Interbasis expansions in the Zernike system

TL;DR

This paper explores new orthogonal polynomial solutions to Zernike's differential equation on the unit disk, revealing their interbasis expansions and connections to special functions and algebraic coefficients.

Contribution

It introduces two novel orthogonal polynomial solutions involving Legendre and Gegenbauer polynomials and details their interbasis expansion coefficients.

Findings

New orthogonal polynomial solutions involving Legendre and Gegenbauer polynomials.

Explicit formulas for interbasis expansion coefficients using hypergeometric functions and special polynomials.

Connections between expansion coefficients and algebraic structures like su(2) Clebsch-Gordan and Racah coefficients.

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.