Power and spherical series over real alternative *-algebras

We study two types of series over a real alternative $^*$-algebra $A$. The first type are 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 $x-y$ w.r.t.\ 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 also 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.