Representation of (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Bernstein polynomials in terms of (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Jacobi polynomials

A representation of (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Bernstein polynomials in terms of (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Jacobi polynomials is obtained.


Introduction
Classical univariate Bernstein polynomials were introduced by Bernstein in a constructive proof for the Stone-Weierstrass approximation theorem [], and they are defined as []   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 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  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 [-] and the references therein.
The (p, q)-power is defined as The (p, q)-hypergeometric series is defined as r s (a p , a q ), . . . , (a rp , a rq ) where (a p , a q ), . . . , (a rp , a rq ); (p, q) j = r s= (a sp , a sq ); (p, q) j , and r, s ∈ Z + and a p , a q , . . . , a rp , where the shift operator is defined by and and can be expanded in the basis {x k } k≥ as From the definition of (p, q)-Bernstein polynomials it is possible to derive the basic properties of (p, q)-Bernstein polynomials.
By using the linear independence of {P k (p  x; α, β; p, q)} we obtain the three-term recurrence relation () for the connection coefficients H k (i, n; α, β; 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.