On the Fueter-Sce theorem and Cauchy-Kovalevskaya extensions over alternative $\ast$-algebras

TL;DR

This paper extends the theory of generalized partial-slice monogenic functions over real alternative $ ext{*}$-algebras, exploring the Fueter-Sce theorem and Cauchy-Kovalevskaya extensions to unify concepts of monogenicity and harmonicity.

Contribution

It develops a comprehensive framework connecting monogenic functions, harmonic functions, and partial-slice monogenic functions over hypercomplex spaces within alternative $ ext{*}$-algebras.

Findings

Established new relationships between monogenicity and harmonicity.

Extended Fueter-Sce theorem to broader algebraic contexts.

Analyzed properties of Cauchy-Kovalevskaya extensions in hypercomplex settings.

Abstract

Recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced and studied over Clifford algebras and octonions, respectively. In this paper, we further develop the theory of generalized partial-slice monogenic functions defined on hypercomplex subspaces and with values in a real alternative $\ast$-algebra and we concentrate on the Fueter-Sce theorem, three types of Cauchy-Kovalevskaya extensions, and their various internal relationships. The paper proposes more bridges between the theories of monogenicity, harmonicity, and generalized partial-slice monogenicity.