# A Note on the Modifiedq-Bernoulli Numbers and Polynomials with Weight  α

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

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## Abstract

A systemic study of some families of the modifiedq-Bernoulli
numbers and polynomials with weight α is presented by using the p-adic q-integration
ℤp. The study of these numbers and polynomials yields an interesting q-analogue related to Bernoulli numbers and polynomials.


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## Body

## 1. Introduction

Letp be a fixed prime number. Throughout this paper ℤp,ℚp,ℂ, and ℂp will, respectively, denote 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. Let vp be the normalized exponential valuation of ℂp with |p|p=p-vp(p)=1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex q∈ℂ, or a p-adic number q∈ℂp. If q∈ℂ, one normally assumes |q|<1. If q∈ℂp, then we assume |q-1|p<p-1/(p-1) so that qx=exp(xlogq) for |x|p≤1.Theq-number [x]q is defined by (1.1)[x]q=1-qx1-q,
see [1–10].We say thatf is a uniformly differentiable function at a point a∈ℤp and denote this property by f∈UD(ℤp), if the difference quotients (1.2)Ff(x,y)=f(x)-f(y)x-y
have a limit l=f′(a) as (x,y)→(a,a). c.f. [11].Forf∈UD(ℤp,ℂp)={f∣f:ℤp→ℂpisuniformlydifferentiablefunctions}, 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  [3]).From (1.3), we can easily derive the following: (1.4)qnIq(fn)=Iq(f)+(q-1)∑l=0n-1f(l)ql+q-1log⁡q∑l=0n-1qlf′(l),
where fn(x)=f(x+n), (see  [5, 12]).In [1, 2], Carlitz defined a set of numbers Bk,q inductively by (1.5)B0,q=1,(qBq+1)k-Bk,q={1,ifk=1,0,ifk>1,
with the usual convention about replacing Bqk by Bk,q.These numbers are theq-extension of ordinary Bernoulli numbers. But they do not remain finite when q=1. So, Carlitz modified (1.5) as follows: (1.6)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.In [1], Carlitz also considered the extension of Carlitz’s q-Bernoulli numbers as follows: (1.7)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.In this paper, we construct the modifiedq-Bernoulli numbers with weight α, which are different Carlitz’s q-Bernoulli numbers, by using p-adic q-integral equation. From (1.4), we derive some interesting identities and relations on the modified q-Bernoulli numbers and polynomials.


