R E S E A R C H
Open Access
A note on Carlitz
q
-Bernoulli numbers and
polynomials
Daeyeoul Kim
1and Min-Soo Kim
2** Correspondence: [email protected]
2Division of Cultural Education,
Kyungnam University, Changwon 631-701, South Korea
Full list of author information is available at the end of the article
Abstract
In this article, we first aim to give simple proofs of known formulae for the
generalized Carlitzq-Bernoulli polynomialsbm,c(x, q) in thep-adic case by means of a method provided by Kim and then to derive a complex, analytic, two-variableq-L -function that is aq-analog of the two-variableL-function. Using this function, we calculate the values of two-variableq-L-functions at nonpositive integers and study their properties whenqtends to 1.
Mathematics Subject Classification (2000): 11B68; 11S80.
Keywords:Carlitz q-Bernoulli numbers, Carlitzq-Bernoulli polynomials, Dirichlet q-L-functions
1. Introduction
Letpbe a fixed prime. We denote byℤp,ℚp, andℂpthe ring ofp-adic integers, the field
ofp-adic numbers, and the completion of the algebraic closure ofℚp, respectively. Let
vpbe the normalized exponential valuation ofℂpwithpp=p−vp(p)=p−1. When one
talks of aq-extension, qcan be variously considered as an indeterminate, a complex numberqÎ ℂ, or ap-adic numberq Îℂp. Ifq Îℂ, one normally assumes |q| < 1.
IfqÎℂp, one normally assumes |1 -q|p<p-1/(p-1), so thatqx= exp(xlogpq) for |x|p≤1.
Letdbe a fixed positive integer. Let
X=Xd= lim←− N
(Z/dpNZ), X1=Zp, X∗= ∪
0<a<dp (a,p)=1
a+dpZp,
a+dpNZp={x∈X|x≡a (moddpN)},
(1:1)
whereaÎℤlies in 0≤a<dpN. We use the following notation:
[x]q=
1−qx
1−q. (1:2)
Hence limq®1 [x]q =xfor anyxÎ ℂin the complex case and anyxwith |x|p≤ 1 in
the presentp-adic case. This is the hallmark of aq-analog: The limit asq®1 recovers the classical object.
In 1937, Vandiver [1] and, in 1941, Carlitz [2] discussed generalized Bernoulli and Euler numbers. Since that time, many authors have studied these and other related
subjects (see, e.g., [3-6]). The final breakthrough came in the 1948 article by Carlitz [7]. He defined inductively newq-Bernoulli numbersbm=bm(q) by
β0(q) = 1, q(qβ(q) + 1)m−βm(q) =
1 ifm= 1
0 ifm>1, (1:3)
with the usual convention ofbibybi. The q-Bernoulli polynomials are defined by
βm(x,q) = (qxβ(q) + [x]q)m= m
i=0
m i
βi(q)qix[x]mq−i. (1:4)
In 1954, Carlitz [8] generalized a result of Frobenius [3] and showed many of the properties of the q-Bernoulli numbers bm(q). In 1964, Carlitz [9] extended the
Ber-noulli, Eulerian, and Euler numbers and corresponding polynomials as a formal Dirich-let series. In what follows, we shall call them the Carlitz q-Bernoulli numbers and polynomials.
Some properties of Carlitz q-Bernoulli numbers bm(q) were investigated by various
authors. In [10], Koblitz constructed a q-analog of p-adicL-functions and suggested two questions. Question (1) was solved by Satoh [11]. He constructed a complex analy-tic q-L-series that is aq-analog of Dirichlet L-function and interpolates Carlitz q-Ber-noulli numbers, which is an answer to Koblitz’s question. By using aq-analog of the p-adic Haar distribution (see (1.6) below), Kim [12] answered part of Koblitz’s question (2) and constructedq-analogs of the p-adic log gamma functions Gp,q(x) onℂp\ℤp.
In [11], Satoh constructed the generating function of the Carlitz q-Bernoulli numbers Fq(t) inℂwhich is given by
Fq(t) = ∞
m=0
qme[m]qt(1−q−qmt) = ∞
m=0
βm(q)t
m
m!, (1:5)
where qis a complex number with 0 < |q| < 1. He could not explicitly determineFq
(t) inℂp, see [11, p.347].
In [12], Kim defined theq-analog of thep-adic Haar distributionμHaar(a+pNℤp) =
1/pNby
μq(a+pNZp) = qa
[pN] q
. (1:6)
Using this distribution, he proved that the Carlitzq-Bernoulli numbersbm(q) can be
represented as the p-adicq-integral onℤpbyμq, that is,
βm(q) =
Zp
[a]mqdμq(a), (1:7)
and found the following explicit formula
βm(q) = 1 (q−1)m
m
i=0
(−1)m−i
m i
i+ 1 [i+ 1]q
, (1:8)
wherem≥0 andq Îℂpwith0<1−q
p<p −p−11.
