Orthogonal Polynomials on the Sphere with Octahedral Symmetry

For any finite reflection group G acting on RN there is a family of G-invariant measures ((h²dω, where h is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for G) on the unit sphere and an associated partial differential operator (Lₕf: = Δ (fh) - fΔ h; Δ is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to h²dω) decomposition of homogeneous polynomials, that is, if p is of degree n then \[ p(x) = ∑ j = 0[n/2] {|x{|²ʲ}{pn - 2j}(x),} \] where Lₕpᵢ = 0 and degpᵢ = i for each i, but with the restriction that p and pᵢ must all be G-invariant. The main topic is the hyperoctahedral group with \[ h(x) = {({x_1}{x_2} ⋯ {x_N})^α }{ ( {∏ i < j {(x_i^2 - x_j^2)} } )^β }.\] The special case $N = 2$ leads to Jacobi polynomials. A detailed study of the case $N = 3$ is made; an important result is the construction of a third-order differential operator that maps polynomials associated to h with indices (α ,β ) to those associated with (α + 2,β + 1).