We define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomi-als attached to χ of higher order and investigate some symmetric properties of them.. Furth
Trang 1Volume 2010, Article ID 801580, 13 pages
doi:10.1155/2010/801580
Research Article
A Note on Symmetric Properties of
L.-C Jang,1 H Yi,2 K Shivashankara,3 T Kim,4 Y H Kim,4 and B Lee5
1 Department of Mathematics and Computer Science, KonKuk University,
Chungju 138-70, Republic of Korea
2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3 Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore 570# 005, India
4 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
5 Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
Correspondence should be addressed to H Yi,hsyi@kw.ac.kr
Received 11 September 2009; Revised 14 April 2010; Accepted 31 May 2010
Academic Editor: Abdelkader Boucherif
Copyrightq 2010 L.-C Jang 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 define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomi-als attached to χ of higher order and investigate some symmetric properties of them Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted q-Bernoulli numbers and polynomials and between twisted generalized q-q-Bernoulli numbers and
polynomials
1 Introduction
Let p be a fixed prime number Throughout this paper Z p , Q p, andCpwill, respectively, denote
the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion
of algebraic closure ofQp Let v pbe the normalized exponential valuation ofCpwith|p| p
p −v p p p−1 When one talks of q-extension, q is variously considered as an indeterminate,
a complex number q ∈ C, or a p-adic number q ∈ C p If q ∈ C, one normally assumes
|q| < 1 If q ∈ C p , then we assume |q − 1| p < p −1/p−1 , so that q x expx log q for |x| p ≤ 1
cf 1 32
Trang 2For N, d ∈ N, we set
X X d lim←NZ
see 1 13 The Bernoulli numbers B n and polynomials B n x are defined by the generating
function as
t
e t− 1
∞
n0
B n t n
t
e t− 1e xt
∞
n0
B n x t n
cf 17,18,21,24,26 Let UDX be the set of uniformly differentiable functions on X For
f ∈ UDX, the p-adic invariant integral on Z pis defined as
I
f
X
f xdx lim
N → ∞
1
dp N
dpN−1
x0
Note that
X fxdx
Zp fxdx see 27 Let f n x be a translation with f n x fx n.
We note that
I
f n
If
n−1
i0
cf 1 32 Kim 18 studied the symmetric properties of the q-Bernoulli numbers and
polynomials as follows:
t log q
qe t− 1 e xt
∞
n0
B n q x t n
In this paper, we define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomials attached to χ of higher order and investigate some
symmetric properties of them Furthermore, using these symmetric properties of them, we
can obtain some relationships between the twisted q-Bernoulli numbers and polynomials and between the twisted generalized q-Bernoulli numbers and polynomials attached to χ of
higher order
Trang 32 The Twisted q-Bernoulli Polynomials
Let C p∞ n≥1 C p n limn → ∞ C p n be the locally constant space, where C p n {ξ | ξ p n
1} is the
cyclic group of order p n For w ∈ C p∞, we denote the locally constant function by
cf 2,3,21,24 If we take fx φ w xq x e tx, then
Zp
e xt w x q x dx log q t
Now we define the q-extension of twisted Bernoulli numbers and polynomials as
follows:
log q t
wqe t− 1
∞
n0
B n,w q t n
log q t
wqe t− 1e tx
∞
n0
B q n,w x t n
see 31 From 1.5, 2.2, 2.3, and 2.4, we can derive
Zp
w y q y
x yn
dy B q n,w x,
Zp
w y q y y n dy B q n,w 2.5
By1.5, we can see that
1
log q t
Zp
w nx q nx e nxt dx −
Zp
w x q x e xt dx
w n q n e nt− 1
t log q
Zp
w x q x e xt dx
w n q n e nt− 1
wqe t− 1
n−1
i0
w i q i e it
∞
k0
n−1
i0
i k w i q i
t k
k! .
2.6
Trang 4In1.4, it is easy to show that
1
log q t
Zp
w nx q nx e nxt dx −
Zp
w x q x e xt dx
n
Zp w x q x e xt dx
Zp w nx q nx e nxt dx . 2.7
For each integer k ≥ 0, let
S q k,w n 0 k 1k wq 2 k w2q2 · · · n k w n q n 2.8 From2.6, 2.7, and 2.8, we derive
1
log q t
Zp
w nx q nx e nxt dx −
Zp
w x q x e xt dx
n
Zp w x q x e xt dx
Zp w nx q nx e nxt dx ∞
k0
S q k,w n − 1 t k
k! .
