Universal Taylor series with respect to a prescribed subsequence

For a holomorphic function f in the open unit disc D and ζ ∈ , S n ( ) denotes -th partial sum of Taylor development at . Given an increasing sequence positive integers μ = we consider classes U functions such that subsequences sums { : 1 2 … } respectively approximate all polynomials uniformly on compact sets K ⊂ z C | ≥ with connected complement. We show these two universal series coincide if only lim sup → + ∞ < In same spirit, prove that, for ≠ 0 equality holds Finally deal case real series.