Doubly universal Taylor series

For a holomorphic function f in the unit disk, Sn(f) denotes the n-th partial sum of the Taylor development of f with center at 0. We show that given a strictly increasing sequence of positive integers (λn), there exists a holomorphic function f on the unit disk such that the pairs of partial sums {(Sn(f),Sλn(f)):n=1,2,…} approximate all plausibly approximable functions uniformly on suitable compact subsets K of the complex plane if and only if lim supnλnn=+∞. This provides a new strong notion of universality for Taylor series.