2.9 From2.9, we note that
B q k,w n − B q
k,w kS q
k−1,w n − 1 log qS q
for all k, n ∈ N Let u1, u2 ∈ N and w ∈ C p∞; then we have
Zp w u1x1u2x2q u1x1u2x2e u1x1u2x2dx1dx2
Zp w u1u2x q u1u2x e u1u2xt dx t log q w u1u2q u1u2e u1t− 1
w u2q u2e u2t− 1 . 2.11
By2.9, we see that
u1
Zp w x q x e xt dx
Zp w u1x q u1x e u1xt dx ∞
l0
u
1 −1
k0
k l w k q k
t l
l!∞
l0
S q l,w u1− 1t l
Let T w u1, u2; x, t be as follows:
T w u1, u2; x, t
Zp w u1x1u2x2q u1x1u2x2e u1x1u2x2u1u2xt dx1dx2
Zp w u1u2x q u1u2x e u1u2xt dx . 2.13 Then we have
T w u1, u2; x, t
t log q
e u1u2t
w u1u2q u1u2e u1u2t− 1
w u1q u1e u1t− 1w u2q u2e u2t− 1 2.14
Trang 5From2.13, we derive
T w u1, u2; x, t
1
u1
Zp
w u1x1q u1x1e u1x1u2xt dx1
⎛
⎝ u1
Zp w u2x2q u2x2e u2x2t
Zp w u1u2x q u1u2x e u1u2xt dx
⎞
⎠ 2.15
By2.4, 2.12, and 2.15, we can see that
T w u1, u2; x, t 1
u1
∞
i0
B i,w q u1 u1 u2xu i1t i
i!
∞
l0
S q l,w u2 u2 u1− 1u l2t l
l!
∞
n0
n
i0
n i
B q i,w u1 u1 u2x S q n−i,w u2 u2 u1− 1u i−1
1 u n−i2
t n n! .
2.16
By the symmetry of p-adic invariant integral on Z p, we also see that
T w u1, u2; x, t
1
u2
Zp
w u2x2q u2x2e u2x2u1xt dx2
⎛
⎝ u2
Zp w u1x1q u1x1e u1x1t
Zp w u1u2x q u1u2x e u1u2xt dx
⎞
⎠
∞
n0
n
i0
n i
B q i,w u2 u2 u1x S q u1
n−i,w u1 u2− 1u i−1
2 u n−i1
t n
n! .
2.17
By comparing the coefficients of tn /n! on both sides of 2.16 and 2.17, we obtain the following theorem
Theorem 2.1 Let u1, u2, n ∈ N Then for all x ∈ Z p ,
n
i0
n
i
B i,w q u1 u1 u2x S q u2
n−i,w u2 u1− 1u i−1
1 u n−i2 n
i0
n i
B i,w q u2 u2 u1x S q u1
n−i,w u1 u2− 1u i−1
2 u n−i1 , 2.18
wheren
i is the binomial coefficient.
FromTheorem 2.1, if we take u2 1, then we have the following corollary
Corollary 2.2 For m ≥ 0, one we has
B q i,w u1x n
i0
n i
B i,w q u1 u1 xS q
n−i,w u1− 1u i−1
wheren
i is the binomial coefficient.
Trang 6By2.17, 2.18, and 2.19, we can see that
T w u1, u2; x, t
e u1u2xt
u1
Zp
w u1x q u1x1e u1x1t dx1
⎛
⎝u1
Zp w u2x2q u2x2e u2x2t dx2
Zp w u1u2x q u1u2x e u1u2xt dx
⎞
⎠
e u1u2xt
u1
Zp
w u1x q u1x1e u1x1t dx1
u
1 −1
i0
w u2i q u2i e u2it
1
u1
u1 −1
i0
w u2i q u2i
Zp
w u1x q u1x e x1u2xu2/u1itu1dx1
∞
n0
u1 −1
i0
B q n,w u1 u1
u2x u2
u1i
u n−11 w u2i q u2i t n
n! .
2.20
From the symmetry of T w u1, u2; x, t, we can also derive
T w u1, u2; x, t ∞
n0
u2 −1
i0
B q n,w u2 u2
u1x u1
u2i
u n−12 w u1i q u1i t n
By comparing the coefficients of tn /n! on both sides of 2.20 and 2.21, we obtain the following theorem
Theorem 2.3 For m ∈ Z, u1, u2∈ N, we have
u1 −1
i0
B n,w q u1 u1
u2x u2
u1i
u n−11 w u2i q u2iu2−1
i0
B n,w q u2 u2
u1x u1
u2i
u n−12 w u1i q u1i 2.22
We note that by setting u2 1 in Theorem 2.3, we get the following multiplication
theorem for the twisted q-Bernoulli polynomials.
