Representation of \((p,q)\) Bernstein polynomials in terms of \((p,q)\) Jacobi polynomials

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R E S E A R C H

Open Access

Representation of

(

p

,

q

)

-Bernstein

polynomials in terms of

(

p

,

q

)

-Jacobi

polynomials

F Soleyman

1

_{, I Area}

2*

_{, M Masjed-Jamei}

1

_{and JJ Nieto}

3

*_{Correspondence: [email protected]} 2_{Departamento de Matemática} Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo, Campus As Lagoas s/n, Ourense, 32004, Spain

Full list of author information is available at the end of the article

Abstract

A representation of (p,q)-Bernstein polynomials in terms of (p,q)-Jacobi polynomials is obtained.

MSC: Primary 34B24; secondary 39A70

Keywords: (p,q)-Bernstein polynoimals; (p,q)-Pearson diﬀerence equation; (p,q)-orthogonal solutions; (p,q)-diﬀerence operator

1 Introduction

Classical univariate Bernstein polynomials were introduced by Bernstein in a constructive proof for the Stone-Weierstrass approximation theorem [], and they are deﬁned as []

bn_i(x) =

n i

xi( –x)n–i, i= , , . . . ,n.

They form a basis of polynomials and satisfy a number of important properties as non-negativity (bn

i(x)≥ for ≤x≤), partition of unity (

n

i=bni(x) = ) or symmetry (bni(x) = bn_n_–_i( –x)).

For a given real-valued deﬁned and bounded functionf on the interval [, ], thenth Bernstein polynomial forf is

Bn(f)(x) = n

k=

bn_k(x)f

k n

.

Then, for each pointxof continuity off, we haveBn(f)(x)→f(x) asn→ ∞. Moreover,

iff is continuous on [, ] thenBn(f) converges uniformly tof asn→ ∞. Also, for each

pointxof diﬀerentiability off, we haveB_n(f)(x)→f(x) asn→ ∞and iffis continuously diﬀerentiable on [, ] thenB_n(f) converges tofuniformly asn→ ∞.

Bernstein polynomials have been generalized in the framework ofq-calculus. More pre-cisely, Lupaş [] initiated the application ofq-calculus in area of the approximation theory, and introduced theq-Bernstein polynomials. Later on, Philips [] proposed and studied otherq-Bernstein polynomials. In both the classical case and in itsq-analogs, expansions

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of Bernstein polynomials have been obtained in terms of appropriate orthogonal bases [, ].

Mursaleenet al.[] recently introduced ﬁrst the concept of (p,q)-calculus in approx-imation theory and studied the (p,q)-analog of Bernstein operators. The approximation properties for these operators based on Korovkin’s theorem and some direct theorems were considered []. Also, many well-known approximation operators have been intro-duced using these techniques, such as Bleimann-Butzer-Hahn operators [] and Szász-Mirakyan operators []. Very recently Milovanovićet al.[] considered a (p,q)-analog of the beta operators and using it proposed an integral modiﬁcation of the generalized Bernstein polynomials. (p,q)-analogs of classical orthogonal polynomials have been char-acterized in [].

The main aim of this work is to obtain a representation of (p,q)-Bernstein polynomials in terms of suitable (p,q)-orthogonal polynomials, where the connection coeﬃcients are proved to satisfy a three-term recurrence relation. For this purpose, we have divided the work in two sections. First, we present the basic deﬁnitions and notations. Later, in Sec-tion  we obtain the main results of this work relating (p,q)-Bernstein polynomials and (p,q)-Jacobi orthogonal polynomials.

2 Basic deﬁnitions and notations

Next, we summarize the basic deﬁnitions and results which can be found in [–] and the references therein.

The (p,q)-power is deﬁned as

(a,b); (p,q)_k=

k–

j=

apj_–_bqj _with₍_a_,_b_{); (}_p_,_q₎

= . ()

The (p,q)-hypergeometric series is deﬁned as

rs

(ap,aq), . . . , (arp,arq)

(bp,bq), . . . , (bsp,bsq)

(p,q);z

= ∞

j=

((ap,aq), . . . , (arp,arq); (p,q))j

((bp,bq), . . . , (bsp,bsq); (p,q))j zj

((p,q); (p,q))j

(–)j(q/p)j(j–) +s–r_, _()

where

(ap,aq), . . . , (arp,arq); (p,q)

j= r

s=

(asp,asq); (p,q)

j,

andr,s∈Z+andap,aq, . . . ,arp,arq,bp,bq, . . . ,bsp,bsq,z∈C.

