On doubly universal functions

Abstract: Let $(\lambda_n)$ 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 $\sum_{k\geq 0}a_kz^k$ with radius of convergence 1 such that the pairs of partial sums ${(\sum_{k=0}^na_kz^k,\sum_{k=0}^{\lambda_n}a_kz^k): n=1,2,\dots}$ approximate all pairs of polynomials uniformly on compact subsets $K\subset{z\in\mathbb{C} :| z|\textgreater{}1},$ with connected complement, if and only if $\limsup_{n}\frac{\lambda_n}{n}=+\infty.$ In the present paper, we give a new proof of this statement avoiding the use of advanced tools of potential theory. It allows to obtain the algebraic genericity of the set of such power series and to study the case of doubly universal infinitely differentiable functions. Further we show that the Ces`aro means of partial sums of power series with radius of convergence 1 cannot be frequently universal.