Properties of Clifford Legendre 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.