Approximation properties of the $q$-Bal\'{a}zs-Szabados operators in the case $q\ge 1$

This paper deals with approximation properties of the newly defined $q$-generalization of the Bal'{a}zs-Szabados operators in the case $q\ge 1$. Quantitative estimates of the convergence and Voronovskaja type theorem are given. In particular, it is proved that the rate of approximation by the $q$-Bal'{a}zs-Szabados ($q>1$) is of order $q^{-n}$ versus $1\mathrm{/}n$ for the classical Bal'{a}zs-Szabados ($q=1$) operators. The results are new even for the classical case $q=1$.