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

Journal of Inequalities and Applications20172017:167

https://doi.org/10.1186/s13660-017-1443-7

• Received: 8 February 2017
• Accepted: 28 June 2017
• Published:

## Abstract

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

## Keywords

• $$(p,q)$$-Bernstein polynoimals
• $$(p,q)$$-Pearson difference equation
• $$(p,q)$$-orthogonal solutions
• $$(p,q)$$-difference operator

• 34B24
• 39A70

## 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 
$$b_{i}^{n}(x)=\binom{n}{i} x^{i}(1-x)^{n-i}, \quad i=0,1,\dots,n.$$
They form a basis of polynomials and satisfy a number of important properties as non-negativity ($$b_{i}^{n}(x) \geq0$$ for $$0 \leq x \leq 1$$), 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
$$B_{n}(f) (x)=\sum_{k=0}^{n} b_{k}^{n}(x) f \biggl(\frac{k}{n} \biggr).$$
Then, for each point x of continuity of f, we have $$B_{n}(f)(x) \to f(x)$$ as $$n \to\infty$$. Moreover, if f is continuous on $$[0,1]$$ then $$B_{n}(f)$$ converges uniformly to f as $$n \to\infty$$. Also, for each point x of differentiability of f, we have $$B_{n}'(f)(x) \to f'(x)$$ as $$n \to\infty$$ and if f is continuously differentiable on $$[0,1]$$ then $$B_{n}'(f)$$ converges to $$f'$$ uniformly as $$n \to \infty$$.

Bernstein polynomials have been generalized in the framework of q-calculus. More precisely, Lupaş  initiated the application of q-calculus in area of the approximation theory, and introduced the q-Bernstein polynomials. Later on, Philips  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.  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 . Also, many well-known approximation operators have been introduced using these techniques, such as Bleimann-Butzer-Hahn operators  and Szász-Mirakyan operators . Very recently Milovanović et 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 characterized 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 Section 3 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
$$\bigl((a,b);(p,q)\bigr)_{k}=\prod _{j=0}^{k-1} \bigl(ap^{j}-bq^{j} \bigr)\quad\text{with } \bigl((a,b);(p,q)\bigr)_{0}=1.$$
(1)
The $$(p,q)$$-hypergeometric series is defined as
\begin{aligned}[b] & {}_{r} \Phi_{s}\left ( \textstyle\begin{array}{c}{(a_{1p},a_{1q}),\dots,(a_{rp},a_{rq})}\\ {(b_{1p},b_{1q}),\dots,(b_{sp},b_{sq})} \end{array}\displaystyle \Big|{(p,q)};{z} \right ) \\ &\quad= \sum_{j=0}^{\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}} \bigl((-1)^{j} (q/p)^{\frac{j(j-1)}{2}} \bigr)^{1+s-r}, \end{aligned}
(2)
where
$$\bigl((a_{1p},a_{1q}),\dots,(a_{rp},a_{rq});(p, q) \bigr)_{j}=\prod_{s=1}^{r} \bigl((a_{sp},a_{sq});(p, q) \bigr)_{j},$$
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. )
$$(\text{{\mathcal {D}}_{p,q}}f) (x)=\frac{\text{{\mathcal {L}}_{p}} f(x)-\text{{\mathcal {L}}_{q}} f(x)}{(p-q)x},\quad x\neq0,$$
(3)
where the shift operator is defined by
$$\text{{\mathcal {L}}_{a}}h(x)=h(ax),$$
(4)
and $$(\text{{\mathcal {D}}_{p,q}}f)(0)=f'(0)$$, provided that f is differentiable at 0.
The $$(p,q)$$-Bernstein polynomials are defined as
$$b_{i}^{n}(x;p,q)=p^{n(1-n)/2} \begin{bmatrix}{n}\\{i} \end{bmatrix} _{p,q} p^{i(i-1)/2} x^{i} \bigl((1,x);(p,q)\bigr)_{n-i},$$
(5)
and can be expanded in the basis $$\{x^{k}\}_{k \geq0}$$ as
$$b_{i}^{n}(x;p,q)=\sum _{k=i}^{n} (-1)^{k-i} q^{(k-i)(k-i-1)/2} p^{\frac{1}{2} ((i-1) i+k (k-2 n+1))} \begin{bmatrix}{n}\\{k} \end{bmatrix} _{p,q} \begin{bmatrix}{k}\\{i} \end{bmatrix} _{p,q} x^{k}.$$
(6)
From the definition of $$(p,q)$$-Bernstein polynomials it is possible to derive the basic properties of $$(p,q)$$-Bernstein polynomials.
1. (1)
Partition of unity
$$\sum_{i=0}^{n} b_{i}^{n}(x;p,q)=1.$$

