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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 quaternionic power series centered at 0 in its domain of convergence (which is a ball B(0,R) centered at 0). At each point p of this ball, f admits expansions in terms of appropriately defined 'regular power series centered at p'. The expansion holds in a ball Sigma(p, R-|p|), defined with respect to a (non-Euclidean) distance sigma. We thus say that f is 'sigma-analytic' in B(0,R). Furthermore, we remark that Sigma(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 only near the real axis. We then relate these notions of anayticity to the class of regular quaternionic functions introduced in Adv. Math. 216 (2007), 279-301, and recently extended. Indeed, sigma-analyticity proves equivalent to regularity, in the same way as complex analyticity is equivalent to holomorphy. Hence the theory of regular quaternionic functions, which is quickly developing, reveals a new feature that reminds the nice properties of holomorphic complex functions.