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

Abstract

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

Introduction

Classical univariate Bernstein polynomials were introduced by Bernstein in a constructive proof for the Stone-Weierstrass approximation theorem [1], and they are defined as [2]

$$b_i^n(x)=\left(\genfrac{}{}{0ex}{}{n}{i}\right)x^i{(1-x)}^{n-i},i=0,1,\dots ,n.$$

They form a basis of polynomials and satisfy a number of important properties as non-negativity ($b_i^n(x)\ge 0$ for $0\le x\le 1$), partition of unity (${\sum }_{i=0}^n{b}_i^n(x)=1$) or symmetry ($b_i^n(x)=b_{n-i}^n(1-x)$).

For a given real-valued defined and bounded function f on the interval $[0,1]$, the nth Bernstein polynomial for f is

$$B_n(f)(x)=\sum _{k=0}^n{b}_k^n(x)f\left(\frac{k}{n}\right).$$

Then, for each point x of continuity of f, we have $B_n(f)(x)\to f(x)$ as $n\to \mathrm{\infty }$. Moreover, if f is continuous on $[0,1]$ then $B_n(f)$ converges uniformly to f as $n\to \mathrm{\infty }$. Also, for each point x of differentiability of f, we have $B_n^{\prime }(f)(x)\to f^{\prime }(x)$ as $n\to \mathrm{\infty }$ and if f is continuously differentiable on $[0,1]$ then $B_n^{\prime }(f)$ converges to $f^{\prime }$ uniformly as $n\to \mathrm{\infty }$.

Bernstein polynomials have been generalized in the framework of q-calculus. More precisely, Lupaş [3] initiated the application of q-calculus in area of the approximation theory, and introduced the q-Bernstein polynomials. Later on, Philips [4] proposed and studied other q-Bernstein polynomials. In both the classical case and in its q-analogs, expansions of Bernstein polynomials have been obtained in terms of appropriate orthogonal bases [5, 6].

Mursaleen et al. [7] recently introduced first the concept of $(p,q)$-calculus in approximation 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 [8]. Also, many well-known approximation operators have been introduced using these techniques, such as Bleimann-Butzer-Hahn operators [9] and Szász-Mirakyan operators [10]. Very recently Milovanović et al. [11] 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 characterized in [12].

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 Section 3 we obtain the main results of this work relating $(p,q)$-Bernstein polynomials and $(p,q)$-Jacobi orthogonal polynomials.

Basic definitions and notations

Next, we summarize the basic definitions and results which can be found in [13–18] and the references therein.

The $(p,q)$-power is defined as

$${((a,b);(p,q))}_k=\prod _{j=0}^{k-1}(a{p}^j-b{q}^j)\text{with }{((a,b);(p,q))}_0=1.$$ 1

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

$$\begin{array}{rl}& {}_r\mathrm{\Phi }_s\left(\begin{array}{c}(a_{1p},a_{1q}),\dots ,(a_{rp},a_{rq})\\ (b_{1p},b_{1q}),\dots ,(b_{sp},b_{sq})\end{array}\right|(p,q);z)\\ & =\sum _{j=0}^{\mathrm{\infty }}\frac{{((a_{1p},a_{1q}),\dots ,(a_{rp},a_{rq});(p,q))}_j}{{((b_{1p},b_{1q}),\dots ,(b_{sp},b_{sq});(p,q))}_j}\frac{z^j}{{((p,q);(p,q))}_j}{\left({(-1)}^j{(q/p)}^{\frac{j(j-1)}{2}}\right)}^{1+s-r},\end{array}$$ 2

where

$${((a_{1p},a_{1q}),\dots ,(a_{rp},a_{rq});(p,q))}_j=\prod _{s=1}^r{((a_{sp},a_{sq});(p,q))}_j,$$

and $r,s\in {\mathbb{Z}}_{+}$ and $a_{1p},a_{1q},\dots ,a_{rp},a_{rq},b_{1p},b_{1q},\dots ,b_{sp},b_{sq},z\in \mathbb{C}$.