2. (2)
End-point properties
$$b_{i}^{n}(0;p,q)= \textstyle\begin{cases} 1, & i=0, \\ 0, & \text{otherwise}, \end{cases}\displaystyle \qquad b_{i}^{n}(1;p,q)= \textstyle\begin{cases} 1, & i=n, \\ 0, & \text{otherwise}. \end{cases}$$

The $$(p,q)$$-Jacobi polynomials are defined by
$$P_{n}(x;\alpha,\beta;p,q)= {}_{2} \Phi_{1}\left ( \textstyle\begin{array}{c}{(p^{-n},q^{-n}),(p ^{\alpha+\beta+n+1},q^{\alpha +\beta+n+1})}\\ {(p ^{\beta+1},q^{\beta+1})} \end{array}\displaystyle \Big|{(p,q)};{ \frac{x q^{-\alpha}}{p}} \right ) ,$$
(7)
and they satisfy the second order $$(p,q)$$-difference equation
\begin{aligned}[b] &\frac{q x (q x-p)}{p^{2}} \bigl({ \mathcal{D}}_{p,q}^{2} y \bigr) (x)+ \biggl( \frac {x (p^{\alpha+\beta+2} q^{-\alpha-\beta}-q^{2} )-p^{\beta+2} q^{-\beta}+p q}{p^{2} (p-q)} \biggr) \text{{\mathcal {L}}_{p}} \bigl((\text{{\mathcal {D}}_{p,q}}y) (x) \bigr) \\ &\quad+ [n]_{p,q} \biggl(\frac{q p^{-n-2}-p^{\alpha+\beta -1} q^{-\alpha -\beta-n}}{p-q} \biggr) \text{{\mathcal {L}}_{pq}} y(x)=0. \end{aligned}
(8)
The $$(p,q)$$-Jacobi polynomials satisfy the three-term recurrence relation
$$\begin{gathered} P_{0}(x;\alpha,\beta;p,q)=1, \qquad P_{1}(x;\alpha,\beta ;p,q)=x-B_{0}(\alpha,\beta;p,q), \\ P_{n+1}(x;\alpha,\beta;p,q)= \bigl(x-B_{n}(\alpha,\beta;p,q) \bigr) P_{n}(x;\alpha,\beta;p,q) - C_{n}(\alpha,\beta;p,q) P_{n-1}(x;\alpha,\beta;p,q), \end{gathered}$$
where
\begin{aligned}[b] B_{n}(\alpha,\beta;p,q)={}& \frac{p^{n+2} q^{\alpha+n+1}}{(p-q)^{2} [\alpha+\beta+2 n]_{p,q} [\alpha+\beta+2 n+2]_{p,q}} \\ &\times \bigl( \bigl(p^{\beta}+q^{\beta} \bigr) q^{\alpha+\beta+2 n+1}-(p+q) \bigl(p^{\alpha}+q^{\alpha} \bigr) p^{\beta+n} q^{\beta +n} \\ &+ \bigl(p^{\beta}+q^{\beta } \bigr) p^{\alpha+\beta+2 n+1} \bigr) \end{aligned}
(9)
and
$$C_{n}(\alpha,\beta;p,q)=\frac{p^{\beta+2 n+3} q^{2 \alpha+\beta+2 n+1} [n]_{p,q} [\alpha+n]_{p,q} [\beta +n]_{p,q} [\alpha +\beta+n]_{p,q}}{[\alpha+\beta +2 n-1]_{p,q} ([\alpha+\beta+2 n]_{p,q})^{2} [\alpha+\beta+2 n+1]_{p,q}}.$$
(10)

