# A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics

## TL;DR

This paper develops a comprehensive framework for hyperholomorphic functions valued in Banach algebras, covering Fueter expansions, rational functions, and modules, unifying various algebraic structures like quaternions and Clifford algebras.

## Contribution

It introduces a general setting for hyperholomorphic functions over Banach algebras, encompassing many classical algebras and extending the theory of Fueter variables.

## Key findings

- Unified framework for hyperholomorphic functions in Banach algebras
- Development of Fueter expansions and rational function theory
- Application to various algebraic structures like quaternions and Clifford algebras

## Abstract

We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field -- assumed to be the real or the complex numbers -- and which contains the field. Notably, we consider Fueter expansions, Gleason's problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.