Generalized Zernike or disc polynomials

Abstract

We investigate generalized Zernike or disc polynomials P m,n α(z,z ∗) which are orthogonal 2D polynomials in the unit disc 0⩽ zz ∗ <1 with weights (1−zz ∗) α in complex coordinates z≡ x+i y, 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 L m,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− r 2) 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.