The (p,q)-diﬀerence operator is deﬁned as (seee.g.[])

(Dp,qf)(x) =Lp

f(x) –Lqf(x)

(p–q)x , x= , ()

where the shift operator is deﬁned by

Lah(x) =h(ax), ()

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The (p,q)-Bernstein polynomials are deﬁned as

bn_i(x;p,q) =pn(–n)/

n i

p,q

pi(i–)/xi(,x); (p,q)_n_–_i, ()

and can be expanded in the basis{xk_} k≥as

bn_i(x;p,q) =

n

k=i

(–)k–iq(k–i)(k–i–)/p((i–)i+k(k–n+))

n k

p,q

k i

p,q

xk. ()

From the deﬁnition of (p,q)-Bernstein polynomials it is possible to derive the basic prop-erties of (p,q)-Bernstein polynomials.

() Partition of unity

n

i=

bn_i(x;p,q) = .

() End-point properties

bn_i(;p,q) = ⎧ ⎨ ⎩

, i= ,

, otherwise, b

n

i(;p,q) =

⎧ ⎨ ⎩

, i=n,

, otherwise.

The (p,q)-Jacobi polynomials are deﬁned by

Pn(x;α,β;p,q) =

(p–n,q–n), (pα+β+n+_,_qα+β+n+₎ (pβ+_,_qβ+₎

(p,q);xq –α

p

, ()

and they satisfy the second order (p,q)-diﬀerence equation

qx(qx–p)

p

D

p,qy

(x) +

x(pα+β+_q–α–β_–_q_{) –}_pβ+_q–β₊_pq

p₍_p_–_q₎

Lp

(Dp,qy)(x)

+ [n]p,q

qp–n–_–_pα+β–_q–α–β–n p–q

Lpqy(x) = . ()

The (p,q)-Jacobi polynomials satisfy the three-term recurrence relation

P(x;α,β;p,q) = , P(x;α,β;p,q) =x–B(α,β;p,q),

Pn+(x;α,β;p,q) =

x–Bn(α,β;p,q)

Pn(x;α,β;p,q) –Cn(α,β;p,q)Pn–(x;α,β;p,q),

where

Bn(α,β;p,q) =

pn+_qα+n+ (p–q)_[_α₊_β_{+ }_n_]

p,q[α+β+ n+ ]p,q

×pβ+qβqα+β+n+– (p+q)pα+qαpβ+nqβ+n

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and

Cn(α,β;p,q) =

pβ+n+_qα+β+n+_[_n_]

p,q[α+n]p,q[β+n]p,q[α+β+n]p,q

[α+β+ n– ]p,q([α+β+ n]p,q)[α+β+ n+ ]p,q

. ()

3 Representation of(p,q)-Bernstein polynomials in terms of(p,q)-Jacobi polynomials

Lemma . The (p,q)-Bernstein polynomials satisfy the following ﬁrst order (p,q) -diﬀerence equation:

(px– )xDp,qbni

(x;p,q) +–p–n[n]p,qx+p–i[i]p,q

bn_i(px;p,q) = . ()

Proof The result can be obtained by equating the coeﬃcients inxj_.

If we introduce the ﬁrst order (p,q)-diﬀerence operator

Li,n= (px– )xDp,q+

–p–n[n]p,qx+p–i[i]p,q

Lp, ()

then

Li,nbni(x;p,q) = .

Lemma . The(p,q)-Jacobi polynomials satisfy the following structure relation:

x(px– )Dp,q

Pn

px;α,β;p,q

= [n]p,qp–n–Pn+

p_x_;_α_,_β_;_p_,_q₊ (n)Pn

p_x_;_α_,_β_;_p_,_q

+(n)Pn–

px;α,β;p,q, ()

where

(n) = –[n]p,q(–(p+q)q

α+n_–_pβ+n₊_pα+β+n+₊_qα+β+n+_)[_α₊_β₊_n_{+ ]}

p,q

(p–q)[α+β+ n]p,q[α+β+ n+ ]p,q

,

(n) =

qα+n_pβ+n+_[_n_]

p,q[α+n]p,q[β+n]p,q[α+β+n]p,q[α+β+n+ ]p,q

[α+β+ n– ]p,q([α+β+ n]p,q)[α+β+ n+ ]p,q

.