## 3 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:
$$(p x-1) x \bigl(D_{p,q}b_{i}^{n} \bigr) (x;p,q) + \bigl(-p^{1-n} [n]_{p,q} x+p^{-i} [i]_{p,q} \bigr) b_{i}^{n}(p x;p,q) =0.$$
(11)

### Proof

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

If we introduce the first order $$(p,q)$$-difference operator
$$L_{i,n}=(p x-1) x D_{p,q}+ \bigl(-p^{1-n} [n]_{p,q} x+p^{-i} [i]_{p,q} \bigr) {\mathcal{L}}_{p},$$
(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{aligned}[b] &x (p x-1 ) D_{p,q} \bigl(P_{n} \bigl(p^{2} x;\alpha,\beta;p,q \bigr) \bigr) \\ &\quad= [n]_{p,q} p^{-n-2} P_{n+1} \bigl(p^{3}x;\alpha, \beta;p,q \bigr) + \varpi_{1}(n) P_{n} \bigl(p^{3}x; \alpha,\beta;p,q \bigr) \\ &\qquad+ \varpi_{2}(n) P_{n-1} \bigl(p^{3}x;\alpha, \beta;p,q \bigr), \end{aligned}
(13)
where
$$\begin{gathered} \varpi_{1}(n)=-\frac{[n]_{p,q} (-(p+q) q^{\alpha +n}-p^{\beta +n}+p^{\alpha+\beta+2 n+1}+q^{\alpha+\beta+2 n+1} ) [\alpha +\beta+n+1]_{p,q}}{(p-q) [\alpha+\beta+2 n]_{p,q} [\alpha +\beta+2 n+2]_{p,q}}, \\ \varpi_{2}(n)=\frac{q^{\alpha+n} p^{\beta+2 n+1} [n]_{p,q} [\alpha+n]_{p,q} [\beta+n]_{p,q} [\alpha+\beta+n]_{p,q} [\alpha+\beta+n+1]_{p,q}}{[\alpha+\beta+2 n-1]_{p,q} ([\alpha+\beta+2 n]_{p,q})^{2} [\alpha +\beta+2 n+1]_{p,q}}. \end{gathered}$$

