On countably universal series in the complex plane

Abstract

ABSTRACTLet be countably many increasing sequences of integers tending to ∞. For a power series , we denote by the nth partial sum of f. Under some condition on , we construct lacunary power series converging in the unit disk , with arbitrarily large lacunes, which satisfy the following property (P): for any countable family of compact sets in the complement of , each of them with connected complement, the set is dense in , endowed with the product topology, inherited from the -norm of the space of all continuous functions on K which are holomorphic in its interior. The set of such countably universal series is shown to be invariant under some summability processes. Our construction also allows us to exhibit power series with Padé approximants enjoying countably universal properties, and to prove that the set of all power series with radius of convergence 1 which satisfy (P) is either void or spaceable. We finally use Ostrowski-gaps to give a very direct proof of two known results: (1) The set of doubly universal Taylor series is non-void only if for some increasing sequence , and (2) The set of frequently universal series is void.