Power and spherical series over real alternative *-algebras

Abstract

We study two types of series over a real alternative $^*$-algebra $A$. The\nfirst type are series of the form $\\sum_{n} (x-y)^{\\punto n}a_n$, where $a_n$\nand $y$ belong to $A$ and $(x-y)^{\\punto n}$ denotes the $n$--th power of $x-y$\nw.r.t.\\ the usual product obtained by requiring commutativity of the\nindeterminate $x$ with the elements of $A$. In the real and in the complex\ncases, the sums of power series define, respectively, the real analytic and the\nholomorphic functions. In the quaternionic case, a series of this type\nproduces, in the interior of its set of convergence, a function belonging to\nthe recently introduced class of slice regular functions. We show that also in\nthe general setting of an alternative algebra $A$, the sum of a power series is\na slice regular function. We consider also a second type of series, the\nspherical series, where the powers are replaced by a different sequence of\nslice regular polynomials. It is known that on the quaternions, the set of\nconvergence of these series is an open set, a property not always valid in the\ncase of power series. We characterize the sets of convergence of this type of\nseries for an arbitrary alternative $^*$-algebra $A$. In particular, we prove\nthat these sets are always open in the quadratic cone of $A$. Moreover, we show\nthat every slice regular function has a spherical series expansion at every\npoint.\n