# On the Modifiedq-Bernoulli Numbers of Higher Order with Weight

Authors: T. Kim; J. Choi; Y.-H. Kim; S.-H. Rim
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

The purpose of this paper is to give some properties of the modifiedq-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic p-adic q-integral on ℤp, we derive new identities of q-Bernoulli numbers and polynomials with weight.


---

## Body

## 1. Introduction

Letp be a fixed odd 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 the algebraic closure of ℚp. Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}. The p-adic norm of ℂp is defined by |p|p=1/p. When one talks of a q-extension, q can be considered as an indeterminate, a complex number q∈ℂ, or a p-adic number q∈ℂp. Throughout this paper we assume that α∈ℚ and q∈ℂp with |1-q|p<p-1/(p-1) so that qx=exp(xlogq).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 (see [1–3]) as follows:(1.1)Iq(f)=∫Zpf(x)dμq(x)=lim⁡N→∞1[pN]q∑x=0pN-1f(x)qx,
where [x]q is the q-number of x which is defined by [x]q=(1-qx)/(1-q).From (1.1), we have(1.2)qnIq(fn)-Iq(f)=(q-1)∑l=0n-1qlf(l)+q-1log⁡q∑l=0n-1qlf′(l),
where fn(x)=f(x+n) (see [2–4]).As is well known, Bernoulli numbers are inductively defined by(1.3)B0=1,(B+1)n-Bn={1ifn=1,0ifn>1,
with the usual convention about replacing Bn by Bn (see [3, 5]).In [2, 5, 6], the q-Bernoulli numbers are defined by(1.4)B0,q=q-1log⁡q,(qBq+1)n-Bn,q={1ifn=1,0ifn>1,
with the usual convention about replacing Bqn by Bn,q. Note that limq→1Bn,q=Bn. In the viewpoint of (1.4), we consider the modified q-Bernoulli numbers with weight.In this paper we study families of the modifiedq-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the multivariate p-adic q-integral on ℤp, we give new identities of the higher-order q-Bernoulli numbers and polynomials with weight.


