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We investigate generalized Zernike or disc polynomials P m; n (z; z \* ) which are orthogonal 2D polynomials in the unit disc 0 6 √ zz \* ¡ 1 with weights (1 -zz \* ) in complex coordinates z ≡ x + iy, z \* ≡ x -iy, where ¿ -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 → ∞ 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 di erential equations, a ÿrst-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 6 r 6 1 with weight factors (1 -r 2 ) are ultraspherical or Gegenbauer polynomials in a new standardization. The lowering and raising operators to the corresp

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Author

Alfred Wünsche

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0377-0427)

Published

2005

Language

EN

Field

Mathematics (Physical Sciences)