Interbasis expansions in the Zernike system

Abstract

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I), serves to define a classical and a quantum system which have been found to be superintegrable. We have determined two new orthogonal polynomial solutions (indicated as II and III) that are separable, and which involve Legendre and Gegenbauer polynomials. Here we report on their three interbasis expansion coefficients: between the I--II and I--III bases they are given by $_3F_2(\cdots|1)$ polynomials that are also special su($2$) Clebsch-Gordan coefficients and Hahn polynomials. Between the II--III bases, we find an xpansion expressed by $_4F_3(\cdots|1)$'s and Racah polynomials that are related to the Wigner $6j$ coefficients.