Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 498173, 7 pages
doi:10.1155/2008/498173
Research Article
A Note on the Multiple Twisted Carlitz’s Type
q-Bernoulli Polynomials
Lee-Chae Jang1 and Cheon-Seoung Ryoo2
1
2
Department of Mathematics and Computer Science, Konkuk University, Chungju 380701, South Korea
Department of Mathematics, Hannam University, Daejeon 306-791, South Korea
Correspondence should be addressed to Lee-Chae Jang, leechae.jang@kku.ac.kr
Received 26 January 2008; Accepted 17 March 2008
Recommended by Ferhan Atici
We give the twisted Carlitz’s type q-Bernoulli polynomials and numbers associated with p-adic qinetgrals and discuss their properties. Furthermore, we define the multiple twisted Carlitz’s type
q-Bernoulli polynomials and numbers and obtain the distribution relation for them.
Copyright q 2008 L.-C. Jang and C.-S. Ryoo. 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.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper Zp , Qp , C, and Cp will, respectively,
be the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number
field, and the p-adic completion of the algebraic closure of Qp . The p-adic absolute value in Cp
is normalized so that |p|p 1/p. When one talks of q-extension, q is variously considered as
an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, one normally
assumes |q| < 1. If q ∈ Cp , one normally assumes that |1−q|p < p−1/p−1 so that qx expx log q
for each x ∈ Zp . We use the notation
xq 1 − qx
,
1−q
x−q 1 − −qx
1q
1.1
cf. 1–20 for all x ∈ Zp . For a fixed odd positive integer d with p, d 1, let
X Xd lim←n Z
dpn Z
,
X1 Zp ,
X∗ 0<a<dp
a,p1
a dpZp ,
a dpn Zp x ∈ X | x ≡ a mod dpn ,
1.2
2
Abstract and Applied Analysis
where a ∈ Z lies in 0 ≤ a < dpn . For any n ∈ N,
qa
μq a dpn Zp n dp q
1.3
is known to be a distribution on X cf. 1–20.
We say that f is uniformly differentiable function at a point a ∈ Zp and denote this
property by f ∈ UDZp , if the difference quotients
fx − fy
x−y
Ff x, y 1.4
have a limit l f a as x, y → a, a cf. 10–13. The p-adic q-integral of a function f ∈
UDZp was defined as
Iq f pn −1
fxdμq x lim n→∞
Zp
1 fxqx .
pn q x0
1.5
By using p-adic q-integrals on Zp , it is well known that
∞
t
et − 1 n0
Zp
xn dμ1 x
tn
,
n!
1.6
where μ1 x pn Zp 1/pn . Then, we note that the Bernoulli numbers Bn were defined as
et
∞
t
tn
Bn ,
− 1 n0 n!
1.7
and hence, we have
Bn Zp
xn dμ1 x
1.8
k
for all n ∈ N ∪ {0}. For k ∈ N and n ∈ N ∪ {0}, the multiple Bernoulli polynomials Bn x were
defined as
t
et − 1
k
ext ∞
k
Bn x
n0
tn
n!
1.9
cf. 2. We note that
t
t
e −1
k
e xt
∞ n0
Zp
···
Zp
k-times
x x1 · · · xk
n
dμ1 x1 · · · dμ1 xk .
1.10
L.-C. Jang and C.-S. Ryoo
3
From 1.9 and 1.10, we obtain
k
Bn x Zp
···
Zp
x x1 · · · xk
n
dμ1 x1 · · · dμ1 xk .
1.11
k-times
In view of 1.11, the multiple Carlitz’s type q-Bernoulli polynomials were defined as
k,q
βn x
Zp
···
Zp
n
x x1 · · · xk q dμq x1 · · · dμq xk .
1.12
k-times
k,q
k,q
In this case, x 0, we write βn 0 βn , which were called the Carlitz’s type q-Bernoulli
numbers. By 1.11 and 1.12, we note that
k,q
limβn
q→1
k,1
Bn
Bnk .
1.13
In Section 2, we give the twisted Carlitz’s type q-Bernoulli polynomials and numbers
associated with p-adic q-inetgrals and discuss their properties. In Section 3, we define the
multiple twisted Carlitz’s type q-Bernoulli polynomials and numbers. We also obtain the
distribution relation for them.
2. Twisted Carlitz’s type q-Bernoulli polynomials
In this section, we assume that q ∈ Cp with |1 − q|p < p−1/p−1 . By using p-adic q-integral on Zp ,
we derive
1
Iq f1 Iq f q
q−1 f 0 q − 1f0 ,
log q
2.1
cf. 8, where f1 x fx 1. From 1.5, we can derive
qq − 1
q Iq fn Iq f log q
n
n−1
i0
f iq log q
i
n−1
fiq
i
,
2.2
i0
cf. 8, where n ∈ N and fn x fx n.