## 2. Modifiedq-Bernoulli Numbers with Weight of Higher Order

Forn∈ℤ+, let us consider the following modified q-Bernoulli numbers with weight α (see [1, 3]):(2.1)B̃n,q(α)=∫Zp[x]qαnq-xdμq(x)=1(1-q)n[α]qn∑l=0n(nl)(-1)lαl[αl]q,B̃n,q(α)(x)=∫Zp[x+y]qαnq-ydμq(y)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlxαl[αl]q.
From (2.1), we note that(2.2)B̃n,q(α)(x)=∑l=0n(nl)[x]qαn-lqαlxB̃l,q(α)
(see [1, 3]).Fork∈ℕ and n∈ℤ+, by making use of the multivariate p-adic q-integral on ℤp, we consider the following modified q-Bernoulli numbers with weight α of order k, B̃n,q(k,α):(2.3)B̃n,q(k,α)=∫Zp⋯∫Zp[x1+⋯+xk]qαnq-x1-⋯-xkdμq(x1)⋯dμq(xk).
Note that B̃n,q(1,α)=B̃n,q(α) and limq→1B̃n,q(k,α)=Bn(k), where Bn(k) are the nth ordinary Bernoulli numbers of order k.Fork,N∈ℕ, we have(2.4)(1-q1-qpN)k∑i1=0pN-1⋯∑ik=0pN-1[i1+⋯+ik]qαn=(1-q1-qpN)k(11-qα)n∑i1,…,ik=0pN-1∑j=0n(nj)(-1)jqα(i1+⋯+ik)j=1(1-q)n[α]qn∑j=0n(nj)(-1)j(1-q)k(1-qpN)k(1-qαpNj1-qαj⋯1-qαpNj1-qαj)︸k-times.
By (1.1), (2.3), and (2.4), we get(2.5)B̃n,q(k,α)=1(1-q)n[α]qn∑j=0n(nj)(-1)j(αj)k[αj]qk.
Therefore, by (2.5), we obtain the following theorem.Theorem 2.1.
Forn≥0, one has
(2.6)B̃n,q(k,α)=1(1-q)n[α]qn∑j=0n(nj)(-1)j(αj)k[αj]qk.Let us consider the modifiedq-Bernoulli and polynomials with weight α of order k as follows:(2.7)B̃n,q(k,α)(x)=∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq-x1-⋯-xkdμq(x1)⋯dμq(xk).
By the same method of (2.5), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, one has
(2.8)B̃n,q(k,α)(x)=1(1-q)n[α]qn∑j=0n(nj)(-1)jqαxj(αj)k[αj]qk.By Theorem2.2, we get(2.9)B̃n,q-1(k,α)(k-x)=1(1-q-α)n∑j=0n(nj)(-1)j(αj)k[αj]q-1kq-αj(k-x)=(-1)nqαn(1-qα)n∑j=0n(nj)(-1)j(q-1(q-1)αj(qαj-1)q-αj)kq-αj(k-x)=(-1)nqαn(1-qα)n∑j=0n(nj)(-1)jqαjxq-k(αj)k[αj]qk=(-1)nqαn-kB̃n,q(k,α)(x).
Therefore, by (2.9), we obtain the following theorem.Theorem 2.3.
Forn∈ℤ+, one has
(2.10)B̃n,q-1(k,α)(k-x)=(-1)nqαn-kB̃n,q(k,α)(x),B̃n,q-1(k,α)(k)=(-1)nqαn-kB̃n,q(k,α).From Theorem2.3, we note that(2.11)lim⁡q→1B̃n,q-1(k,α)(k-x)=Bn(k)(k-x),lim⁡q→1B̃n,q-1(k,α)(k)=(-1)nBn(k).
Thus, we have Bn(k)(k)=(-1)nBn(k), where Bn(k) are the nth Bernoulli numbers of order k.From (2.3) and (2.7), we can derive the following equations:(2.12)B̃k,q(l,α)(x)=lim⁡N→∞1[m]ql[pN]qml∑i1,…,il=0m-1∑n1,…,nl=0pN-1[x+i1+⋯+il+m(n1+⋯+nl)]qαk=[m]qαk[m]ql∑i1,…,il=0m-1∫Zp⋯∫Zp[x+i1+⋯+ilm+x1+⋯+xl]qαmk×q-mx1-⋯-mxldμqm(x1)⋯dμqm(xk)=[m]qαk[m]ql∑i1,…,il=0m-1B̃k,qm(l,α)(x+i1+⋯+ilm).
Therefore, by (2.12), we obtain the following theorem.Theorem 2.4.
Fork∈ℤ+ and  l,m∈ℕ, one has
(2.13)B̃k,q(l,α)(x)=[m]qαk[m]ql∑i1,…,il=0m-1B̃k,qm(l,α)(x+i1+⋯+ilm).In particular,(2.14)B̃k,q(l,α)(mx)=[m]qαk[m]ql∑i1,…,il=0m-1B̃k,qm(l,α)(x+i1+⋯+ilm).From (1.2), we can derive the following integral:(2.15)∫Zpf(x+1)q-xdμq(x)=∫Zpf(x)q-xdμq(x)+q-1log⁡qf′(0),∫Zpf(x+2)q-xdμq(x)=∫Zpf1(x)q-xdμq(x)+q-1log⁡qf′(1)=∫Zpf(x)q-xdμq(x)+q-1log⁡⁡q(f′(0)+f′(1)).
Continuing this process, we obtain(2.16)∫Zpf(x+n)q-xdμq(x)=∫Zpf(x)q-xdμq(x)+q-1log⁡q∑l=0n-1f′(l).
By (2.16), we get(2.17)∫Zp[x+n]qαmq-xdμq(x)=∫Zp[x]qαmq-xdμq(x)+mα[α]q∑l=0n-1[l]qαm-1qαl.
Therefore, by (2.1) and (2.17), we obtain the following theorem.Theorem 2.5.
Forn∈ℕ and  m∈ℤ+, one has
(2.18)B̃m,q(α)(n)-B̃m,q(α)=mα[α]q∑l=0n-1[l]qαmqαl.In an analogues manner as the previous investigation [7–10], we can define a further generalization of modified q-Bernoulli numbers with weight. Let χ be the Dirichlet character with conductor d∈ℕ. Then the generalized q-Bernoulli numbers with weight attached to χ can be defined as follows:(2.19)B̃n,χ,q(α)=∫Xχ(x)[x]qαnq-xdμq(x)=[d]qαn[d]q∑a=0d-1χ(a)B̃n,qd(α)(ad).
We expect to investigate these objects in future papers. This definition B̃n,q(α) was also given in a previous paper (see [9]).



---

*Source: 948050-2012-02-20.xml*