External Ellipsoidal Harmonics for the Dunkl-Laplacian

TL;DR

This paper develops external ellipsoidal and sphero-conal $h$-harmonics for the Dunkl-Laplacian, providing integral representations and connections to classical special functions, advancing the mathematical theory of Dunkl operators.

Contribution

It introduces and characterizes external $h$-harmonics for the Dunkl-Laplacian, linking them to integral formulas and classical Jacobi functions, which is a novel extension in harmonic analysis.

Findings

External $h$-harmonics admit integral representations.

External $h$-harmonics are connected by Niven's type formula.

In the plane, they relate to Jacobi polynomials and functions.

Abstract

The paper introduces external ellipsoidal and external sphero-conal $h$-harmonics for the Dunkl-Laplacian. These external $h$-harmonics admit integral representations, and they are connected by a formula of Niven's type. External $h$-harmonics in the plane are expressed in terms of Jacobi polynomials $P_n^{\alpha,\beta}$ and Jacobi's functions $Q_n^{\alpha,\beta}$ of the second kind.