A generalization of universal Taylor series in simply connected domains

Let Ω be a simply connected proper subdomain of the complex plane and z 0 be a point in Ω. It is known that there are holomorphic functions f on Ω for which the partial sums ( S n ( f , z 0 ) ) of the Taylor series about z 0 have universal approximation properties outside Ω. In this paper we investigate what can be said for the sequence ( β n S n ( f , z 0 ) ) when ( β n ) is a sequence of complex numbers. We also study a related analogue of a classical theorem of Seleznev concerning the case where the radius of convergence of the universal power series is zero.