# New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion

## TL;DR

This paper introduces new polynomial solutions to Zernike's differential equation on the unit disk, involving Legendre and Gegenbauer polynomials, and explores their interbasis expansions with overlaps expressed as Clebsch-Gordan coefficients.

## Contribution

It presents novel orthonormal polynomial bases for the Zernike system using Legendre and Gegenbauer polynomials, expanding the mathematical tools for wavefront analysis.

## Key findings

- New polynomial bases involving Legendre and Gegenbauer polynomials
- Explicit overlap calculations as Clebsch-Gordan coefficients
- Enhanced understanding of interbasis expansions in Zernike systems

## Abstract

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the the unit disk to classify wavefront aberrations in circular pupils, is shown to have a set of new orthonormal solution bases, involving Legendre and Gegenbauer polynomials, in non-orthogonal coordinates close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch-Gordan coefficients.