Generalized Zernike or disc polynomials

Abstract

We investigate generalized Zernike or disc polynomials Pm,nα(z,z*) which are orthogonal 2D polynomials in the unit disc 0≤ zz* <1 with weights (1-zz*)α in complex coordinates z ≡ x+iy, z* ≡ x-iy, where α > -1 is a free parameter. These polynomials can be expressed by Jacobi polynomials of transformed arguments in connection with a simple angle dependence. A limiting procedure α→∞ leads to Laguerre 2D polynomials Lm,n(z,z*). Furthermore, we introduce the corresponding orthonormalized disc functions. The disc polynomials and disc functions obey two differential equations, a first-order and a second-order one with a certain degree of freedom, and the operators of lowering and raising of the indices are found. These operators can be closed to a Lie algebra su(1,1)⊕su(1,1). New generating functions are derived from an operational representation which is alternative to the Rodrigues-type representation. The one-dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0≤r≤1 with weight factors (1-r2)α are ultraspherical or Gegenbauer polynomials in a new standardization. The lowering and raising operators to the corresponding orthonormalized functions form a simple su(1,1) Lie algebra. This is given in the appendix in sketched form. © 2004 Elsevier B.V. All rights reserved.