Hermite and Gegenbauer polynomials in superspace using Clifford analysis

TL;DR

This paper extends Clifford-Hermite and Clifford-Gegenbauer polynomials to superspace, establishing their fundamental properties and exploring a physical application related to super-dimension, without requiring an integration theory in superspace.

Contribution

It introduces a symbolic framework for Clifford analysis in superspace, generalizing key polynomials and proving their properties without relying on integration theory.

Findings

Orthogonality relations established

Differential equations derived

Recursion formulae proven

Abstract

The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore a lot of basic properties, such as orthogonality relations, differential equations and recursion formulae are proven. Finally, an interesting physical application of the super Clifford-Hermite polynomials is discussed, thus giving an interpretation to the super-dimension.