# A Note on the Generalizedq-Bernoulli Measures with Weight α

Authors: T. Kim; S. H. Lee; D. V. Dolgy; C. S. Ryoo
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We discuss a new concept of theq-extension of Bernoulli measure. From those measures, we derive some interesting properties on the generalized
q-Bernoulli numbers with weight α attached to χ.


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## Body

## 1. Introduction

Letp 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 ofq-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 (1.1)[x]q=1-qx1-q,
(see [1–14]). Thus, we note that limq→1[x]q=x.In [2], Carlitz defined a set of numbers ξk=ξk(q) inductively by (1.2)ξ0=1,(qξ+1)k-ξk={1,ifk=1,0,ifk>1,
with the usual convention of replacing ξk by ξk.These numbers areq-extension of ordinary Bernoulli numbers Bk. But they do not remain finite when q=1. So he modified (1.2) as follows:(1.3)β0,q=1,q(qβ+1)k-βk,q={1,ifk=1,0,ifk>1,
with the usual convention of replacing βk by βk,q.The numbersβk,q are called the k-th Carlitz q-Bernoulli numbers.In [1], Carlitz also considered the extended Carlitz’s q-Bernoulli numbers as follows:(1.4)β0,qh=h[h]q,qh(qβh+1)k-βk,qh={1,ifk=1,0,ifk>1,
with the usual convention of replacing (βh)k by βk,qh.Recently, Kim consideredq-Bernoulli numbers, which are different extended Carlitz’s q-Bernoulli numbers, as follows: for α∈ℕ and n∈ℤ+,(1.5)β̃0,q(α)=1,q(qαβ̃(α)+1)n-β̃n,q(α)={α[α]q,ifn=1,0,ifn>1,
with the usual convention of replacing (β̃(α))k by β̃k,q(α) (see [3]).The numbersβ̃k,q(α) are called the k-th q-Bernoulli numbers with weight α.For fixedd∈ℤ+ with (p,d)=1, we set (1.6)X=Xd=lim⁡←N⁡(ZdpNZ),X1=Zp,X*=⋃0<a<dp(a,p)=1(a+dpZp),a+dpNZp={x∈X∣x≡a(mod⁡dpN)},
where a∈ℤ satisfies the condition 0≤a<dpN.LetUD(ℤp) be the space of uniformly differentiable functions on ℤp. For f∈UD(ℤp), the p-adic q-integral on ℤp is defined by Kim as follows: (1.7)Iq(f)=∫Zpf(x)dμq(x)=lim⁡N→∞1[pN]q∑x=0pN-1f(x)qx,
(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 (1.8)∫Zp[x]qαndμq(x)=β̃n,q(α),wheren∈Z+.Theq-Bernoulli polynomials with weight α are also defined by (1.9)β̃n,q(α)(x)=∑l=0n(nl)[x]qαn-lqαlxβ̃l,q(α).From (1.7), (1.8), and (1.9), we can derive the Witt’s formula for β̃n,q(α)(x) as follows: (1.10)∫Zp[x+y]qαndμq(y)=β̃n,q(α)(x),wheren∈Z+.Forn∈ℤ+ and d∈ℕ, the distribution relation for the q-Bernoulli polynomials with weight α are known that (1.11)β̃n,q(α)(x)=[d]qαn[d]q∑a=0d-1qaβ̃n,qd(α)(x+ad),
(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

