  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.
