Fueter trees for Dunkl-regular functions over alternative *-algebras

TL;DR

This paper establishes a general Fueter Theorem for Dunkl-regular functions over real alternative *-algebras, linking hypercomplex function theories with Dunkl monogenic functions and introducing Fueter trees to describe their interactions.

Contribution

It introduces the concept of Fueter trees and provides the most general form of the Laplacian's action on hypercomplex function spaces, unifying various Fueter-type results.

Findings

A suitable power of the Laplacian maps Dunkl-regular functions to Dunkl monogenic functions.

The number of Fueter trees on an (n+1)-dimensional space equals the number of partitions of n into odd parts.

The work unifies and generalizes existing Fueter-type theorems in hypercomplex analysis.

Abstract

We prove a general Fueter Theorem over real alternative *-algebras. We show that a suitable power of the Laplacian maps Dunkl-regular functions to Dunkl monogenic functions with axial symmetries. Using the embedding of hypercomplex function theories in the class of Dunkl monogenic functions, we subsume several Fueter-type results known in the literature and obtain the most general form for the action of the Laplacian on function spaces over hypercomplex subspaces. We show that Fueter Theorems are in a one-to-one correspondence with a class of graphs, the Fueter trees, that describe the interactions between Dunkl-regular function spaces and the relation with the iterated Laplacian. We obtain that the number of distinct Fueter trees on a hypercomplex space of dimension $n+1$ is equal to the number of partitions in odd parts of the integer $n$.