The $(p,q)$-difference operator is defined as (see e.g. [14])

$$({\mathcal{D}}_{p,q}f)(x)=\frac{{\mathcal{L}}_{p}f(x)-{\mathcal{L}}_{q}f(x)}{(p-q)x},x\ne 0,$$ 3

where the shift operator is defined by

$${\mathcal{L}}_{a}h(x)=h(ax),$$ 4

and $({\mathcal{D}}_{p,q}f)(0)=f^{\prime }(0)$, provided that f is differentiable at 0.

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

$$b_i^n(x;p,q)=p^{n(1-n)/2}{\left[\begin{array}{c}n\\ i\end{array}\right]}_{p,q}{p}^{i(i-1)/2}{x}^i{((1,x);(p,q))}_{n-i},$$ 5

and can be expanded in the basis ${\{x^k\}}_{k\ge 0}$ as

$$b_i^n(x;p,q)=\sum _{k=i}^n{(-1)}^{k-i}{q}^{(k-i)(k-i-1)/2}{p}^{\frac{1}{2}((i-1)i+k(k-2n+1))}{\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}{\left[\begin{array}{c}k\\ i\end{array}\right]}_{p,q}{x}^k.$$ 6

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

1. Partition of unity

   $$\sum _{i=0}^n{b}_i^n(x;p,q)=1.$$
2. End-point properties

   $$b_i^n(0;p,q)=\{\begin{array}{cc}1,\hfill & i=0,\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}{b}_i^n(1;p,q)=\{\begin{array}{cc}1,\hfill & i=n,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

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

$$P_n(x;\alpha ,\beta ;p,q)={}_2{\mathrm{\Phi }}_1\left(\begin{array}{c}(p^{-n},q^{-n}),(p^{\alpha +\beta +n+1},q^{\alpha +\beta +n+1})\\ (p^{\beta +1},q^{\beta +1})\end{array}\right|(p,q);\frac{x{q}^{-\alpha }}{p}),$$ 7

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

$$\begin{array}{rl}& \frac{qx(qx-p)}{p^2}\left({\mathcal{D}}_{p,q}^{2}y\right)(x)+\left(\frac{x(p^{\alpha +\beta +2}{q}^{-\alpha -\beta }-q^2)-p^{\beta +2}{q}^{-\beta }+pq}{p^2(p-q)}\right){\mathcal{L}}_p(({\mathcal{D}}_{p,q}y)(x))\\ & +{[n]}_{p,q}\left(\frac{q{p}^{-n-2}-p^{\alpha +\beta -1}{q}^{-\alpha -\beta -n}}{p-q}\right){\mathcal{L}}_{pq}y(x)=0.\end{array}$$ 8

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

$$\begin{array}{c}{P}_0(x;\alpha ,\beta ;p,q)=1,P_1(x;\alpha ,\beta ;p,q)=x-B_0(\alpha ,\beta ;p,q),\\ P_{n+1}(x;\alpha ,\beta ;p,q)=(x-B_n(\alpha ,\beta ;p,q))P_n(x;\alpha ,\beta ;p,q)-C_n(\alpha ,\beta ;p,q)P_{n-1}(x;\alpha ,\beta ;p,q),\end{array}$$

where

$$\begin{array}{rl}{B}_n(\alpha ,\beta ;p,q)=& \frac{p^{n+2}{q}^{\alpha +n+1}}{{(p-q)}^2{[\alpha +\beta +2n]}_{p,q}{[\alpha +\beta +2n+2]}_{p,q}}\\ & \times ((p^{\beta }+q^{\beta })q^{\alpha +\beta +2n+1}-(p+q)(p^{\alpha }+q^{\alpha })p^{\beta +n}{q}^{\beta +n}\\ & +(p^{\beta }+q^{\beta })p^{\alpha +\beta +2n+1})\end{array}$$ 9

and

$$C_n(\alpha ,\beta ;p,q)=\frac{p^{\beta +2n+3}{q}^{2\alpha +\beta +2n+1}{[n]}_{p,q}{[\alpha +n]}_{p,q}{[\beta +n]}_{p,q}{[\alpha +\beta +n]}_{p,q}}{{[\alpha +\beta +2n-1]}_{p,q}{({[\alpha +\beta +2n]}_{p,q})}^2{[\alpha +\beta +2n+1]}_{p,q}}.$$ 10

