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

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.

Submitted to JOSA A, 2 figures

References in corpus (1)

• Separation of variables and Lie algebra contractions. Applications to special functions

Cited by in corpus (4)

• Interbasis expansions in the Zernike system
• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum