# Some Identities on theq-Bernoulli Numbers and Polynomials with Weight 0

Authors: T. Kim; J. Choi; Y. H. Kim
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Recently, Kim (2011) has introduced theq-Bernoulli numbers with weight
α. In this paper, we consider the q-Bernoulli numbers and polynomials with weight α=0 and give p-adic q-integral representation of Bernstein polynomials associated with q-Bernoulli numbers and polynomials with weight 0. From these integral representation on ℤp, we derive some interesting identities on the q-Bernoulli numbers and polynomials with weight 0.


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## Body

## 1. Introduction

Letp be a fixed prime number. Throughout this paper, ℤp,ℚp, and ℂp will denote the ring of  p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ℚp, respectively. Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}.Let|·|p be a p-adic norm with |x|p=p-r, where x=prs/t and (p,s)=(p,t)=(s,t)=1, r∈ℚ. In this paper, we assume that q∈ℂp with |1-q|p<p-1/(p-1) so that qx=exp(xlogq), and [x]q=(1-qx)/(1-q).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:
(1.1)Iq(f)=∫Zpf(x)dμq(x)=lim⁡N→∞∑x=0pN-1f(x)μq(x+pNZp)=lim⁡N→∞1[pN]q∑x=0pN-1f(x)qx,
(see [1–5]). For n∈ℕ, let fn(x)=f(x+n). From (1.1), we note that
(1.2)qnIq(fn)-Iq(f)=(q-1)∑l=0n-1qlf(l)+q-1log⁡q∑l=0n-1qlf′(l),
where f′(l)=df(x)/dx|x=l, (see [3, 6, 7]). In the special case, n=1, we get
(1.3)q∫Zpf(x+1)dμq(x)-∫Zpf(x)dμq(x)=(q-1)f(0)+q-1log⁡qf′(0).Throughout this paper, we assume thatα∈ℚ.Theq-Bernoulli numbers with weight α are defined by Kim [8] as follows:
(1.4)β̃0,q(α)=1,q(qαβ̃q(α)+1)n-β̃n,q(α)={α[α]qifn=1,0ifn>1,
with the usual convention about replacing (β̃q(α))n with β̃n,q(α). From (1.4), we can derive the following equation:
(1.5)β̃n,q(α)=1(1-q)n[α]qn∑l=0n(nl)(-1)lαl+1[αl+1]q=-nα[α]q∑m=0∞qmα+m[m]qαn-1+(1-q)∑m=0∞qm[m]qαn.
By (1.1), (1.4), and (1.5), we get(1.6)β̃n,q(α)=∫Zp[x]qαndμq(x)=1(1-q)n[α]qn∑l=0n(nl)(-1)lαl+1[αl+1]q.
The q-Bernoulli polynomials with weight α are defined by
(1.7)β̃n,q(α)(x)=∫Zp[x+y]qαndμq(y)=∑l=0n(nl)qαlx[x]qαn-lβ̃l,q(α).
By (1.6) and (1.7), we easily see that
(1.8)β̃n,q(α)(x)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlxαl+1[αl+1]q.LetC(ℤp) be the set of continuous functions on ℤp. For f∈C(ℤp), the p-adic analogue of Bernstein operator of order n for f is given by
(1.9)Bn,q(f∣x)=∑k=0nf(kn)Bk,n(x)=∑k=0nf(kn)(nk)xk(1-x)n-k,
where n,k∈ℤ+ (see [1, 9, 10]). For n,k∈ℤ+, the p-adic Bernstein polynomials of degree n are defined by Bk,n(x)=(nk)xk(1-x)n-k for x∈ℤp, (see [1, 10, 11]).In this paper, we consider Bernstein polynomials to express thep-adic q-integral on ℤp and investigate some interesting identities of Bernstein polynomials associated with the q-Bernoulli numbers and polynomials with weight 0 by using the expression of p-adic q-integral on ℤp of these polynomials.