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

Lemma 3.1

The $(p,q)$-Bernstein polynomials satisfy the following first order $(p,q)$-difference equation:

$$(px-1)x\left(D_{p,q}{b}_i^n\right)(x;p,q)+(-p^{1-n}{[n]}_{p,q}x+p^{-i}{[i]}_{p,q})b_i^n(px;p,q)=0.$$ 11

Proof

The result can be obtained by equating the coefficients in $x^j$. □

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

$$L_{i,n}=(px-1)x{D}_{p,q}+(-p^{1-n}{[n]}_{p,q}x+p^{-i}{[i]}_{p,q}){\mathcal{L}}_p,$$ 12

then

$$L_{i,n}{b}_i^n(x;p,q)=0.$$

Lemma 3.2

The $(p,q)$-Jacobi polynomials satisfy the following structure relation:

$$\begin{array}{rl}& x(px-1)D_{p,q}\left(P_n(p^{2}x;\alpha ,\beta ;p,q)\right)\\ & ={[n]}_{p,q}{p}^{-n-2}{P}_{n+1}(p^{3}x;\alpha ,\beta ;p,q)+{\varpi }_1(n)P_n(p^{3}x;\alpha ,\beta ;p,q)\\ & +{\varpi }_2(n)P_{n-1}(p^{3}x;\alpha ,\beta ;p,q),\end{array}$$ 13

where

$$\begin{array}{c}{\varpi }_1(n)=-\frac{{[n]}_{p,q}(-(p+q)q^{\alpha +n}-p^{\beta +n}+p^{\alpha +\beta +2n+1}+q^{\alpha +\beta +2n+1}){[\alpha +\beta +n+1]}_{p,q}}{(p-q){[\alpha +\beta +2n]}_{p,q}{[\alpha +\beta +2n+2]}_{p,q}},\\ {\varpi }_2(n)=\frac{q^{\alpha +n}{p}^{\beta +2n+1}{[n]}_{p,q}{[\alpha +n]}_{p,q}{[\beta +n]}_{p,q}{[\alpha +\beta +n]}_{p,q}{[\alpha +\beta +n+1]}_{p,q}}{{[\alpha +\beta +2n-1]}_{p,q}{({[\alpha +\beta +2n]}_{p,q})}^2{[\alpha +\beta +2n+1]}_{p,q}}.\end{array}$$

Proof

The result follows from (7) by equating the coefficients in $x^j$. □

Theorem 3.1

The $(p,q)$-Bernstein polynomials defined in (5) have the following representation in terms of $(p,q)$-Jacobi polynomials defined in (7):

$$b_i^n(x;p,q)=\sum _{k=0}^n{H}_k(i,n;\alpha ,\beta ;p,q)P_k(p^{2}x;\alpha ,\beta ;p,q),$$ 14

where the connection coefficients $H_k(i,n;\alpha ,\beta ;p,q)$ satisfy the following three-term recurrence relation:

$$\begin{array}{rl}& H_{k-1}(i,n;\alpha ,\beta ;p,q){\mathrm{\Lambda }}_1(k-1,i,n;\alpha ,\beta ;p,q)+H_k(i,n;\alpha ,\beta ;p,q){\mathrm{\Lambda }}_2(k,i,n;\alpha ,\beta ;p,q)\\ & +H_{k+1}(i,n;\alpha ,\beta ;p,q){\mathrm{\Lambda }}_3(k+1,i,n;\alpha ,\beta ;p,q)=0,\end{array}$$ 15

valid for $1\le k\le n-1$ with initial conditions

$$H_{n+1}(i,n;\alpha ,\beta ;p,q)=0,$$ 16

$$H_n(i,n;\alpha ,\beta ;p,q)={(-1)}^{n+1}{q}^{-\frac{1}{2}(1-n)n}{p}^{-n(n+3)/2+k(k+1)/2}{\left[\begin{array}{c}n\\ i\end{array}\right]}_{p,q},$$ 17

and

$$\{\begin{array}{c}{\mathrm{\Lambda }}_1(k,i,n;\alpha ,\beta ;p,q)=p^{-k-2}{[k]}_{p,q}-p^{-n-2}{[n]}_{p,q},\hfill \\ {\mathrm{\Lambda }}_2(k,i,n;\alpha ,\beta ;p,q)=p^{-i}{[i]}_{p,q}-p^{-2-n}{[n]}_{p,q}{B}_k(\alpha ,\beta ;p,q)+{\varpi }_1(k),\hfill \\ {\mathrm{\Lambda }}_3(k,i,n;\alpha ,\beta ;p,q)=-p^{-n-2}{[n]}_{p,q}{C}_k(\alpha ,\beta ;p,q)+{\varpi }_2(k).\hfill \end{array}$$ 18

