Generalized Zernike or disc polynomials

On the construction of uniformly convergent disk polynomial expansions

“…Disk polynomials appear quite frequently in problems where a given complex function defined in B[0, 1] needs to be expanded with respect to an orthonormal set of functions over B [0,1]. Moving to applications, the interested reader can ratify the use of disk polynomials in geometric and wave optics for systems with circular apertures, as described in [12] and references included there.…”

“…(1.2) We refer the reader to [5,6] where a very nice account on these polynomials is developed. Other relevant references are Boyd's dissertation [2], Wünche's recent paper [12] and references included there. In the last reference mentioned above, disk polynomials are called generalized Zernicke polynomials.…”

Generalized Zernike or disc polynomials: An application in quantum optics

“…In a recent paper [18], Wünsche has introduced the "generalized Zernike or disc polynomials" P α m,n (z, z * ), generalizing to pairs of complex conjugate variables (z, z * ), the real-domain relation of the Zernike polynomials to the Jacobi polynomials.…”

“…Accordingly, the generalized Zernike polynomials P α m,n (z, z * ) are understood in [18] as 2D polynomials of the independent variables z and z * , involving as well the Jacobi polynomials in the form z * n−m P (α,n−m) m (2zz * − 1). In this connection, it is worth noting that the use of complex variables (z, z * ) instead of cartesian ones to achieve results connected with Zernike polynomials was already suggested in [8].…”

Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces

“…He also mentioned that this method is applied to solve the Neumann problem of differential equations. By studying the algebraic structure of differential operators on the complex unit disk, Wünsche 21 generalized these discussions and derived a generalized form of orthogonal polynomials in the analogy of quantum theory.…”