Biaxial monogenic functions from Funk-Hecke's formula combined with Fueter's theorem

TL;DR

This paper presents a novel method combining Funk-Hecke's formula and Fueter's theorem to construct biaxial monogenic functions that are invariant under specific symmetry groups, advancing the theory of monogenic functions.

Contribution

It introduces a new approach to generate biaxial monogenic functions from holomorphic functions by combining two classical mathematical results.

Findings

Derived a method to construct biaxial monogenics from holomorphic functions.

Extended the application of Funk-Hecke's formula to biaxial invariance.

Demonstrated the theoretical framework for combining Funk-Hecke's formula with Fueter's theorem.

Abstract

Funk-Hecke's formula allows a passage from plane waves to radially invariant functions. It may be adapted to transform axial monogenics into biaxial monogenics that are monogenic functions invariant under the product group SO(p)xSO(q). Fueter's theorem transforms holomorphic functions in the plane into axial monogenics, so that by combining both results, we obtain a method to construct biaxial monogenics from holomorphic functions.