# Onp-Adic Analogue of q-Bernstein Polynomials and Related Integrals

Authors: T. Kim; J. Choi; Y. H. Kim; L. C. Jang
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

Recently, Kim's work (in press) introducedq-Bernstein polynomials which are different Phillips' q-Bernstein polynomials introduced in the work by (Phillips, 1996; 1997). The purpose of this paper is to study some properties of several type Kim's q-Bernstein polynomials to express the p-adic q-integral of these polynomials on ℤp associated with Carlitz's q-Bernoulli numbers and polynomials. Finally, we also derive some relations on the p-adic q-integral of the products of several type Kim's q-Bernstein polynomials and the powers of them on ℤp.


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## Body

## 1. Introduction

LetC[0,1]  denote the set of continuous functions on [0,1]. For 0<q<1 and f∈C[0,1], Kim introduced the q-extension of Bernstein linear operator of order n for f as follows: (1.1)Bn,q(f∣x)=∑k=0nf(kn)(nk)[x]qk[1-x]1/qn-k=∑k=0nf(kn)Bk,n(x,q),
where [x]q=(1-qx)/(1-q) (see [1]). Here 𝔹n,q(f∣x) is called Kim's q-Bernstein operator of order n for f. For k,n∈ℤ+(=ℕ∪{0}), Bk,n(x,q)=(nk)[x]qk[1-x]1/qn-k are called the Kim's q-Bernstein polynomials of degree n (see [2–6]).In [7], Carlitz defined a set of numbers ξk=ξk(q) inductively by (1.2)ξ0=1,(qξ+1)k-ξk={1ifk=1,0ifk>1,
with the usual convention of replacing ξk by ξk. These numbers are q-analogues of ordinary Bernoulli numbers Bk, but they do not remain finite for q=1. So he modified the definition as follows: (1.3)β0,q=1,q(qβ+1)k-βk,q={1ifk=1,0ifk>1,
with the usual convention of replacing βk by βk,q (see [7]). These numbers βn,q are called the nth Carlitz q-Bernoulli numbers. And Carlitz's q-Bernoulli polynomials are defined by (1.4)βk,q(x)=(qxβ+[x]q)k=∑i=0k(ki)βi,qqix[x]qk-i.
As q→1, we have βk,q→Bk and βk,q(x)→Bk(x), where Bk and Bk(x) are the ordinary Bernoulli numbers and polynomials, respectively.Letp be a fixed prime number. Throughout this paper, ℤ, ℚ, ℤp, ℚp, and ℂp will denote the ring of rational integers, the field of rational numbers, the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of ℚp, respectively. Let νp be the normalized exponential valuation of ℂp such that |p|p=p-νp(p)=1/p.Letq be regarded as either a complex number q∈ℂ or a p-adic number q∈ℂp. If q∈ℂ, we assume |q|<1, and if q∈ℂp, we normally assume |1-q|p<1.We say thatf is a uniformly differentiable function at a point a∈ℤp and denote this property by f∈UD(ℤp) if the difference quotient Ff(x,y)=(f(x)-f(y))/(x-y) has a limit f′(a) as (x,y)→(a,a) (see [1, 3, 8–13]).Forf∈UD(ℤp), let us begin with the expression(1.5)1[pN]q∑0≤x<pNqxf(x)=∑0≤x<pNf(x)μq(x+pNZp),
representing a q-analogue of the Riemann sums for f (see [11]). The integral of f on ℤp is defined as the limit as n→∞ of the sums (if exists). The p-adic q-integral on a function f∈UD(ℤp) is defined by (1.6)Iq(f)=∫Zpf(x)dμq(x)=lim⁡N→∞1[pN]q∑x=0pN-1f(x)qx,
(see [11]).As was shown in [3], Carlitz's q-Bernoulli numbers can be represented by p-adic q-integral on ℤp as follows:(1.7)∫Zp[x]qmdμq(x)=βm,q,form∈Z+.
Also, Carlitz's q-Bernoulli polynomials βk,q(x) can be represented(1.8)βm,q(x)=∫Zp[x+y]qmdμq(y),form∈Z+,
(see [3]).In this paper, we consider thep-adic analogue of Kim's q-Bernstein polynomials on ℤp and give some properties of the several type Kim's q-Bernstein polynomials to represent the p-adic q-integral on ℤp of these polynomials. Finally, we derive some relations on the p-adic q-integral of the products of several type Kim's q-Bernstein polynomials and the powers of them on ℤp.