## 2.q-Bernoulli Numbers with Weight 0 and Bernstein Polynomials

In the special case,α=0, the q-Bernoulli numbers with weight 0 will be denoted by β̃n,q(0)=β̃n,q. From (1.4), (1.5), and (1.6), we note that
(2.1)∑n=0∞β̃n,qtnn!=∑n=0∞∫Zpxndμq(x)tnn!=∫Zpextdμq(x)=(q-1log⁡q)(t+log⁡qqet-1).It is easy to show that(2.2)t+log⁡qqet-1=tq-1(1-q-1et-q-1)+log⁡qq-1(1-q-1et-q-1)=tq-1∑n=0∞Hn(q-1)tnn!+log⁡qq-1∑n=0∞Hn(q-1)tnn!=1q-1∑n=1∞nHn-1(q-1)tnn!+log⁡qq-1∑n=0∞Hn(q-1)tnn!,
where Hn(q-1) are the nth Frobenius-Euler numbers.By (2.1) and (2.2), we get
(2.3)β̃n,q={1ifn=0,nlog⁡qHn-1(q-1)+Hn(q-1)ifn>0.
Therefore, we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+, we have
(2.4)β̃n,q={1ifn=0,nlog⁡qHn-1(q-1)+Hn(q-1)ifn>0,
where Hn(q-1) are the nth Frobenius-Euler numbers.From (1.5), (1.6), and (1.7), we have
(2.5)β̃0,q=1,q(β̃q+1)n-β̃n,q={q-1log⁡qifn=1,0ifn>1,
with the usual convention about replacing (β̃q)n with β̃n,q. By (1.7), the nth q-Bernoulli polynomials with weight 0 are given by
(2.6)β̃n,q(x)=∫Zp(x+y)ndμq(y)=∑l=0n(nl)xn-lβ̃l,q.
From (2.6), we can derive the following function equation:
(2.7)(q-1log⁡q)(t+log⁡qqet-1)ext=∑n=0∞β̃n,q(x)tnn!.
Thus, by (2.7), we get that
(2.8)β̃n,q-1(1-x)=(-1)nβ̃n,q(x),forn∈Z+.
By the definition of p-adic q-integral on ℤp, we see that (2.9)∫Zp(1-x)ndμq(x)=(-1)n∫Zp(x-1)ndμq(x)=(-1)nβ̃n,q(-1).
Therefore, by (2.8) and (2.9), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, we have
(2.10)(-1)nβ̃n,q(x)=β̃n,q-1(1-x).
In particular, x=-1, we get
(2.11)∫Zp(1-y)ndμq(y)=(-1)nβ̃n,q(-1)=β̃n,q-1(2).From (2.5), we can derive the following equation:
(2.12)q2β̃n,q(2)=q2+nqq-1log⁡q-q+β̃n,q,ifn>1.
Therefore, by (2.12), we obtain the following theorem.Theorem 2.3.
Forn∈ℕ with n>1, we have
(2.13)β̃n,q(2)=1+nq(q-1log⁡q)-1q+1q2β̃n,q.Taking thep-adic q-integral on ℤp for one Bernstein polynomials in (1.9), we get
(2.14)∫ZpBk,n(x)dμq(x)=(nk)∫Zpxk(1-x)n-kdμq(x)=(nk)∑l=0n-k(n-kl)(-1)l∫Zpxk+ldμq(x)=(nk)∑l=0n-k(n-kl)(-1)lβ̃k+l,q.
From the symmetry of Bernstein polynomials, we note that
(2.15)∫ZpBk,n(x)dμq(x)=∫ZpBn-k,n(1-x)dμq(x)=(nk)∑l=0k(kl)(-1)k+l∫Zp(1-x)n-ldμq(x).Letn>k+1. Then, by Theorem 2.3 and (2.15), we get
