1 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]

$$b_{i}^{n}(x)=\binom{n}{i} x^{i}(1-x)^{n-i}, \quad i=0,1,\dots,n. $$

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 [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.

2 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

$$ \bigl((a,b);(p,q)\bigr)_{k}=\prod _{j=0}^{k-1} \bigl(ap^{j}-bq^{j} \bigr)\quad\text{with } \bigl((a,b);(p,q)\bigr)_{0}=1. $$
(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. [14])

$$ (\text{${\mathcal {D}}_{p,q}$}f) (x)=\frac{\text{${\mathcal {L}}_{p}$} f(x)-\text{${\mathcal {L}}_{q}$} f(x)}{(p-q)x},\quad x\neq0, $$
(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.