## 2.q-Bernstein Polynomials Associated with p-Adic q-Integral on ℤp

In this section, we assume thatq∈ℂp with |1-q|p<1.From (1.5), (1.7) and (1.8), we note that(2.1)∫Zp[1-x+x1]1/qndμ1/q(x1)=qn(q-1)n-1∑l=0n(nl)(-1)lqlxl+1ql+1-1,∫Zp[x+x1]qndμq(x1)=1(1-q)n-1∑l=0n(nl)(-1)lqlxl+11-ql+1.
By (2.1), we get (2.2)(-1)nqn∫Zp[x+x1]qndμq(x1)=∫Zp[1-x+x1]1/qndμ1/q(x1).
Therefore, we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+, one has
(2.3)∫Zp[1-x+x1]1/qndμ1/q(x1)=(-1)nqn∫Zp[x+x1]qndμq(x1).By the definition of Carlitz'sq-Bernoulli numbers and polynomials, we get (2.4)q2βn,q(2)-(n+1)q2+q=q(qβ+1)n=βn,qifn>1.
Thus, we have the following proposition.Proposition 2.2.
Forn∈ℕ with n>1, one has
(2.5)βn,q(2)=1q2βn,q+n+1-1q.It is easy to show that(2.6)[1-x]1/qn=(1-[x]q)n=(-1)nqn[x-1]qn.
Hence, we have (2.7)∫Zp[1-x]1/qndμq(x)=(-1)nqn∫Zp[x-1]qndμq(x).
By (1.8), we get(2.8)∫Zp[1-x]1/qndμq(x)=(-1)nqnβn,q(-1).
By Theorem 2.1 and (2.8), we see that(2.9)∫Zp[1-x]1/qndμq(x)=(-1)nqnβn,q(-1)=βn,1/q(2).
From (2.9) and Proposition 2.2, we have(2.10)∫Zp[1-x]1/qndμq(x)=βn,1/q(2)=q2βn,1/q+n+1-q.
By (1.7) and (2.10), we obtain the following theorem.Theorem 2.3.
Forn∈ℕ with n>1, one has
(2.11)∫Zp[1-x]1/qndμq(x)=q2∫Zp[x]1/qndμ1/q(x)+n+1-q.Taking thep-adic q-integral on ℤp for one Kim's q-Bernstein polynomials, we get(2.12)∫ZpBk,n(x,q)dμq(x)=(nk)∫Zp[x]qk[1-x]1/qn-kdμq(x)=(nk)∑l=0n-k(n-kl)(-1)l∫Zp[x]qk+ldμq(x)=(nk)∑l=0n-k(n-kl)(-1)lβk+l,q,
and, by the q-symmetric property of Bk,n(x,q), we see that(2.13)∫ZpBk,n(x,q)dμq(x)=∫ZpBn-k,n(1-x,1q)dμq(x)=(nk)∑l=0k(kl)(-1)k+l∫Zp[1-x]1/qn-ldμq(x).Forn>k+1, by Theorem 2.3 and (2.13), one has(2.14)∫ZpBk,n(x,q)dμq(x)=(nk)∑l=0k(-1)k+l(kl)[n-l+1-q+q2∫Zp[x]1/qn-ldμ1/q(x)]=(nk)∑l=0k(-1)k+l(kl)[n-l+1-q+q2βn-l,1/q].Letm,n,k∈ℤ+ with m+n>2k+1. Then the p-adic q-integral for the multiplication of two Kim's q-Bernstein polynomials on ℤp can be given by the following relation:(2.15)∫ZpBk,n(x,q)Bk,m(x,q)dμq(x)=(nk)(mk)∫Zp[x]q2k[1-x]1/qn+m-2kdμq(x)=(nk)(mk)∑l=02k(2kl)(-1)l+2kq∫Zp[1-x]1/qn+m-ldμq(x).By Theorem2.3 and (2.15), we get(2.16)∫ZpBk,n(x,q)Bk,m(x,q)dμq(x)=(nk)(mk)∑l=02k(2kl)(-1)l+2k[n+m-l+1-q+q2∫Zp[x]1/qn+m-ldμ1/q(x)]=(nk)(mk)∑l=02k(2kl)(-1)l+2k[n+m-l+1-q+q2βn+m-l,1/q].By the simple calculation, we easily get(2.17)∫ZpBk,n(x,q)Bk,m(x,q)dμq(x)=(nk)(mk)∫Zp[x]q2k[1-x]1/qn+m-2kdμq(x)=(nk)(mk)∑l=0n+m-2k(n+m-2kl)(-1)l∫Zp[x]ql+2kdμq(x)=(nk)(mk)∑l=0n+m-2k(n+m-2kl)(-1)lβl+2k,q.
