Orthogonal polynomials on the sphere with octahedral symmetry

Abstract:

For any finite reflection group $G$ acting on ${{\mathbf {R}}^N}$ there is a family of $G$-invariant measures ($({h^2}d\omega$, 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 (${L_h}f: = \Delta (fh) - f\Delta h$; $\Delta$ is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to ${h^2}d\omega$) decomposition of homogeneous polynomials, that is, if $p$ is of degree $n$ then \[ p(x) = \sum \limits _{j = 0}^{[n/2]} {|x{|^{2j}}{p_{n - 2j}}(x),} \] where ${L_h}{p_i} = 0$ and ${\operatorname {deg}}{p_i} = i$ for each $i$, but with the restriction that $p$ and ${p_i}$ must all be $G$-invariant. The main topic is the hyperoctahedral group with \[ h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left ( {\prod \limits _{i < j} {(x_i^2 - x_j^2)} } \right )^\beta }.\] 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 $(\alpha ,\beta )$ to those associated with $(\alpha + 2,\beta + 1)$.