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Generalized Bernstein polynomials. (English) Zbl 1045.41003

The authors define generalized Bernstein polynomials of degree \(n\), for \(n \in \mathbb{N}\) and \(i \in \{0,1,\dots,n\}\), by \[ B_i^n(x;\omega| q):= \frac{1}{(\omega;q)_n} \begin{bmatrix} n \\i \end{bmatrix}_q x^i(\omega x^{-1};q)_i(x;q)_{n-i}. \] Here \(q\) and \(\omega\) are real parameters such that \(q \neq 1\) and \(\omega \neq 1,q^{-1},\dots,q^{1-n}\). In addition, \[ (c;q)_k := \prod_{j=0}^{k-1} (1-cq^j) \] and \[ \begin{bmatrix} n \\i \end{bmatrix}_q := \frac{(q;q)_n}{(q;q)_i(q;q)_{n-i}}. \] These generalized Bernstein polynomials include as special or limiting cases the classical Bernstein polynomials, discrete Bernstein polynomials, and \(q\)-Bernstein polynomials. The authors deduce a number of recurrences and other basic properties of the generalized Bernstein polynomials and show how to express them in terms of big \(q\)-Jacobi and \(q\)-Hahn (or dual \(q\)-Hahn) polynomials. Their results contain as special or limiting cases many known results for classical, discrete, and/or \(q\)-Bernstein polynomials.

MSC:

41A10 Approximation by polynomials
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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