Weak slice regular functions on the $n$-dimensional quadratic cone of octonions

TL;DR

This paper introduces weak slice regular functions on the $n$-dimensional quadratic cone of octonions, highlighting differences from traditional slice regular functions and exploring their properties, including Taylor expansions.

Contribution

It extends the concept of slice regular functions to the first approach in hypercomplex variables, revealing new functions not captured by the second approach.

Findings

Weak slice regular functions differ from classical slice regular functions on certain domains.

Existence of weak slice regular functions that are not slice regular in the second approach.

Development of a Taylor expansion for weak slice regular functions.

Abstract

In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called $n$-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion.