Zernike System Stems from Free Motion on the 3-Sphere

Systems that stem from projection of free motion on a manifold are the best candidates to exhibit remarkable symmetry properties. This is the case of free motion on the 3-sphere which, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system. This exhibits separability in a variety of coordinate systems, polynomial solutions, and interbasis expansion coefficients that are special Clebsch–Gordan coefficients and Hahn orthogonal polynomials.