Spherical geometry, Zernike’s separability, and interbasis expansion coefficients

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include separability in a variety of coordinate systems, polynomial solutions, and a particular subset of Clebsch-Gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the Askey scheme. Deriving these results from the initial formulation in spherical geometry provides the Zernike system with interest beyond its optical applications.