Almansi-type decomposition for slice regular functions of several quaternionic variables

TL;DR

This paper introduces an Almansi-type decomposition for slice regular functions of multiple quaternionic variables, providing explicit, unique components with harmonic and circular properties, and offers new proofs and formulas related to these functions.

Contribution

It develops a novel Almansi-type decomposition for multivariable slice regular quaternionic functions, with explicit components and applications to Fueter's theorem and harmonic analysis.

Findings

Provides 2^n distinct decompositions for slice functions in $\

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Abstract

In this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields $2^n$ distinct and unique decompositions for any slice function with domain in $\mathbb{H}^n$. Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter's Theorem in $\mathbb{H}^n$, establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.