A generalization of universal Taylor series in simply connected domains

Let Ω be a simply connected proper subdomain of the complex plane and z0 be a point in Ω. It is known that there are holomorphic functions f on Ω for which the partial sums (Sn(f,z0)) of the Taylor series about z0 have universal approximation properties outside Ω. In this paper we investigate what can be said for the sequence (βnSn(f,z0)) when (βn) 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.