(2.16)∫ZpBk,n(x)dμq(x)=(nk)∑l=0k(kl)(-1)k+l(1-n-lq-1(q-1-1log⁡q)-q+q2β̃n-l,q-1)={1+n(q-1log⁡q)-q+q2β̃n,q-1ifk=0,n(1-qlog⁡q)+nq2β̃n,q-1+nq2β̃n-1,q-1ifk=1,(nk)q2∑l=0k(kl)(-1)k+lβ̃n-l,q-1ifk>1.By comparing the coefficients on the both sides of (2.14) and (2.16), we obtain the following theorem.Theorem 2.4.
Forn,k∈ℤ+ with n>k+1, we have
(2.17)∑l=0n-1(n-1l)(-1)lβ̃1+l,q=1-qlog⁡q+q2β̃n,q-1+q2β̃n-1,q-1,∑l=0n-k(n-kl)(-1)lβ̃k+l,q=q2∑l=0k(kl)(-1)k+lβ̃n-l,q-1,ifk>1.
In particular, when k=0, we have
(2.18)∑l=0n(nl)(-1)lβ̃l,q=1+nq-1log⁡q-q+q2β̃n,q-1.Letm,n,k∈ℤ+ with m+n>2k+1. Then we see that
(2.19)∫ZpBk,n(x)Bk,m(x)dμq(x)=(nk)(mk)∫Zpx2k(1-x)n+m-2kdμq(x)=(nk)(mk)∑l=02k(2kl)(-1)l+2k∫Zp(1-x)n+m-ldμq(x)=(nk)(mk)∑l=02k(2kl)(-1)l+2k(1-(n+m-l)(1-qlog⁡q)-q+q2β̃n+m-l,q-1)={1+(n+m)(q-1log⁡q)-q+q2β̃n+m,q-1ifk=0,(nk)(mk)q2∑l=02k(2kl)(-1)l+2kβ̃n+m-l,q-1ifk>0.
For m,n,k∈ℤ+, we have
(2.20)∫ZpBk,n(x)Bk,m(x)dμq(x)=(nk)(mk)∫Zpx2k(1-x)n+m-2kdμq(x)=(nk)(mk)∑l=0n+m-2k(n+m-2kl)(-1)l∫Zpx2k+ldμq(x)=(nk)(mk)∑l=0n+m-2k(n+m-2kl)(-1)lβ̃l+2k,q.By comparing the coefficients on the both sides of (2.19) and (2.20), we obtain the following theorem.Theorem 2.5.
Form,n,k∈ℤ+ with m+n>2k+1, we have
(2.21)∑l=0n+m(n+ml)(-1)lβ̃l,q=1+(n+m)(q-1log⁡q)-q+q2β̃n+m,q-1.
In particular, when k≠0, we have
(2.22)∑l=0n+m-2k(n+m-2kl)(-1)lβ̃l+2k,q=q2∑l=02k(2kl)(-1)l+2kβ̃n+m-l,q-1.Fors∈ℕ, let k,n1,…,ns∈ℤ+ with n1+n2+⋯+ns>sk+1. By the same method above, we get
(2.23)∫Zp(∏i=1sBk,ni(x))dμq(x)={1+(∑i=1sni)(q-1log⁡q)-q+q2β̃n1+n2+⋯+ns,q-1ifk=0,(∏i=1s(nik))q2∑l=0sk(skl)(-1)l+skβ̃n1+n2+⋯+ns-l,q-1ifk>0.
From the binomial theorem, we note that (2.24)∫Zp(∏i=1sBk,ni(x))dμq(x)=(∏i=1s(nik))∑l=0n1+⋯+ns-sk(n1+⋯+ns-skl)(-1)lβ̃l+sk,q.By comparing the coefficients on the both sides of (2.23) and (2.24), we obtain the following theorem.Theorem 2.6.
Fors∈ℕ, let k,n1,…,ns∈ℤ+ with n1+n2+⋯+ns>sk+1. Then, we have
(2.25)∑l=0n1+⋯+ns(n1+⋯+nsl)(-1)lβ̃l,q=1+(∑i=1sni)(q-1log⁡q)-q+q2β̃n1+⋯+ns,q-1.
In particular, when k≠0, we have
(2.26)∑l=0n1+⋯+ns-sk(n1+⋯+ns-skl)(-1)lβ̃l+sk,q=q2∑l=0sk(skl)(-1)l+skβ̃n1+⋯+ns-l,q-1.



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*Source: 361484-2011-11-17.xml*