Algebraic approach to slice monogenic functions

TL;DR

This paper develops an algebraic framework for slice monogenic functions by extending the Dirac operator, revealing Lie superalgebra structures, and constructing Hermite-like functions within this setting.

Contribution

It introduces an extended Dirac operator for slice monogenic functions, uncovering underlying Lie superalgebra structures and developing associated Hermite-type functions.

Findings

Lie superalgebra structure established for slice monogenic functions

Inner product defined with polynomial null-solutions characterized

Hermite-like polynomials and functions constructed and analyzed

Abstract

In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.