On doubly universal functions

Let (λn) be a strictly increasing sequence of positive integers. Inspired by the notions of topological multiple recurrence and disjointness in dynamical systems, Costakis and Tsirivas have recently established that there exist power series ∑k≥0akzk with radius of convergence 1 such that the pairs of partial sums {(∑k=0nakzk,∑k=0λnakzk):n=1,2,…} approximate all pairs of polynomials uniformly on compact subsets K⊂{z∈C:|z|≥1}, with connected complement, if and only if lim supnλnn=+∞. In the present paper, we give a new proof of this statement avoiding the use of tools of potential theory. It allows to study the case of doubly universal infinitely differentiable functions. We obtain also the algebraic genericity of the set of such power series. Further we show that the Cesàro means of partial sums of power series with radius of convergence 1 cannot be frequently universal.