Proof The result follows from () by equating the coeﬃcients inxj.

Theorem . The(p,q)-Bernstein polynomials deﬁned in()have the following represen-tation in terms of(p,q)-Jacobi polynomials deﬁned in():

bn_i(x;p,q) =

n

k=

Hk(i,n;α,β;p,q)Pk

px;α,β;p,q, ()

where the connection coeﬃcients Hk(i,n;α,β;p,q)satisfy the following three-term recur-rence relation:

Hk–(i,n;α,β;p,q)(k– ,i,n;α,β;p,q) +Hk(i,n;α,β;p,q)(k,i,n;α,β;p,q)

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valid for≤k≤n– with initial conditions

Hn+(i,n;α,β;p,q) = , ()

Hn(i,n;α,β;p,q) = (–)n+q–



(–n)n_p–n(n+)/+k(k+)/

n i

p,q

, () and ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(k,i,n;α,β;p,q) =p–k–[k]p,q–p–n–[n]p,q,

(k,i,n;α,β;p,q) =p–i[i]p,q–p––n[n]p,qBk(α,β;p,q) +(k),

(k,i,n;α,β;p,q) = –p–n–[n]p,qCk(α,β;p,q) +(k).

()

Proof In order to obtain the result we shall apply the so-called Navima algorithm (seee.g.

[, ] and the references therein) for solving connection problems. If we apply the ﬁrst order linear operatorLi,ndeﬁned in () to both sides of () we have

 =

n

k=

Hk(i,n;α,β;p,q)Li,nPk

px;α,β;p,q

=

n

k=

Hk(i,n;α,β;p,q)

(px– )xDp,q

Pk

px;α,β;p,q

+–p–n[n]p,qx+p–i[i]p,q

Pk

px;α,β;p,q.

From the three-term recurrence relation for (p,q)-Jacobi polynomials it yields

–p–n[n]p,qx+p–i[i]p,q

Pk

px;α,β;p,q

= –p–n–[n]p,qPk+

px;α,β;p,q

+p––n–i–pn+[i]p,q+pi[n]p,qBk(α,β;p,q)

Pk

px;α,β;p,q

–p–n–[n]p,qCk(α,β;p,q)Pk–

px;α,β;p,q.

Therefore, by using the structure relation for (p,q)-Jacobi polynomials () we have

(px– )xDp,q

Pk

px;α,β;p,q+–p–n[n]p,qx+p–i[i]p,q

Pk

px;α,β;p,q

=(k,i,n;α,β;p,q)Pk+

px;α,β;p,q+(k,i,n;α,β;p,q)Pk

px;α,β;p,q

+(k,i,n;α,β;p,q)Pk–

px;α,β;p,q,

wherei(k,i,n;α,β;p,q) are given in ().

As a consequence,

 =

n

k=

Hk(i,n;α,β;p,q)

(k,i,n;α,β;p,q)Pk+

px;α,β;p,q

+(k,i,n;α,β;p,q)Pk

px;α,β;p,q+(k,i,n;α,β;p,q)Pk–

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By using the linear independence of{Pk(px;α,β;p,q)}we obtain the three-term

recur-rence relation () for the connection coeﬃcientsHk(i,n;α,β;p,q), where the initial

con-ditions are obtained by equating the highest power inxk.

4 Conclusions

In this work we have obtained a three-term recurrence relation for the coeﬃcients in the expansion of (p,q)-Bernstein polynomials in terms of (p,q)-Jacobi polynomials. For our purposes some auxiliary results both for (p,q)-Bernstein polynomials and (p,q)-Jacobi polynomials have been derived.

Acknowledgements

The authors thank both reviewers for their comments. This work has been partially supported by the Ministerio de Ciencia e Innovación of Spain under grant MTM2016-75140-P, co-ﬁnanced by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015-004 and R 2016/022. The ﬁrst author thanks the hospitality of Departamento de Estatística, Análise Matemática e Optimización of Universidade de Santiago de Compostela, and Departamento de Matemática Aplicada II of Universidade de Vigo during her visit.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, FS, IA, MMJ, and JJN contributed to each part of this study equally and read and approved the ﬁnal version of the manuscript.

Author details

1_{Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.}2_{Departamento de} Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo, Campus As Lagoas s/n, Ourense, 32004, Spain. 3_{Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain.}

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Received: 8 February 2017 Accepted: 28 June 2017

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