The Saturation of Convergence for the Complex q-Durrmeyer Polynomials

The aim of this paper is to establish a saturation result for the complex q-Durrmeyer polynomials \((D_{n,q}f)(z)\) $(D_{n,q}f)(z)$ , where \(q \in (0,1)\) $q\in (0,1)$ , \(f \in C[0,1].\) $f\in C[0,1].$ It is known that the sequence \(\{(D_{n,q}f)(z)\}_{n \in {\mathbb {N}}}\) ${\{(D_{n,q}f)(z)\}}_{n\in \mathbb{N}}$ converges uniformly on any compact set in \({\mathbb {C}}\) $\mathbb{C}$ to the limit function \((D_{\infty ,q}f)(z)\) $(D_{\infty ,q}f)(z)$ , which, therefore, is entire. Previously, the rate of this convergence has been estimated as \(O(q^n)\) $O(q^n)$ , \(n \rightarrow \infty .\) $n\to \infty .$ In the present article, this result is refined to derive Voronovskaya-type formula and to demonstrate that this rate is \(o(q^n)\) $o(q^n)$ , \(n \rightarrow \infty \) $n\to \infty$ on a set possessing an accumulation point if and only if f takes on the same value at all \(q^j\) $q^j$ , \(j \in {\mathbb {N}}_{0}\) $j\in {\mathbb{N}}_0$ .