LetX be any compact-open subset of ℚp, such as ℤp or ℤp*. 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:(2.1)μ(⋃k=1nUk)=∑k=1nμ(Uk)(additivity),
where {U1,U2,…,Un} is any collection of disjoint compact opensets in X.The setℤp has a topological basis of compact open sets of the form a+pnℤp.Consequently, ifU is any compact open subset of ℤp, it can be written as a finite disjoint union of sets (2.2)U=⋃j=1k(aj+pnZp),
where n∈ℕ and a1,a2,…,ak∈ℤ with 0≤ai<pn for i=1,2,…k.Indeed, thep-adic ball a+pnℤp can be represented as the union of smaller balls (2.3)a+pnZp=⋃b=0p-1(a+bpn+pn+1Zp).Lemma 2.1.
Every mapμ from the collection of compact-open sets in X to ℚp for which
(2.4)μ(a+pNZp)=⋃b=0p-1(a+bpN+dpN+1Zp)
holds whenever a+pNℤp⊂X, extends to a p-adic distribution (=p-adic unbounded measure) on X.Now we define a mapμk,q(α) on the balls in ℤp as follows:(2.5)μk,q(α)(a+pnZp)=[pn]qαk[pn]qqafk,qpn(α)({a}npn),
where {a}n is the unique number in the set {0,1,2,…,pn-1} such that {a}n≡a(modpn).Ifa∈{0,1,2,…,pn-1}, then(2.6)∑b=0p-1μk,q(α)(a+bpn+pn+1Zp)=∑b=0p-1[pn+1]qαk[pn+1]qqa+bpnfk,qpn+1(α)(a+bpnpn+1)=qa[pn]qαk[pn]q[p](qpn)αk[p]qpn∑b=0p-1qbpnfk,(qpn)p(α)((a/pn)+bp).From (2.6), we note that μk,q(α) is p-adic distribution on ℤp if and only if (2.7)[p](qpn)αk[p]qpn∑b=0p-1qbpnfk,(qpn)p(α)((a/pn)+bp)=fk,qpn(α)(apn).Theorem 2.2.
Letα∈ℕ and k∈ℤ+. Then we see that μk,q(α)(a+pnℤp) is p-adic distribution on ℤp if and only if
(2.8)[p](qpn)αk[p]qpn∑b=0p-1qbpnfk,(qpn)p(α)((a/pn)+bp)=fk,qpn(α)(apn).
One sets(2.9)fk,qpn(α)(x)=β̃k,qpn(α)(x).
From (2.5) and (2.9), one gets
(2.10)μk,q(α)(a+pnZp)=[pn]qαk[pn]qqaβ̃k,qpn(α)(apn).By (1.11), (2.10), and Theorem 2.2, we obtain the following theorem.Theorem 2.3.
Letμk,q(α) be given by
(2.11)μk,q(α)(a+dpNZp)=[dpN]qαk[dpN]qqaβ̃k,qdpN(α)(adpN).
Then μk,q(α) extends to a ℚ(q)-valued distribution on the compact open sets U⊂X.
From (2.11), one notes that
(2.12)∫Xdμk,q(α)(x)=lim⁡N→∞∑x=0dpN-1μk,q(α)(x+dpNZp)=lim⁡N→∞[dpN]qαk[dpN]q∑a=0dpN-1qaβ̃k,qdpN(α)(adpN).
By (1.11) and (2.12), one gets
(2.13)∫Xdμk,q(α)(x)=β̃k,q(α).Therefore, we obtain the following theorem.Theorem 2.4.
Forα∈ℕ and k∈ℤ+, one has
(2.14)∫Xdμk,q(α)(x)=β̃k,q(α).
Letχ be Dirichlet character with conductor d∈ℕ. Then one defines the generalized q-Bernoulli numbers attached to χ as follows:
(2.15)β̃n,χ,q(α)=∫Xχ(x)[x]qαndμq(x)=[d]qαn[d]q∑a=0d-1qaχ(a)β̃n,qd(α)(ad).
