Generalized universal series

We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form $\sum_{k=0}^n a_k x_{n,k}$ for given sequences of vectors $(x_{n,k})_{n\geq k\geq 0}$ in a topological vector space X. The algebraic and topological genericity as well as the spaceability are discussed. Then we provide various examples of such generalized universal series which do not proceed from the classical theory. In particular, we build universal series involving Bernstein's polynomials, we obtain a universal series version of MacLane's Theorem, and we extend a result of Tsirivas concerning universal Taylor series on simply connected domains, exploiting Bernstein- Walsh quantitative approximation theorem.