On the convergence of the \(q\)-Bernstein polynomials for power functions (Q2026814)

The authors establish three theorems. The first theorem is an auxiliary result on the sum of the series \(S_{\alpha}:=\sum_{k=1}^{\infty} (-1)^k \frac{q^k (q^k-1)^{\alpha}}{(q-1)\cdots(q^{k}-1)}; q>1, \alpha>0\) of which the sum does not vanish for \(\alpha >1\) (not an integer) and \(q>1\). Generalizing Theorem 2.5 in [Abstr. Appl. Anal. 2012, Article ID 185948, 19 p. (2012; Zbl 1254.41013)], where the case of real \(q\)-Bernstein polynomials with \(q \geq 2\) and \(0<\alpha\leq 1/2\) is considered, the authors obtain as second theorem a result related to the convergence of the sequence of the complex \(q\)-Bernstein polynomials \(B_{n,q}(f_{\alpha};z)\), where \(0 < q \neq 1\) and \(f_{\alpha}=x^{\alpha}, \alpha \geq0\), is a power function on \([0,1]\). In the third theorem, the authors deal with the asymptotic behaviour of \(q\)-Bernstein polynomials \(B_{n,q}(f_{\alpha};z); q>1\) for linear combinations of power functions. The authors also provide some numerical examples in order to explain rapid changes of the \(q\)-Bernstein polynomials associated with non-polynomial power functions for \(q>1\).