  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.
