Power and spherical series over real alternative *-algebras

TL;DR

This paper extends the theory of power and spherical series to real alternative *-algebras, showing that these series define slice regular functions and characterizing their convergence sets.

Contribution

It generalizes the concept of power and spherical series from quaternions to arbitrary real alternative *-algebras, establishing their properties and convergence behavior.

Findings

Power series sum to slice regular functions in general algebras.

Spherical series convergence sets are always open in the quadratic cone.

Every slice regular function admits a spherical series expansion at any point.

Abstract

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.