Continuing this process, we obtain(2.18)∫Zp(∏i=1sBk,ni(x,q))dμq(x)=(∏i=1s(nik))∫Zp[x]qsk[1-x]1/qn1+⋯+ns-skdμq(x)=(∏i=1s(nik))∑l=0sk(skl)(-1)sk+l∫Zp[1-x]1/qn1+⋯+ns-ldμq(x).Lets∈ℕ and n1,…,ns, k∈ℤ+ with n1+n2+⋯+ns>sk+1. By Theorem 2.3 and (2.18), we get(2.19)∫Zp(∏i=1sBk,ni(x,q))dμq(x)=(∏i=1s(nik))∑l=0sk(skl)(-1)sk+l{∑i=1sni-l+1-q+q2∫Zp[x]1/qn1+⋯+ns-ldμ1/q(x)}=(∏i=1s(nik))∑l=0sk(skl)(-1)sk+l{∑i=1sni-l+1-q+q2βn1+⋯+ns-l,1/q}.
From the definition of binomial coefficient, we note that(2.20)∫Zp(∏i=1sBk,ni(x,q))dμq(x)=(∏i=1s(nik))∫Zp[x]qsk[1-x]1/qn1+⋯+ns-skdμq(x)=(∏i=1s(nik))∑l=0n1+⋯+ns-sk(n1+⋯+ns-skl)(-1)l∫Zp[x]qsk+ldμq(x)=(∏i=1s(nik))∑l=0n1+⋯+ns-sk(n1+⋯+ns-skl)(-1)lβsk+l,q,
where s∈ℕ and n1,…,ns, k∈ℤ+.By (2.19) and (2.20), we obtain the following theorem.Theorem 2.4.
(I) Fors∈ℕ and n1,…,ns, k∈ℕ with n1+n2+⋯+ns>sk+1, one has
(2.21)∫Zp(∏i=1sBk,ni(x,q))dμq(x)=(∏i=1s(nik))∑l=0sk(skl)(-1)sk+l{∑i=1sni-l+1-q+q2βn1+⋯+ns-l,1/q}.
(II) Fors∈ℕ and n1,…,ns, k∈ℤ+, one has(2.22)∫Zp(∏i=1sBk,ni(x,q))dμq(x)=(∏i=1s(nik))∑l=0n1+⋯+ns-sk(n1+⋯+ns-skl)(-1)lβsk+l,q.By Theorem2.4, we obtain the following corollary.Corollary 2.5.
Fors∈ℕ and n1,…,ns, k∈ℕ with n1+n2+⋯+ns>sk+1, one has
(2.23)∑l=0sk(skl)(-1)sk+l{∑i=1sni-l+1-q+q2βn1+⋯+ns-l,1/q}=∑l=0n1+⋯+ns-sk(n1+⋯+ns-skl)(-1)lβsk+l,q.Lets∈ℕ and m1,…,ms, n1,…,ns, k∈ℤ+ with m1n1+⋯+msns>(m1+⋯+ms)k+1. Then one has(2.24)∫Zp(∏i=1sBk,nimi(x,q))dμq(x)=(∏i=1s(nik)mi)∑l=0k∑i=1smi(k∑i=1smil)(-1)k∑i=1smi-l×∫Zp[1-x]q∑i=1snimi-ldμq(x)=(∏i=1s(nik)mi)∑l=0k∑i=1smi(k∑i=1smil)(-1)k∑i=1smi-l×{(∑i=1smini-l+1)-q+q2∫Zp[x]1/q∑i=1snimi-ldμ1/q(x)}=(∏i=1s(nik)mi)∑l=0k∑i=1smi(k∑i=1smil)(-1)k∑i=1smi-l×{(∑i=1smini-l+1)-q+q2βn1m1+⋯+nsms-l,1/q}.From the definition of binomial coefficient, one has(2.25)∫Zp(∏i=1sBk,nimi(x,q))dμq(x)=(∏i=1s(nik)mi)∑l=0∑i=1snimi-k∑i=1smi(∑i=1snimi-k∑i=1smil)(-1)l×∫Zp[x]q(m1+⋯+ms)k+ldμq(x)=(∏i=1s(nik)mi)∑l=0∑i=1snimi-k∑i=1smi(∑i=1snimi-k∑i=1smil)×(-1)lβ(m1+⋯+ms)k+l,q.By (2.24) and (2.25), we obtain the following theorem.Theorem 2.6.
Fors∈ℕ and m1,…,ms, n1,…,ns, k∈ℤ+ with m1n1+⋯+msns>(m1+⋯+ms)k+1, one has
(2.26)∑l=0k∑i=1smi(k∑i=1smil)(-1)k∑i=1smi-l{(∑i=1smini-l+1)-q+q2βn1m1+⋯+nsms-l,1/q}=∑l=0∑i=1snimi-k∑i=1smi(∑i=1snimi-k∑i=1smil)(-1)lβ(m1+⋯+ms)k+l,q.



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