Properties of Clifford Legendre Polynomials

TL;DR

This paper studies Clifford-Legendre and Clifford-Gegenbauer polynomials, deriving new recurrence relations, explicit representations, and analyzing their properties in Clifford algebra context, with special cases for dimension two.

Contribution

It introduces new recurrence and Bonnet type formulas, computes Fourier transforms, and provides explicit representations of Clifford polynomials, advancing understanding of their structure and zeros.

Findings

Derived new recurrence relations for Clifford polynomials

Computed Fourier transforms of these polynomials

Provided explicit representations involving spherical monogenics and Jacobi polynomials

Abstract

Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra ${\mathbb R}_m$. New recurrence and Bonnet type formulae for these polynomials are proved, as their Fourier transforms are computed. Explicit representations in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of the zeros. In the case $m=2$ we describe a degeneracy between the even- and odd-indexed polynomials.