Theorem 2.4 For m ∈ Z, u1∈ N, one has
B n,w q u1x u n−1
1
u1 −1
i0
B n,w q u1 u1
x i
u1
Remark 2.5. 18, Kim suggested open questions related to finding symmetric properties for
Carlitz q-Bernoulli numbers In this paper, we give the symmetric property for q-Bernoulli
numbers in the viewpoint to give the answer of Kim’s open questions
3 The Twisted Generalized Bernoulli Polynomials
In this section, we consider the generalized Bernoulli numbers and polynomials and then
define the twisted generalized Bernoulli polynomials attached to χ of higher order by using
Trang 7multivariate p-adic invariant integrals on Z p Let χ be Dirichlet’s character with conductor
d ∈ N Then the generalized Bernoulli numbers B n,χ and polynomials B n,χ x attached to χ
are defined as
td−1
a0 χ ae at
e dt− 1
∞
n0
B n,χ t n
td−1
a0 χ ae at
e dt− 1 e xt
∞
n0
B n,χ x t n
cf 2,18,23,27
Let C p∞ n≥1 C p n limn → ∞ C p n be the locally constant space, where C p n {w | w p n
1} is the cyclic group of order pn For w ∈ C p∞, we denote the locally constant function by
cf 2,3,21,23,24 If we take fx χxe tx φ w xq x , for q ∈ C pwith|q − 1| p < 1, then it is
obvious from3.1 that
X
χ xe tx w x q x dx
t log q d−1
a0 χ aw a q a e at
Now we define the twisted generalized Bernoulli numbers B q n,χ,w and polynomials B n,χ,w q x attached to χ as follows:
t log q d−1
a0 χ aw a q a e at
w d q d e dt− 1
∞
n0
B q n,χ,w t n
t log q d−1
a0 χ aw a q a e at e xt
∞
n0
B n,χ,w q x t n
for each w ∈ C p∞see 31,32 By 3.5 and 3.6,
X
χ xx n w x q x dx B n,χ,w q ,
X
χ
y
x yn
w y q y dy B q n,χ,w x.
3.7
Trang 8Thus we have
1
log q t
X
χ xe ndxt w nx q nx dx −
X
χ xe xt w x q x dx
nd
X χ xe xt w x q x dx
X e ndxt w ndx q ndx dx
w nd q nd e ndt− 1
w d q d e dt− 1
d−1
i0
χ ie it w i q i
3.8
Then
1
log q t
X
χ xe ndxt w nx q nx dx −
X
χ xe xt w x q x dx
nd−1
l0
χ le lt w l q l∞
k0
nd−1
l0
χ ll k w l q l t
k
k! .
3.9
Let us define the p-adic twisted q-function T k,w q χ, n as follows:
T k,w q
χ, n
n
l0
By3.9 and 3.10, we see that
1
log q t
X
χ xe ndxt w ndx q ndx dx −
X
χ xe xt w x q x dx
∞
k0
T k,w q
χ, nd − 1 t k
k! . 3.11
Thus,
X
χ xnd x k
w nx q nx dx −
X
χ xx k w x q x dx
t log q
T k,w q
χ, nd − 1
, 3.12
for all k, n, d ∈ N This means that
B k,χ,w q nd − B q
n,χ,wt log q
T k,w q
χ, nd − 1
Trang 9for all k, n, d ∈ N For all u1, u2, d ∈ N, we have
d
X
X χ x1χx2e w1x1w2x2t w u1x1u2x2q u1x1u2x2dx1dx2
X e du1u2xt w du1u2x q du1u2x dx
t log q
e du1u2t w du1u2q du1u2− 1
e du1t w du1q du1− 1e du2t w du2q du2− 1
×
d−1
a0
χ ae u1at w u1a q u1a
d−1
b0
χ be u2bt w u2b q u2b
.
3.14
The twisted generalized Bernoulli numbers B k,q n,χ,w and polynomials B n,χ,w k,q x attached
to χ of order k are defined as
t log q d−1
a0 χ aw a q a e at
w d q d e dt− 1
k
∞
n0
B k,q n,χ,w t n
t log q d−1
a0 χ aw a q a e at
w d q d e dt− 1
k
e xt∞
n0
B n,χ,w k,q x t n
for each w ∈ C p∞ For u1, u2∈ N, we set
K w q
m, χ; u1, u2
d
X m
m
i1 χ x i em
i1 x i u2xu1t wm i1 x i u2xu1qm i1 x i u2xu1dx1· · · dx m
X e du1u2xt w du1u2x q du1u2x dx
×
X m
m
i1
χ x i em
i1 x i u1yu2t wm i1 x i u1yu2qm i1 x i u1yu1dx1· · · dx m ,
3.17
where
X m fx1· · · x m dx1· · · dx m X· · ·X fx1, , x m dx1· · · dx m In3.17, we note that
Kq w m, χ; u1, u2 is symmetric in u1, u2 From3.17, we have
K q w
m, χ; u1, u2
X m
m
i1
χ x i em
i1 x i u2t wm i1 x i u2qm i1 x i u2dx1· · · dx m
× e u1u2xt w u1u2x q u1u2x
d
X χ x m e u2x m t w u2x m q u2x m dx m
X e du1u2x q du1u2x dx
×
X m−1
m−1
i1
χ x i em−1
i1 x i u2t wm−1 i1 x i u2qm−1 i1 x i u2dx1· · · dx m−1
× e u1u2yt w u1u2y q u1u2y
3.18
Trang 10Thus we can obtain
u1d
X χ xe xt w x q x dx
X e du2xt w du2x q du2x dx ∞
k0
u
1d−1
i0
χ ii k w i q i
t k
k! ∞
k0
T k,w q
χ, u1d − 1 t k
k! ,
e u1u2xt w u1u2x q u1u2x
X m
m
i1
χ x i em
i1 x i u1t wm i1 x i u1qm i1 x i u1dx1· · · dx m
e u1u2xt w u1u2x q u1u2x
u1
e du1t w du1q du1− 1
d−1
a0
χ ae u1at w u1a q u1a
∞
n0
B m,q n,χ,w u2x u n
1
t n
n! .
