# Identities on the Weightedq-Bernoulli Numbers of Higher Order

Authors: T. Kim; D. V. Dolgy; B. Lee; S.-H. Rim
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

We give a new construction of the weightedq-Bernoulli
numbers and polynomials of higher order by using multivariate p-adic
q-integral on ℤp.


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## Body

## 1. Introduction

Letp be a fixed prime number. Throughout this paper, ℤ,ℤp,ℚp,ℂ, and ℂp will, respectively, denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of ℚp. The p-adic norm of ℂp is defined by |x|p=1/pr where x=prs/t with (p,s)=(p,t)=(s,t)=1,r∈ℚ. In this paper, we assume α∈ℚ and ℤ+=ℕ∪{0}. Let q∈ℂp with |q-1|p<p-1/(p-1) and let [x]q=(1-qx)/(1-q). Note that   limq→1[x]q=x (see [1–13]). Recently, the q-Bernoulli numbers with weight α are defined by (1.1)β̃0,q(α)=1,q(qαβ̃qα+1)n-β̃n,q(α)={α[α]qifn=1,0ifn>1,
with the usual convention about replacing (β̃(α))n by β̃n,q(α), (see [4]).Theq-Bernoulli polynomials with weight α are also defined by as (1.2)β̃n,q(α)(x)=∑l=0n(nl)[x]qαn-lqαlxβ̃l,q(α)=([x]qα+qαxβ̃q(α))n,forn≥0.
For f∈UD(ℤp)= the space of uniformly differentiable functions on ℤp, the p-adic q-integral on ℤp is defined by (1.3)Iq(f)=∫Zpf(x)dμq(x)=lim⁡N→∞1[pN]q∑x=0pN-1f(x)qx,
(see [4–12]). From (1.3), we note that (1.4)qnIq(fn)=Iq(f)+(q-1)∑l=0n-1qlf(l)+q-1log⁡q∑l=0n-1qlf′(l),
where fn(x)=f(x+n), (see [1–12]).We have the Witt formula for theq-Bernoulli numbers and polynomials with weight α as follows (see [4, 5, 12]): (1.5)β̃n,q(α)=∫Zp[x]qαndμq(x),β̃n,q(α)(x)=∫Zp[x+y]qαndμq(x).
From (1.4) and (1.5), we have (1.6)β̃n,q(α)(x)=1(1-q)n[α]qn∑l=0n(-1)lqαlxαl+1[αl+1]q,
(see [4]). By (1.6), we easily get limq→1β̃n,q(α)(x)=Bn(x), where Bn(x) are the Bernoulli polynomials of degree n.To give the new construction of the weightedq-Bernoulli numbers and polynomials of higher order, we first use the multivariate p-adic q-integral on ℤp. The purpose of this paper is to give the higher-order q-Bernoulli numbers and polynomials with weight α and to derive a new explicit formulas by these numbers and polynomials.