From (2.11) and (2.15), one can derive the following equation;
(2.16)∫Xχ(x)dμk,q(α)(x)=lim⁡N→∞∑x=0dpN-1χ(x)μk,q(α)(x+dpNZp)=lim⁡N→∞[dpN]qαk[dpN]q∑x=0dpN-1χ(x)qxβ̃k,qdpN(α)(xdpN)=[d]qαk[d]q∑a=0d-1qaχ(a){lim⁡N→∞[pN]qαdk[pN]qd∑x=0pN-1qdxβ̃k,qdpN((a/d)+xpN)}=[d]qαk[d]q∑a=0d-1qaχ(a)β̃k,qd(α)(ad)=β̃k,χ,q(α),∫pXχ(x)dμk,q(α)(x)=lim⁡N→∞[dpN+1]qαk[dpN+1]q∑x=0dpN-1χ(px)qpxβ̃k,qdpN+1(α)(pxdpN+1)=[p]qαk[p]q[d]qpαk[d]qp∑a=0d-1χ(pa)qpalim⁡N→∞[pN]qdpαk[pN]qdp∑x=0pN-1qpdxβ̃k,qpdpN(α)(p(xd+a)pdpN)=[p]qαk[p]q[d]qαpk[d]qp∑a=0d-1χ(p)χ(a)qpaβ̃k,qpd(α)(ad)=χ(p)[p]qαk[p]qβ̃k,χ,qp(α).
Forβ(≠1)∈X*, one has
(2.17)∫pXχ(x)dμk,q1/β(α)(βx)=χ(pβ)[p]qα/βk[p]q1/ββ̃k,χ,qp/β(α),∫Xχ(x)dμk,q1/β(α)(βx)=χ(1β)β̃k,χ,q1/β(α).Therefore, we obtain the following theorem.Theorem 2.5.
Forβ(≠1)∈X*, one has
(2.18)∫Xχ(x)dμk,q(α)(x)=β̃k,χ,q(α),∫pXχ(x)dμk,q(α)(x)=χ(p)[p]qαk[p]qβ̃k,χ,qp(α),∫pXχ(x)dμk,q1/β(α)(βx)=χ(pβ)[p]qα/βk[p]q1/ββ̃k,χ,qp/β(α),∫Xχ(x)dμk,q1/β(α)(βx)=χ(1β)β̃k,χ,q1/β(α).
Define
(2.19)μk,β,q(α)(U)=μk,q(α)(U)-β-1[β-1]qαk[β-1]qμk,q1/β(α)(βU).
By a simple calculation, one gets
(2.20)∫X*χ(x)dμk,β,q(α)(x)=∫Xχ(x)dμk,q(α)(x)-β-1[β-1]qαk[β-1]q∫pXχ(x)μk,q1/β(α)(x)=β̃k,χ,q(α)-χ(p)[p]qαk[p]qβ̃k,χ,qp(α),[β-1]qαk[β-1]qk∫X*χ(x)dμk,q1/β(α)(βx)=[1/β]qαk[1/β]qχ(1β)β̃k,χ,q1/β(α)-χ(pβ)[p/β]qαk[p/β]qβ̃k,χ,qp/β(α).
By (2.19) and (2.20), one gets
(2.21)∫X*χ(x)dμk,β,q(α)(βx)=∫Xχ(x)dμk,q(α)(x)-β-1[β-1]qαk[β-1]q∫pXχ(x)μk,q1/β(α)(βx)=β̃k,χ,q(α)-χ(p)[p]qαk[p]qβ̃k,χ,qp(α)-1β[1/β]qαk[1/β]qχ(1β)β̃k,χ,q1/β(α)+χ(pβ)[p/β]qαk[p/β]qβ̃k,χ,qp/β(α).
Now one defines the operatorχy=χy,k,α:q on f(q) by
(2.22)χyf(q)=χy,k,α:qf(q)=[y]qαk[y]qχ(y)f(qy).
Thus, by (2.22), one gets
(2.23)χx,k,α:q∘χy,k,α:qf(q)=χx,k,α:q[y]qαk[y]qχ(y)f(qy)=[y]qαk[y]qχ(y)χ(x)[y]qαyk[y]qyχ(y)f(qxy)=[xy]qαk[xy]qχ(xy)f(qxy)=χxy,k,α:qf(q)=χxyf(q).
Let us define χxχy=χx,k,α:q∘χy,k,α:q. Then one has
(2.24)χxχy=χxy.
From the definition of χx, one can easily derive the following equation;
(2.25)(1-χp)(1-1βx1/β)=1-1βx1/β-χp+1βxp/β.
Let f(q)=β̃k,χ,q(α). Then one gets
(2.26)(1-χp)(1-1βx1/β)β̃k,χ,q(α)=β̃k,χ,q(α)-1β[1/β]qαk[1/β]qχ(1β)β̃k,χ,q(α)-[p]qαk[p]qχ(p)β̃k,χ,qp(α)+1β[p/β]qαk[p/β]qχ(pβ)β̃k,χ,qp/β(α).
By (2.21) and (2.26), one obtains the following equation:
(2.27)∫X*χ(x)dμk,β,q(α)(βx)=(1-χp)(1-1βx1/β)β̃k,χ,q(α),
where β(≠1)∈X*.



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