Fq(t) =e t
1−q ∞
j=0
j+ 1 [j+ 1]q(−1)
j
1 1−q
j tj
j!, (1:9)
whereqÎ ℂpwith0<1−q
p<p −p−11.
This article is organized as follows.
In Section 2, we consider the generalized Carlitzq-Bernoulli polynomials in thep-adic case by means of a method provided by Kim. We obtain the generating functions of the generalized Carlitzq-Bernoulli polynomials. We shall provide some basic formulas for the generalized Carlitzq-Bernoulli polynomials which will be used to prove the main results of this article.
In Section 3, we construct the complex, analytic, two-variableq-L-function that is a q-analog of the two-variableL-function. Using this function, we calculate the values of two-variableq-L-functions at nonpositive integers and study their properties whenqtends to 1.
2. Generalized Carlitz q-Bernoulli polynomials in the p-adic (and complex) case
For any uniformly differentiable function f:ℤp® ℂp, the p-adicq-integral onℤpis
defined to be the limit [p1N] q
pN−1
a=0 f(a)qaasN®∞. The uniform differentiability
guar-antees the limit exists. Kim [12,14-16] introduced this construction, denoted Iq(f),
where |1 -q|p<p-1/(p-1).
The construction ofIq(f) makes sense for manyqinℂpwith the weaker condition |1 -q|
p< 1. Indeed, when |1 -q|p< 1 the functionqxis uniformly differentiable and the space of
uniformly differentiable functionsℤp®ℂpis closed under multiplication, so we can make
sense of itsp-adicq-integralIq(f) for |1 -q|p< 1.
Lemma 2.1.For qÎℂpwith0 < |1 -q|p< 1and xÎ ℤp,we have
lim
N→∞
1 1−qpN
pN−1
a=0
qax= x 1−qx.
Proof. We assume that q Î ℂp satisfies the condition 0 < |1 - q|p< 1. Then it is
known that
qx=
∞
m=0
x m
(q−1)m
for any xÎℤp(see [[17], Lemma 3.1 (iii)]). Therefore, we obtain
lim
N→∞
1 1−qpN
pN−1
a=0
qax= 1 1−qxNlim→∞
qpN
x −1
qpN
−1
= 1 1−qxNlim→∞
∞
m=1
x m
qpN−1
m
qpN
−1 = 1
1−qxNlim→∞ ∞
m=0
x m+ 1 q
pN
−1m
This completes the proof.
Definition 2.2 ([12, §2, p. 323]). Let cbe a primitive Dirichlet character with
con-ductor dÎN and let xÎℤp. For q Îℂpwith 0 < |1 - q|p< 1 and an integerm≥ 0,
the generalized Carlitz q-Bernoulli polynomialsbm,c(x, q) are defined by
βm,χ(x,q) = d= 1. In particular, Kim [12] defined a class ofp-adic interpolation functionsGp,q(x) of
the Carlitzq-Bernoulli numbers bm(q) and gave several interesting applications of these
functions.
(where we use Lemma 2.1).
This completes the proof.
Remark2.5. We note here that similar expressions to those of Proposition 2.4 withc = c0 are given by Kamano [[18], Proposition 2.6] and Kim [12, §2]. Also, Ryoo et al. [19, Theorem 4] gave the explicit formula ofbm,c(0,q) inℂform≥0.
Lemma 2.6.Letcbe a primitive Dirichlet character with conductor dÎN.Then for
We now consider the case:
q∈Q∩Cp, 0<q<1, 0<1−qp<1. (2:2)
For instance, if we set
q= 1
1−pz∈Q∩Cp
for eachz≠0Îℤandp> 3, we find 0 < |q| < 1, 0 < |1 - q|p< 1.
LetFq,c(t, x) be the generating function ofbm,c(x, q) defined in Definition 2.2. From Proposition 2.4, we have
Fq,χ(t,x) =
Then, noting that
e
we see that
Moreover, (2.4) now becomes (where we use Lemma 2.6)
=−te
(cf. [13,16,20]). Similar arguments apply to the case Qq,c(t, x). We can rewrite
Qq,χ(t,x) =e
Then, by (2.4), (2.5), (2.6), and (2.7), we have the following theorem.