Proof

In order to obtain the result we shall apply the so-called Navima algorithm (see e.g. [19, 20] and the references therein) for solving connection problems. If we apply the first order linear operator $L_{i,n}$ defined in (12) to both sides of (14) we have

$$\begin{array}{rl}0=& \sum _{k=0}^n{H}_k(i,n;\alpha ,\beta ;p,q)L_{i,n}{P}_k(p^{2}x;\alpha ,\beta ;p,q)\\ =& \sum _{k=0}^n{H}_k(i,n;\alpha ,\beta ;p,q)((px-1)x{D}_{p,q}\left(P_k(p^{2}x;\alpha ,\beta ;p,q)\right)\\ & +(-p^{1-n}{[n]}_{p,q}x+p^{-i}{[i]}_{p,q})P_k(p^{3}x;\alpha ,\beta ;p,q)).\end{array}$$

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

$$\begin{array}{c}(-p^{1-n}{[n]}_{p,q}x+p^{-i}{[i]}_{p,q})P_k(p^{3}x;\alpha ,\beta ;p,q)\\ =-p^{-n-2}{[n]}_{p,q}{P}_{k+1}(p^{3}x;\alpha ,\beta ;p,q)\\ +p^{-2-n-i}(-p^{n+2}{[i]}_{p,q}+p^i{[n]}_{p,q}{B}_k(\alpha ,\beta ;p,q))P_k(p^{3}x;\alpha ,\beta ;p,q)\\ -p^{-n-2}{[n]}_{p,q}{C}_k(\alpha ,\beta ;p,q)P_{k-1}(p^{3}x;\alpha ,\beta ;p,q).\end{array}$$

Therefore, by using the structure relation for $(p,q)$-Jacobi polynomials (13) we have

$$\begin{array}{c}(px-1)x{D}_{p,q}\left(P_k(p^{2}x;\alpha ,\beta ;p,q)\right)+(-p^{1-n}{[n]}_{p,q}x+p^{-i}{[i]}_{p,q})P_k(p^{3}x;\alpha ,\beta ;p,q)\\ ={\mathrm{\Lambda }}_1(k,i,n;\alpha ,\beta ;p,q)P_{k+1}(p^{3}x;\alpha ,\beta ;p,q)+{\mathrm{\Lambda }}_2(k,i,n;\alpha ,\beta ;p,q)P_k(p^{3}x;\alpha ,\beta ;p,q)\\ +{\mathrm{\Lambda }}_3(k,i,n;\alpha ,\beta ;p,q)P_{k-1}(p^{3}x;\alpha ,\beta ;p,q),\end{array}$$

where ${\mathrm{\Lambda }}_i(k,i,n;\alpha ,\beta ;p,q)$ are given in (18).

As a consequence,

$$\begin{array}{rl}0=& \sum _{k=0}^n{H}_k(i,n;\alpha ,\beta ;p,q)({\mathrm{\Lambda }}_1(k,i,n;\alpha ,\beta ;p,q)P_{k+1}(p^{3}x;\alpha ,\beta ;p,q)\\ & +{\mathrm{\Lambda }}_2(k,i,n;\alpha ,\beta ;p,q)P_k(p^{3}x;\alpha ,\beta ;p,q)+{\mathrm{\Lambda }}_3(k,i,n;\alpha ,\beta ;p,q)P_{k-1}(p^{3}x;\alpha ,\beta ;p,q)).\end{array}$$

By using the linear independence of $\{P_k(p^{3}x;\alpha ,\beta ;p,q)\}$ we obtain the three-term recurrence relation (15) for the connection coefficients $H_k(i,n;\alpha ,\beta ;p,q)$, where the initial conditions are obtained by equating the highest power in $x^k$. □

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.