A de Casteljau algorithm for generalized Bernstein polynomials (Q678231)

This paper is concerned with generalized Bernstein polynomials of the form \[ B_n(f;x)= \sum^n_{r=0} f_r{n\brack r} x^r\prod^{n-r-1}_{s=0} (1-q^sr), \] where an empty product is equal to \(1\) and \(f_r=f([r]/[n])\). Further, \(q\) is a positive real number and \[ [r]=\begin{cases} (1-q)^r/(1-q) &\text{for }q\neq 1,\\ r &\text{for }q=1.\end{cases} \] The so-called \(q\)-factorials \([r]!\) are defined by \[ [r]!=\begin{cases} [r][r- 1]\cdots[1] &\text{for }r\in\mathbb{N},\\ 1 &\text{for }r=0\end{cases} \] and hence the \(q\)-binomial coefficients \({n\brack r}\) by \[ {n\brack r}= {[n]!\over [r]![n- r]!}\quad\text{for }n\geq r\geq 0. \] It is shown that these polynomials can be generated by a de Casteljau algorithm which generalizes that for the classical case.