An efficient algofithm for the generation of homogeneous harmonic polynomials

Abstract

In a previous paper the authors have developed a method for exact computation of basis systems of homogeneous harmonic polynomials of degree n = 0,1,... in IR^q using exclusively integer operations. The critical point in that approach is that all members of the basis system of degree n are involved in the computations if one is interested in finding an orthonormalized system of spherical harmonics of degree n on the unit sphere Ω _q in IR^q. In this paper it is shown that the amount of computational work can be reduced drastically if the orthonormalization process is based on the fact that any homogeneous harmonic polynomial of degree n in IR^q can be recovered from any two homogeneous harmonic polynomials inR ^{q -1} of degree n and n - 1.