Due to the fact that in the case q>1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 01) is still open.In this paper, the q-Bernstein polynomials Bn,q(fa;z) of the Cauchy kernel ⧹fa=1/(z-a),a∈C⧹[0,1] are found explicitly and their properties are investigated. In particular, it is proved that if q>1, then polynomials Bn,q(fa;z) converge to fa uniformly on any compact set K⊂{z:|z|<|a|}. This result is sharp in the following sense: on any set with an accumulation point in {z:|z|>|a|}, the sequence {Bn,q(fa;z)} is not even uniformly bounded.
