ABSTRACT
Let 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.
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Acknowledgements
The author is grateful to the anonymous referee for her/his careful reading of the manuscript and valuable comments.
Disclosure statement
No potential conflict of interest was reported by the author.