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 provided, and their Fourier transforms are computed. Explicit representations in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of zeros. In the case $$m=2$$ we describe a degeneracy between the even- and odd-indexed polynomials.