## 2. On the Higher Orderq-Bernoulli Numbers with Weightα

Forhi(i=1,2,…,k)∈ℤ+, we consider a sequence of p-adic rational numbers as expansion of the weighted q-Bernoulli numbers and polynomials of order k as follows: (2.1)β̃n,q(k,α)(h1,h2,…,hk)=∫Zp⋯∫Zp[x1+⋯+xk]qαnq∑j=1kxj(hj-1)dμq(x1)⋯dμq(xk),(2.2)β̃n,q(k,α)(h1,h2,…,hk∣x)=∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq∑j=1kxj(hj-1)dμq(x1)⋯dμq(xk).
From (2.1) and (2.2), we can derive the following equations: (2.3)β̃n,q(k,α)(h1,h2,…,hk)=1(1-q)n[α]qn∑l=0n(nl)(-1)l(αl+h1)(αl+h2)⋯(αl+hk)[αl+h1]q[αl+h2]q⋯[αl+hk]q,(2.4)β̃n,q(k,α)(h1,h2,…,hk∣x)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αl+h1)(αl+h2)⋯(αl+hk)[αl+h1]q[αl+h2]q⋯[αl+hk]q,=∑l=0n(nl)[x]qαn-lqαlxβ̃l,q(k,α)(h1,h2,…,hk).
By (2.3) and (2.4), we get (2.5)∑l=0n(nl)(-1)lqαlx(αl+h1)(αl+h2)⋯(αl+hk)[αl+h1]q[αl+h2]q⋯[αl+hk]q,=∑l=0n(nl)[x]qαn-lqαlx(1-q)n-l[α]qn-l∑s=0l(ls)(-1)s(αs+h1)⋯(αs+hk)[αs+h1]q⋯[αs+hk]q.
Therefore, by (2.5), we obtain the following theorem.Theorem 2.1.
Forhi(i=1,2,…,k)∈ℤ+, and k∈ℕ, we have
(2.6)∑l=0n(nl)(-1)lqαlx(αl+h1)(αl+h2)⋯(αl+hk)[αl+h1]q[αl+h2]q⋯[αl+hk]q=∑l=0n(nl)[x]qαn-lqαlx(1-q)n-l[α]qn-l∑s=0l(ls)(-1)s(αs+h1)⋯(αs+hk)[αs+h1]q⋯[αs+hk]q.From (1.3) and (1.4), we note that
(2.7)qh1∫Zp[x1+x+1]qαnqx1(h1-1)dμq(x1)=∫Zp[x1+x]qαnqx1(h1-1)dμq(x1)+[x]qαnh1(q-1)+n[x]qαn-1α[α]q.
Therefore, by (2.7), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, we have
(2.8)qh1β̃n,q(1,α)(h1∣x+1)-β̃n,q(1,α)(h1∣x)=h1(q-1)[x]qαn+n[x]qαn-1α[α]q.From (2.2) and (2.3), we have
(2.9)qαxβ̃n,q(k,α)(αh1+α,αh2+α,…,αhk+α∣x)=qαx∫Zp⋯∫Zp[x1+⋯+xk+x]qαnq∑j=1kxj(αhj+α-1)dμq(x1)⋯dμq(xk)=(qα-1)∫Zp⋯∫Zp[x1+⋯+xk+x]qαn+1q∑j=1kxj(αhj-1)dμq(x1)⋯dμq(xk)+∫Zp⋯∫Zp[x1+⋯+xk+x]qαnq∑j=1kxj(αhj-1)dμq(x1)⋯dμq(xk).
Therefore, by (2.9), we obtain the following proposition.Proposition 2.3.
Forn∈ℤ+, we have
(2.10)qαxβ̃n,q(k,α)(αh1+α,αh2+α,…,αhk+α∣x)=(qα-1)β̃n+1,q(k,α)(αh1,…,αhk∣x)+β̃n,q(k,α)(αh1,…,αhk∣x).By (2.2), we get
(2.11)β̃n,q(k,α)(h1,h2,…,hk∣x)=∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq∑j=1kxj(hj-1)dμq(x1)⋯dμq(xk)=[d]qαn[d]qk∑s1,…,sk=0d-1q∑j=1khjsj∫Zp⋯∫Zp[s1+⋯+sk+xd+x1+⋯+xk]qdαn×qd∑j=1kxj(hj-1)dμqd(x1)⋯dμqd(xk)=[d]qαn[d]qk∑s1,…,sk=0d-1q∑j=1khjsjβ̃n,qd(k,α)(h1,…,hk∣s1+⋯+sk+xd),
where d∈ℕ.Therefore, by (2.11), we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+ and d∈ℕ, we have
(2.12)β̃n,q(k,α)(h1,h2,…,hk∣x)=[d]qαn[d]qk∑s1,…,sk=0d-1q∑j=1khjsjβ̃n,qd(k,α)(h1,…,hk∣s1+⋯+sk+xd).Letd(>0) be a fixed integer. For N∈ℕ, we set (2.13)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.Letχ be a primitive Dirichlet character with conductor d∈ℕ. Then we consider the generalized q-Bernoulli numbers with weight α of order k as follows:(2.14)β̃n,χ,q(k,α)(h1,h2,…,hk)=∫X⋯∫X∏i=1kχ(xi)[x1+⋯+xk]qαnq∑i=1kxi(hi-1)dμq(x1)⋯dμq(xk).
From (2.14), we have (2.15)β̃n,χ,q(k,α)(h1,h2,…,hk)=[d]qαn[d]qk∑s1,…,sk=0d-1q∑j=1khjsj(∏j=1kχ(sj))β̃n,qd(k,α)(h1,…,hk∣s1+⋯+skd).



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*Source: 918364-2011-10-20.xml*