Generalized Zernike or disc polynomials

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.