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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 $C[0,1]$. Quantitativeestimates of the convergence, the Voronovskaja type theorem and saturation ofconvergence for complex genuine $q$-Bernstein-Durrmeyer polynomials attached toanalytic functions in compact disks are given. In particular, it is proved thatfor functions analytic in $\left\{ z\in\mathbb{C}:\left\vert z\right\vert<R\right\} $, $R>q,$ the rate of approximation by the genuine$q$-Bernstein-Durrmeyer polynomials ($q>1$) is of order $q^{-n}$ versus $1/n$for the classical genuine Bernstein-Durrmeyer polynomials. We give explicitformulas of Voronovskaja type for the genuine $q$-Bernstein-Durrmeyer for$q>1$.

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