Approximation by Genuine $q$-Bernstein-Durrmeyer Polynomials in Compact Disks in the case $q>1$

This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, whcih are no longer positive linear operators on $C[0,1]$. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex genuine $q$-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in $\left{ z\in$\mathbb{C}$:\left\vert z\right\vert <R\right} $,$R>q,$therateofapproximationbythegenuine$q$-Bernstein-Durrmeyerpolynomials($q>1$)isoforder$q^{-n}$versus$1/n$ for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine $q$-Bernstein-Durrmeyer for $q>1$.