Power and spherical series over real alternative *-algebras Riccardo GhiloniAlessandro Perotti Primary 30G35Secondary 30B1030G3032A30Power seriesslice regular functionsquaternionsClifford algebrasalternative algebras We study two types of series over a real alternative $^{*}$-algebra $A$. The first type comprises series of the form $\sum_n(x-y)^{\punto n}a_n$, where $a_n$ and $y$ belong to $A$, and $(x-y)^{\punto n}$ denotes the $n$-th power of $x-y$ with respect to the usual product obtained by requiring commutativity of the indeterminate $x$ with the elements of $A$. In the real and in the complex cases, the sums of power series define, respectively, the real analytic and the holomorphic functions. In the quaternionic case, a series of this type produces, in the interior of its set of convergence, a function belonging to the recently introduced class of slice-regular functions. We show that, additionally, in the general setting of an alternative algebra $A$, the sum of a power series is a slice-regular function. We consider also a second type of series, the spherical series, where the powers are replaced by a different sequence of slice-regular polynomials. It is known that, on the quaternions, the set of convergence of these series is an open set, a property not always valid in the case of power series. We characterize the sets of convergence of this type of series for an arbitrary alternative $^{*}$-algebra $A$. In particular, we prove that these sets are always open in the quadratic cone of $A$. Moreover, we show that every slice-regular function has a spherical series expansion at every point. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5227 10.1512/iumj.2014.63.5227 en Indiana Univ. Math. J. 63 (2014) 495 - 532 state-of-the-art mathematics http://iumj.org/access/