In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$-Bernstein polynomials for a function analytic in the polydisc $D_{R_{1}}\times D_{R_{2}}=\{z\in C\colon \vert z\vert <R_{1}\} \times \{ z\in C\colon \vert z\vert <R_{1}\}$ for arbitrary fixed $q>1$. We give quantitative Voronovskaja type estimates for the bivariate $q$-Bernstein polynomials for $q>1$. In the univariate case the similar results were obtained by S. Ostrovska: $q$-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.
Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q<1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.