The quaternionic Cullen-regular product for a larger class of functions

TL;DR

This paper introduces a new regular product for Cullen-regular quaternionic functions that does not rely on power series representations, expanding the class of functions with a well-defined product structure.

Contribution

It defines a novel regular product for Cullen-regular functions based on a weaker representation, and analyzes its properties and relation to quaternionic power series.

Findings

The regular product is independent of power series representation.

The regular ring of quaternionic power series is a subring of the Hyperholomorphic functions.

The new product extends the class of functions with a consistent algebraic structure.

Abstract

We introduce the regular product for Cullen-regular quaternionic functions in a manner that does not depend upon a representation in power series but upon another, weaker kind of representation. The special case when the functions are represented as quaternionic power series is studied. We show that the regular ring of quaternionic power series is a subring of the regular associative ring of real-analytical Hyperholomorphic functions.