Let Tp n≥1 Cpn limn→∞ Cpn Cp∞ be the locally constant space, where Cpn {w |
n
wp 1} is the cyclic group of order pn . For w ∈ Tp , we denote the locally constant function by
φw : Zp → Cp , x → wx . If we take fx φw x wx , then we have
Zp
etx φw xdμq x log q t qq − 1
q
≡ Fw t.
log q
qwet − 1
2.3
4
Abstract and Applied Analysis
Now we define the twisted q-Bernoulli polynomials as follows:
∞
log q t qq − 1 xt tn
q
q
Fw x, t e
.
B
x
n,w
log q
n!
qwet − 1
n0
q
2.4
q
We note that Bn,w 0 Bn,w are called the twisted q-Bernoulli numbers and by substituting
q
w 1, limq→1 Bn,1 Bn are the familiar Bernoulli numbers. By 2.3, we obtain the following
Witt’s type formula for the twisted q-Bernoulli polynomials and numbers.
Theorem 2.1. For n ∈ N and w ∈ Tp , one has
q
t xn wt dμq t Bn,w x.
Zp
2.5
From 2.5, we consider the twisted Carliz’s type q-Bernoulli polynomials by using padic q-integrals. For w ∈ Tp , we define the twisted Carlitz’s type q-Bernoulli polynomials as
follows:
1
q
βn,w x t xnq wt dμq t.
2.6
1 − q Zp
q
q
When x 0, we write βn,w 0 βn,w which are called twisted Carlitz’s type q-Bernoulli
q
numbers. Note that if w 1, then limq→1 βn,1 Bn . From 2.6, we can see that
n
1
1
n
q
qix −1i
βn,w x .
2.7
n
i
1 − qi1 w
1 − q i0
From 2.7, we can derive the generating function for the twisted Carlitz’s type q-Bernoulli
polynomials as follows:
∞
tn
q
q
Gw x, t βn,w x
n!
n0
∞
n0
∞
n0
n
1
n
1 − q i0
tn
1
n
i
ix
q −1
i
1 − qi1 w n!
n
1
1 − qn i0
∞
tn
n
i
ix
i1l l
q
w
q −1
i
n!
l0
∞
∞
n
ql w l tn
n
i
xli
q
−1
n
i
n!
n0 1 − q i0
l0
2.8
∞ 1 − qxl n n
t
qw
n
n!
n0 1 − q
l0
∞
∞
l
l
ql wl exlq t .
l0
Then it is easily to see that
q
Gw x, t Zp
etxq t wt dμq t.
2.9
L.-C. Jang and C.-S. Ryoo
5
By the kth differentiation on both sides of 2.8 at t 0, we also have
∞
dn q
G
x,
t
ql wl x lnq
w
n
dt
t0
l0
q
βn,w x 2.10
for n ∈ N ∪ {0}. We note that
q
q
βn,w βn,w 0 ∞
ql wl lnq .
2.11
l0
In view of 2.10, we define twisted Carlitz’s type q-zeta function as follows:
q
ζw s, x ∞
ql w l
x lsq
l0
2.12
q
for all s ∈ C and Rex > 0. We note that ζw s, x is analytic function in the whole complex
s-plane. We also have the following theorem in which twisted Carlitz’s type q-zeta functions
interpolate twisted Carlitz’s type q-Bernoulli numbers and polynomials.
Theorem 2.2. For k ∈ N ∪ {0} and w ∈ Tp , one has
q
q
q
q
ζw −k, x βk,w x,
2.13
ζw −k, 0 βk,w .
From 2.11, we obtain the following distribution relation for the twisted q-Bernoulli
polynomials.
Theorem 2.3. For r ∈ N, n ∈ N ∪ {0}, and w ∈ Tp , one has
q
βn,w x rn−1
q
r−1
qr
wi qi βn,wr
i0
ix
.
r
2.14
Proof. If we put i rl j and i 1 · · · r and l 0, 1, . . ., then by 2.11, we have
q
βn,w x ∞
j0
wj qj x jnq
r−1
∞ wirl qirl x i rlnq
l0 i0
1 − qr
1−q
rn−1
q
n r−1
i0
∞
wi qi wrl qrl
l0
r−1
ix
qr
.
wi qi βn,wr
r
i0
1 − qrix/rl
1 − qr
n
2.15
6
Abstract and Applied Analysis
3. Multiple twisted Carlitz’s type q-Bernoulli polynomials
In this section, we consider the multiple twisted Carlitz’s type q-Bernoulli polynomials as
follows:
n
h,q
βk,w x ···
x1 · · · xh x q wx1 ···xh dμq x1 · · · dμq xh
Zp
Zp
3.1
h-times
p
−1
n
1
lim h
x x1 · · · xh q wx1 ···xh qx1 ···xh ,
→∞
p q x1 ···xh 0
h,q
h,q
where h ∈ N, k ∈ N ∪ {0}, and w ∈ Tp . We note that βn,w 0 βn,w are called the multiple
twisted Carlitz’s type q-Bernoulli numbers. We also obtain the generating function of the
multiple twisted Carlitz’s type q-Bernoulli polynomials as follows:
h,q
Gw x, t ···
ex1 ···xh xq t wx1 ···xh dμq x1 · · · dμq xh
Zp
Zp
h-times
∞ l
tl
x1 · · · xh x q wx1 ···xh dμq x1 · · · dμq xh
l!
