  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.
