Universality properties of the quaternionic power series and entire functions

In this paper we establish the existence of “almost universal” quaternionic power series and entire functions. Denoting by B (0, 1) the open unit ball in H , this means that there exists a quaternionic power series S ( q ) = ∑ k = 0 ∞ q k a kwith radius of convergence 1 such that, denoting byS n ( q )the n ‐th partial sum∑ k = 0 n q k a kof S , for every K ∈ F H ∖ B ( 0 ; 1 ) ¯, for every axially symmetric open subset Ω of H containing K and every f slice regular on Ω, there exists a subsequence( S n k( q ) ) k ∈ Nof the partial sums of S such thatS n k( q ) → f ( q )uniformly on K , as k → ∞ . The symbol F H ∖ B ( 0 ; 1 ) ¯denotes the set of axially symmetric compact sets in H ∖ B ( 0 ; 1 ) ¯such thatC I ∖ ( K ∩ C I )is connected for some I ∈ S . This is a slightly weaker property than the classical universal power series phenomenon obtained for f : K → C analytic only on the interior of K and continuous on K . We also generalize a result originally proven by Birkhoff and finally we show that there exists an entire quaternionic function whose set of derivatives is dense in the class of entire quaternionic functions.