## 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.)
Full Text: