Power series and analyticity over the quaternions

We study power series and analyticity in the quaternionic setting. We first consider a function f defined as the sum of a power series P n∈N qnan in its domain of convergence, which is a ball B(0, R) centered at 0. At each p ∈ B(0, R), f admits expansions in terms of appropriately defined regular power series centered at p, P n∈N (q−p)bn. The expansion holds in a ball Σ(p,R− |p|) defined with respect to a (non-Euclidean) distance σ. We thus say that f is σ-analytic in B(0, R). Furthermore, we remark that Σ(p,R − |p|) is not always an Euclidean neighborhood of p; when it is, we say that f is quaternionic analytic at p. It turns out that f is quaternionic analytic in a neighborhood A of B(0, R)∩R, with A strictly contained in B(0, R) unless R is infinite. We then extend these results to the larger class of quaternionic regular functions, enriching the theory recently introduced in [9]. Indeed, regularity proves equivalent to σ-analyticity and regular functions are quaternionic analytic only in a neighborhood of the real axis.