Theorem 2.7. Let q∈Q∩Cp, 0<q<1, 0<1−qp<1. Then the generalized
Carlitz q-Bernoulli polynomialsbm,c(x, q)for m≤0is given by equating the coefficients of powers of t in the following generating function:
Fq,χ(t,x) =e
3. q-analog of the two-variable L-function (in ℂ) From Theorem 2.7, fork≥0, we obtain the following
Hence we can define aq-analog of theL-function as follows:
Definition 3.1. Suppose thatcis a primitive Dirichlet character with conductordÎ
N. Let qbe a complex number with 0 < |q| < 1, and letLq(s, x, c) be a function of
two-variable (s,x)Îℂ×ℝdefined by
Lq(s,x,χ) =
1−q s−1
∞
m=0
χ(m)qm
[m+x]sq−1
+
∞
m=0
χ(m)qm+2x
[m+x]s q
(3:2)
for 0 <x≤1 (cf. [11,13,14,21-25]).
In particular, the two-variable functionLq(s, x,c) is a generalization of the
one-vari-ableLq(s,c) of Satoh [11], yielding the one-variable function when the second variable
vanishes.
Proposition 3.2.For kÎℤ,k≥1,the limiting valuelims®kLq(1 - s, x,c) = Lq(1
-k, x,c)exists and is given explicitly by
Lq(1−k,x,χ) =−
1
kβk,χ(x,q).
Proof. The proof is clear by Proposition 2.4, Theorem 2.7 and (3.1).
The formula of Proposition 3.2 is slight extension of the result in [19] and [11, Theo-rem 2].
Theorem 3.3.For any positive integer k, we have
lim
q→1βk,χ(x,q) = limq→1
1 (1−q)m
d−1
k=0
χ(k)qk m
i=0
m i
(−1)iqi(x+k) i+ 1
[d(i+ 1)]q =Bk,χ(x),
where the Bk,c(x)are the kth generalized Bernoulli polynomials.
Proof. We follow the proof in [[26], Theorem 1] motivated by the study of a simple q-analog of the Riemann zeta function. Recall that the ordinary Bernoulli polynomials Bk(x) are defined by
i qi−1q
ix= 1
logq ilogq eilogq−1e
x(ilogq)= 1
logq
∞
k=0
Bk(x)ik
(logq)k
k! , (3:3)
where it is noted that in this instance, the notation Bk(x) is used to replace Bk(x)
symbolically. For eachm≥1, let
(et−1)m=
∞
k=0
d(km)t k
k!. (3:4)
Note that
(et−1)m=
m
i=0
(−1)m−i
m i
eit=
∞
k=0
m
i=0
m i
(−1)m−iik
tk
k!. (3:5)
From (3.4) and (3.5), we obtain
d(km)=
⎧ ⎨ ⎩
m
i=0(−1)
m−i
m i
ik,m≤k
0, 0≤k<m.
It is also clear from the definition thatd(0)0 = 1,d(0)k = 0anddk(k)=k!forkÎN. From
where theBm,c(x) are the mth generalized Bernoulli polynomials (e.g., [14,19]). This completes the proof.
Corollary 3.4.For any positive integer k, we have
lim
q→1Lq(1−k,x,χ) =−
1
kBk,x(x).
Remark 3.5. The formula of Theorem 3.3 is slight extension of the result in [[26], Theorem 1].
Fχ(t,x) := lim
q→1Fq,χ(t,x)
=−t d
a=1 ∞
l=0
χ(a+dl)e(a+dl)text
=
d
a=1
χ(a)te(a+x)t edt−1
=
∞
m=0
Bm,χ(x)
tm m!.
Remark3.7. If we substitutec=c0, the trivial character, in Definition 3.1 and Corol-lary 3.4, we can also define aq-analog of the Hurwitz zeta function
ζ(s,x) =
∞
m=0
1 (m+x)s
by
ζq(s,x) =Lq(s,x,χ0) =
1−q s−1
∞
m=0
qm+x
[m+x]sq−1
+
∞
m=0
q2(m+x)
[m+x]sq and obtain the identity
lim
q→1ζq(s,x) =ζ(s,x)
for all s≠1, as well as the formula
lim
q→1ζq(1−k,x) =−
1
kBk(x)
for integersk≥1 (cf. [11,13,19,22,24,25]).
Acknowledgements
This work was supported by the Kyungnam University Foundation Grant, 2012.
Author details 1
National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon 305-340, South Korea2Division of Cultural Education, Kyungnam University, Changwon 631-701, South Korea
Authors’contributions
The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 21 December 2011 Accepted: 13 April 2012 Published: 13 April 2012
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doi:10.1186/1687-1847-2012-44
Cite this article as:Kim and Kim:A note on Carlitzq-Bernoulli numbers and polynomials.Advances in Difference Equations20122012:44.
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