
1
αl k · · · αl 1
,
−1l qαlx
n n
αl kq · · · αl 1q
1 − q αq l0 l
k|α
βn,q x Zp
···
Zp
5.9
k − x x1 · · · xk nq−α q−k−1x1 −···−k−kxk dμq−1 x1 · · · dμq−1 xk αn−k k1
n
n
2
−1n q
n n
1 − q αq
−1n q
l0
l
−1l qαlx
αl k · · · αl 1
αl kq · · · αl 1q
5.10
αn−k k1 k|α
2 βn,q x.
Therefore, by 5.9 and 5.10, we obtain the following theorem.
Theorem 5.3. For n ∈ Z and k ∈ N, we have
αn−k
k|α
βn,q−1 k − x −1n q
k1
2
k|α
βn,q x.
5.11
Let x k in Theorem 5.3. Then we see that
αn−k
k|α
βn,q−1 −1n q
k1
2
k|α
βn,q k.
5.12
From Theorem 5.1, we can derive the following equation:
k−1|α
α k,k−1|α
k|α
k|α
βn,q
qk βn,q x 1 − βn,q x k q − 1 βn,q x nqαx
x.
αq
5.13
By 5.9, we easily get
n
n
1
αl k · · · αl 1
.
−1l
n n
kq · · · αl 1q
αl
1 − q αq l0 l
k|α
βn,q 5.14
From the definition of p-adic q-integral on Zp , we note that
n
n l0
l
l
qα − 1
Zp
···
Zp
x1 · · · xk lqα q
αn k!
αn k · · · αn 1
αn k
.
αn kq · · · αn 1q k q kq !
k
l0 k−lxl
dμq x1 · · · dμq xk 5.15
Abstract and Applied Analysis
13
Thus, by 5.15, we get
n
n l0
l
l k|α
qα − 1 βn,q αn
k k!
.
αn
k q kq !
5.16
k|α
By the definition of polynomial βn,q x, we see that
k|α
βn,q x Zp
···
Zp
n
n
l0
l
x x1 · · · xk nqα qk−1x1 ···k−kxk dμq x1 · · · dμq xk k|α
qαlx βl,q xn−l
q
k|α
qαx βq
xqα
n
5.17
,
where n ∈ Z ,
n
k|α
k|α
with the usual convention about replacing βq with βn,q .
Let x 0 in 5.13. Then we have
k−1|α
nα k,k−1|α
k|α
k|α
qk βn,q 1 − βn,q k q − 1 βn,q
.
βn,q
αq
5.18
Acknowledgments
The present research has been conducted by the Research Grant of Kwangwoon University
in 2011. The authors would like to express their gratitude to referees for their valuable
suggestions and comments.
References
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2 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000,
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