3.19
From3.19, we derive
K q w
m, χ; u1, u2
∞
l0
B m,q l,χ,w u1x u l
1
t l
l!
∞
k0
T k,w q
χ, u1d − 1 t k
k!
∞
i0
B m−1,q i,χ,w
u1y u i
2t i
i!
1
u1
∞
n0
n
j0
n j
u j2u n−j−11 B n−j,χ,w m,q u2x ×
j
k0
T k,w q
χ, u1d − 1 j
k
B j−k,χ,w m−1,q
u1y t n
n! .
3.20
By the symmetry of K q w m, χ; u1, u2 in u1and u2, we can see that
K q w
m, χ; u1, u2
∞
n0
n
j0
n j
u j1u n−j−12 B n−j,χ,w m,q u1x ×
j
k0
T k,w q
χ, u2d − 1 j
k
B j−k,χ,w m−1,q
u2y t n
n! .
3.21
By comparing the coefficients on both sides of 3.20 and 3.21, we see the following theorem
Theorem 3.1 For d, u1, u2, m ∈ N, n ∈ Z, one has
n
j0
n
j
u j2u n−j−11 B m,q n−j,χ,w u2x
j
k0
T k,w q
χ, u1d − 1 j
k
B j−k,χ,w m−1,q
u1y
n
j0
n j
u j1u n−j−12 B n−j,χ,w m,q u1xj
k0
T k,w q
χ, u2d − 1 j
k
B j−k,χ,w m−1,q
u2y
.
3.22
Trang 11Remark 3.2 If we take y 0 and m 1 in 3.22, then we have
n
j0
n j
u j2u n−j−11 B n−j,χ,w q u2x
j
k0
T k,w q
χ, u1d − 1 j
k
n
j0
n j
u j1u n−j−12 B q n−j,χ,w u1x
j
k0
T k,w q
χ, u2d − 1 j
k
.
3.23
Now we can also calculate
K q w
m, χ; u1, u2
∞
n0
n
k0
n k
u k−11 u n−k2 B n−k,χ,w m−1,q
u1ydu1 −1
i0
B i,χ,w m,q
u2x u2
u1i
t n
n! . 3.24
From the symmetric property of K w q m, χ; u1, u2 in u1and u2, we derive
K q w
m, χ; u1, u2
∞
n0
n
k0
n k
u k−12 u n−k1 B n−k,χ,w m−1,q
u2ydu2 −1
i0
B i,χ,w m,q
u1x u1
u2i
t n
n! . 3.25
By comparing the coefficients on both sides of 3.24 and 3.26, we obtain the following theorem
Theorem 3.3 For d, u1, u2, m ∈ N, n ∈ Z, we have
n
k0
n k
u k−11 u n−k2 B n−k,χ,w m−1,q
u1ydu1 −1
i0
B m,q k,χ,w
u2x u2
u1i
n
k0
n k
u k−12 u n−k1 B m−1,q n−k,χ,w
u2ydu2 −1
i0
B k,χ,w m,q
u1x u1
u2i
.
3.26
Remark 3.4 If we take y 0 and m 1 in 3.26, then one has
u n−11
du1 −1
i0
B q n,χ,w
u2x u2
u1i
u n−1
2
du2 −1
i0
B n,χ,w q
u1x u1
u2i
Remark 3.5 In our results for q 1, we can also derive similar results, which were treated
in 27 In this paper, we used the p-adic integrals to derive the symmetric properties of the q-Bernoulli polynomials By using the symmetric properties of p-adic integral on X, we
can easily derive many interesting symmetric properties related to Bernoulli numbers and polynomials
Trang 12The authors express Their sincere gratitude to referees for their valuable suggestions and comments This work has been conducted by the Research Grant of Kwangwoon University
in 2010
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