Zp
l0 Zp
∞
···
3.2
h-times
tl
h,q
βl,w x .
l!
l0
Finally, we have the following distribution relation for the multiple twisted q-Bernoulli
polynomials.
Theorem 3.1. For each w ∈ Tp , h, r ∈ N, n ∈ N ∪ {0}, and w ∈ Tp ,
h,q
βn,w x rn−h
q
r−1
h,qr wj1 ···jh qj1 ···jh βn,wr
j1 ,...,jh 0
x j1 · · · jh
.
r
3.3
Proof. If we put jk rlk xk , jk 0, 1, . . . r − 1, and k 1 · · · h, then by 3.1, we have
n
h,q
βk,w x ···
x1 · · · xh x q wx1 ···xh dμq x1 · · · dμq xh
Zp
Zp
h-times
rp
−1 n
1
lim h
x x1 · · · xh q wx1 ···xh qx1 ···xh
→∞
rp q x1 ···xh 0
−1
p
r−1
n
1
lim rn−h
x j1 rl1 · · · jh rlh q · wj1 rl1 ···jh rlh qj1 rl1 ···jh rlh
q h
→∞
p qr j1 ,...,jh 0 l1 ···lh 0
L.-C. Jang and C.-S. Ryoo
7
n
p
−1 x j1 · · · jh
1
wj1 ···jh qj1 ···jh · lim h
l1 · · · lh wrl1 ···lh qrl1 ···lh →∞
r
qr
p qr l1 ···lh 0
j1 ,...,jh 0
rn−h
q
rn−h
q
r−1
r−1
h,qr wj1 ···jh qj1 ···jh βn,wr
j1 ,...,jh 0
x j1 · · · jh
.
r
3.4
Question 1. Are there the analytic multiple twisted Carlitz’s type q-zeta functions which
interpolate multiple twisted Carlitz’s type q-Bernoulli polynomials?
References
1 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, pp. 987–
1000, 1948.
2 L.-C. Jang, “On a q-analogue of the p-adic generalized twisted L-functions and p-adic q-integrals,”
Journal of the Korean Mathematical Society, vol. 44, no. 1, pp. 1–10, 2007.
3 L.-C. Jang, T. Kim, and D.-W. Park, “Kummer congruence for the Bernoulli numbers of higher order,”
Applied Mathematics and Computation, vol. 151, no. 2, pp. 589–593, 2004.
4 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
5 T. Kim, “On a q-analogue of the p-adic log gamma functions and related integrals,” Journal of Number
Theory, vol. 76, no. 2, pp. 320–329, 1999.
6 T. Kim, “Some formulae for the q-Bernoulli and Euler polynomials of higher order,” Journal of
Mathematical Analysis and Applications, vol. 273, no. 1, pp. 236–242, 2002.
7 T. Kim, “On p-adic q-L-functions and sums of powers,” Discrete Mathematics, vol. 252, no. 1–3, pp.
179–187, 2002.
8 T. Kim, “An invariant p-adic q-integrals on Zp ,” Applied Mathematics Letters, vol. 21, pp. 105–108, 2008.
9 T. Kim and H. S. Kim, “Remark on p-adic q-Bernoulli numbers,” Advanced Studies in Contemporary
Mathematics, vol. 1, pp. 127–136, 1999.
10 T. Kim and J.-S. Cho, “A note on multiple Dirichlet’s q-L-function,” Advanced Studies in Contemporary
Mathematics, vol. 11, no. 1, pp. 57–60, 2005.
11 T. Kim, “Sums of products of q-Bernoulli numbers,” Archiv der Mathematik, vol. 76, no. 3, pp. 190–195,
2001.
12 M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple p-adic q-L-function of two variables,”
Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007.
13 Y. Simsek, “Twisted h, q-Bernoulli numbers and polynomials related to twisted h, q-zeta function
and L-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.
14 Y. Simsek, “The behavior of the twisted p-adic h, q-L-functions at s 0,” Journal of the Korean
Mathematical Society, vol. 44, no. 4, pp. 915–929, 2007.
15 Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted h, qBernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44–
56, 2007.
16 Y. Simsek, “On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers,”
Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006.
17 Y. Simsek, “Twisted h, q-L-functions,” to appear in Journal of Nonlinear Mathematical Physics.
18 Y. Simsek, “p-adic Dedekind and Hardy-Berndt type sum related to Volkenborn integral on Zp ,” to
appear in Journal of Nonlinear Mathematical Physics.
19 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of h, q-twisted Euler
polynomials and numbers,” Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages,
2008.
20 H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order q-Euler
numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages,
2008.