### 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):
$$b_{i}^{n}(x;p,q)=\sum _{k=0}^{n} H_{k}(i,n;\alpha,\beta ;p,q)P_{k} \bigl(p^{2}x;\alpha,\beta;p,q \bigr),$$
(14)
where the connection coefficients $$H_{k}(i,n;\alpha,\beta;p,q)$$ satisfy the following three-term recurrence relation:
\begin{aligned}[b] &H_{k-1}(i,n;\alpha, \beta;p,q) \Lambda_{1}(k-1,i,n; \alpha,\beta;p,q)+ H_{k}(i,n; \alpha,\beta;p,q) \Lambda_{2}(k,i,n; \alpha,\beta;p,q) \\ &\quad+ H_{k+1}(i,n;\alpha,\beta;p,q) \Lambda_{3}(k+1,i,n; \alpha, \beta;p,q)=0, \end{aligned}
(15)
valid for $$1 \leq k \leq n-1$$ with initial conditions
\begin{aligned}& H_{n+1}(i,n;\alpha,\beta;p,q)=0, \end{aligned}
(16)
\begin{aligned}& 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} \begin{bmatrix}{n}\\{i} \end{bmatrix} _{p,q}, \end{aligned}
(17)
and
$$\textstyle\begin{cases} \Lambda_{1}(k,i,n;\alpha,\beta;p,q)=p^{-k-2} [k]_{p,q}-p^{-n-2}[n]_{p,q},\\ \Lambda_{2}(k,i,n;\alpha,\beta;p,q)=p^{-i} [i]_{p,q} - p^{-2-n} [n]_{p,q} B_{k}(\alpha,\beta;p,q) + \varpi_{1}(k), \\ \Lambda_{3}(k,i,n;\alpha,\beta;p,q)=-p^{-n-2} [n]_{p,q} C_{k}(\alpha,\beta;p,q) + \varpi_{2}(k). \end{cases}$$
(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{aligned} 0={}&\sum_{k=0}^{n} H_{k}(i,n; \alpha,\beta;p,q)L_{i,n} P_{k} \bigl(p^{2}x;\alpha, \beta;p,q \bigr) \\ ={}&\sum_{k=0}^{n} H_{k}(i,n; \alpha,\beta;p,q) \bigl((px-1) x D_{p,q} \bigl(P_{k} \bigl(p^{2}x;\alpha,\beta;p,q \bigr) \bigr) \\ &+ \bigl(-p^{1-n} [n]_{p,q} x+p^{-i} [i]_{p,q} \bigr) P_{k} \bigl(p^{3}x; \alpha,\beta;p,q \bigr) \bigr). \end{aligned}
From the three-term recurrence relation for $$(p,q)$$-Jacobi polynomials it yields
$$\begin{gathered} \bigl(-p^{1-n} [n]_{p,q} x+p^{-i} [i]_{p,q} \bigr) P_{k} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) \\ \quad=-p^{-n-2} [n]_{p,q} P_{k+1} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) \\ \qquad{}+ p^{-2-n-i} \bigl( -p^{n+2} [i]_{p,q}+p^{i} [n]_{p,q} B_{k}(\alpha,\beta;p,q) \bigr) P_{k} \bigl(p^{3}x; \alpha,\beta;p,q \bigr) \\ \qquad{}-p^{-n-2} [n]_{p,q} C_{k}(\alpha, \beta;p,q)P_{k-1} \bigl(p^{3}x;\alpha,\beta;p,q \bigr). \end{gathered}$$
Therefore, by using the structure relation for $$(p,q)$$-Jacobi polynomials (13) we have
$$\begin{gathered} (px-1) x D_{p,q} \bigl(P_{k} \bigl(p^{2}x;\alpha, \beta;p,q \bigr) \bigr) + \bigl(-p^{1-n} [n]_{p,q} x+p^{-i} [i]_{p,q} \bigr) P_{k} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) \\ \quad= \Lambda_{1}(k,i,n;\alpha,\beta;p,q) P_{k+1} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) +\Lambda_{2}(k,i,n; \alpha,\beta;p,q) P_{k} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) \\ \qquad{}+\Lambda_{3}(k,i,n;\alpha,\beta;p,q) P_{k-1} \bigl(p^{3}x;\alpha,\beta;p,q \bigr), \end{gathered}$$
where $$\Lambda_{i}(k,i,n;\alpha,\beta;p,q)$$ are given in (18).
As a consequence,
\begin{aligned} 0={}&\sum_{k=0}^{n} H_{k}(i,n; \alpha,\beta;p,q) \bigl(\Lambda_{1}(k,i,n; \alpha,\beta;p,q) P_{k+1} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) \\ &+ \Lambda_{2}(k,i,n;\alpha,\beta;p,q) P_{k} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) + \Lambda_{3}(k,i,n; \alpha,\beta;p,q) P_{k-1} \bigl(p^{3}x;\alpha,\beta;p,q \bigr) \bigr). \end{aligned}
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 $$x^{k}$$. □

## 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.

## Declarations

### 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.

## Authors’ Affiliations

(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

## References 