## 2. The Modifiedq-Bernoulli Numbers and Polynomials with Weight α

In this section, we assumeα∈ℚ. Now, we define the modified q-Bernoulli numbers with weight α (=B̃n,q(α) ) as follows: (2.1)B̃n,q(α)=∫Zpq-x[x]qαndμq(x)=1(1-qα)n∑l=0n(nl)(-1)lαl[αl]q.
Thus, by (2.1), we have (2.2)B̃n,q(α)=1(1-qα)nq-1log⁡q-nα[α]q∑m=0∞qαm[m]qαn-1.
Therefore, by (2.1) and (2.2), we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+=ℕ∪{0}, one has
(2.3)B̃n,q(α)=α(1-qα)n∑l=0n(nl)(-1)ll[αl]q=1(1-qα)nq-1log⁡q-nα[α]q∑m=0∞qαm[m]qαn-1.Let us define the generating function of the modifiedq-Bernoulli numbers with weight α as follows: (2.4)Fq(α)(t)=∑n=0∞B̃n,q(α)tnn!.
Then, by (2.3) and (2.4), we get (2.5)Fq(α)(t)=q-1log⁡qe(1/(1-qα))t-tα[α]q∑m=0∞qαme[m]qαt.
In the viewpoint of (2.1), we define the modified q-Bernoulli numbers with weight α as follows: (2.6)B̃n,q(α)(x)=∫Zpq-y[x+y]qαndμq(y)=∑l=0n(nl)[x]qαn-lqαlxB̃l,q(α)=([x]qα+qαxB̃q(α))n,forn∈Z+,
with the usual convention of replacing (B̃q(α))n by B̃n,q(α).From (2.6), we note that (2.7)B̃n,q(α)(x)=α(1-qα)n∑l=0n(nl)(-1)lqαlxl[αl]q=1(1-qα)nq-1log⁡q-nα[α]qqαx∑m=0∞qαm[m+x]qαn-1.
Therefore, by (2.7), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, one has
(2.8)B̃n,q(α)(x)=α(1-qα)n∑l=0n(nl)(-1)lqαlxl[αl]q=1(1-qα)nq-1log⁡q-nα[α]qqαx∑m=0∞qαm[m+x]qαn-1.LetFq(α)(t,x)=∑n=0∞B̃n,q(α)(x)(tn/n!) be the generating function of the modified q-Bernoulli polynomials with weight α.Then, by (2.7), we get (2.9)Fq(α)(t,x)=q-1log⁡qe(1/(1-qα))t-tα[α]q∑m=0∞qα(m+x)e[m+x]qαt.
Therefore, by (2.9), we obtain the following corollaryCorollary 2.3.
LetFq(α)(t,x)=∑n=0∞B̃n,q(α)(x)(tn/n!). Then one has
(2.10)Fq(α)(t,x)=q-1log⁡qe(1/(1-qα))t-tα[α]q∑m=0∞qα(m+x)e[m+x]qαt.In particular,Fq(α)(t,0)=Fq(α)(t).From Corollary2.3, we can derive the following equation:(2.11)Fqα(t,1)-Fqα(t)=tα[α]q.By (2.5) and (2.11), we get(2.12)B̃0,q(α)=q-1log⁡q,B̃n,q(α)(1)-B̃n,q(α)={α[α]q,ifn=1,0,ifn>1.Therefore, by (2.12), we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+, one has
(2.13)B̃0,q(α)=q-1log⁡q,B̃n,q(α)(1)-B̃n,q(α)={α[α]q,ifn=1,0,ifn>1.By using (2.6), we obtain the following corollary.Corollary 2.5.
Forn∈ℤ+, one has
(2.14)B̃0,q(α)=q-1log⁡q,(qαB̃q(α)+1)n-B̃n,q(α)={α[α]q,ifn=1,0,ifn>1.
with the usual convention of replacing (B̃q(α))n by B̃n,q(α).From (1.4), we can derive the following equation:(2.15)∫Zpf(x+n)q-xdμq(x)=∫Zpf(x)q-xdμq(x)+q-1log⁡q∑l=0n-1f′(l).Thus, by (1.6), (2.6), and (2.15), we get(2.16)B̃m,q(α)(n)-B̃m,q(α)=α[α]qm∑l=0n-1[l]qαm-1qαl,n∈N,m∈Z+.Therefore, by (2.16), we obtain the following theorem.Theorem 2.6.
Forn∈ℕ,m∈ℤ+, one has
(2.17)B̃m,q(α)(n)-B̃m,q(α)=α[α]qm∑l=0n-1[l]qαm-1qαl.From (2.6), we note that(2.18)B̃n,q(α)(x)=∫Zp[x+y]qαnq-ydμq(y)=lim⁡N→∞1[pN]q∑y=0pN-1[x+y]qαn=1-q1-qd∑a=0d-1lim⁡N→∞1[pN]qd∑y=0pN-1[a+x+dy]qαn=[d]qαn[d]q∑a=0d-1∫Zp[a+xd+y]qαdnq-dydμqd(y)=[d]qαn[d]q∑a=0d-1B̃n,qd(α)(x+ad).Therefore, by (2.18), we obtain the following distribution relation for the modified q-Bernoulli polynomials with weight α.Theorem 2.7.
Ford∈ℕ,n∈ℤ+, one has
(2.19)B̃n,q(α)(x)=[d]qαn[d]q∑a=0d-1B̃n,qd(α)(x+ad).To derive the relation of reflection symmetry of the modifiedq-Bernoulli polynomials with weight α, we evaluate the following p-adic q-integral on ℤp:(2.20)B̃n,q-1(α)(1-x)=∫Zp[1-x+x1]q-αnqx1dμq-1(x1)=1(1-q-α)n∑l=0n(nl)(-1)lqαlx-1αl[αl]q=(-1)nqqαn(1-qα)n∑l=0n(nl)(-1)lqαlxαl[αl]q=qαn-1(-1)nB̃n,q(α)(x).Therefore, by (2.20), we obtain the following reflection symmetry relation of the modified q-Bernoulli polynomials with weight α.Theorem 2.8.
Forn∈ℤ+, one has
(2.21)B̃n,q-1(α)(1-x)=qαn-1(-1)nB̃n,q(α)(x).From (1.3), we note that(2.22)1q∫Zp[1-x]q-αnq-xdμq(x)=(-1)nqαn-1∫Zp[x-1]qαnq-xdμq(x)=(-1)nqαn-1B̃n,q(α)(-1)=B̃n,q-1(α)(2),
and, by (2.6), we get(2.23)B̃n,q(α)(2)=(q2αB̃q(α)+[2]qα)n=(qα(qαB̃q(α)+1)+1)n=∑l=0n(nl)qαl(qαB̃q(α)+1)l=B̃0,q(α)+nqα(qαB̃q(α)+1)1+∑l=2n(nl)qαl(qαB̃q(α)+1)l=(q-1)log⁡q+nqα(α[α]q+B̃1,q(α))+∑l=2n(nl)qαlB̃l,q(α)=nqαα[α]q+∑l=0n(nl)qαlB̃l,q(α).Letn∈ℕ    with   n≥2. Then, by (2.12) and (2.23), we obtain the following theorem.Theorem 2.9.
Forn∈ℕ    with   n≥2, one has
(2.24)B̃n,q(α)(2)-nqαα[α]q=(qαB̃q(α)+1)n=B̃n,q(α).
In particular,
(2.25)1q∫Zp[1-x]q-αnq-xdμq(x)=B̃n,q-1(α)(2)=nqα[α]q+B̃n,q-1(α).



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