Convergence domains of sedenionic star-power series

TL;DR

This paper extends the theory of slice regular functions to the sedenion algebra, a non-associative and non-alternative algebra, and characterizes the convergence domains of sedenionic power series.

Contribution

It introduces the first analysis of slice regularity in sedenions, including explicit solution characterizations and convergence domain descriptions with two radii.

Findings

Convergence domain determined by two radii due to zero divisors.

Explicit characterization of slice-solutions and hyper-solutions in sedenions.

Extension of slice regular function theory to non-alternative algebra.

Abstract

The theory of slice regular (also called hyperholomorphic) functions is a generalization of complex analysis originally given in the quaternionic framework, and then further extended to Clifford algebras, octonions, and to real alternative algebras. Recently, we have extended this theory to the case of (real) even-dimensional Euclidean space. We provided several tools for studying slice regular functions in this context, such as a path-representation formula, slice-solutions, hyper-solutions and hyper-sigma balls. When adding an algebra structure to Euclidean space we have more properties related to the possibility of multiplying elements in the algebra. In this paper, we focus on exploring the properties specific to the context of the algebra of sedenions which is rather peculiar, being the algebra non associative and non alternative. The consideration of a non alternative algebra is a novelty in the context of slice regularity which opens the way to a number of further progresses. We offer an explicit characterization for slice-solutions and hyper-solutions, and we determine the convergence domain of the power series in the form $q\mapsto\sum_{n\in\mathbb{N}}(q-p)^{*n}a_n$ with sedenionic coefficients. Unlike the case of complex numbers and quaternions, the convergence domain in this context is determined by two convergence radii instead of one. These results are closely tied to the presence of zero divisors in the algebra of sedenions.