Generalized disk polynomial via Laplace integral representation

Abstract

The present paper investigates a class of disk functions via Laplace integral representation, where the disk polynomials appear as special cases. Recurrence relations involving the first-order derivative for them will be obtained. The connection of these functions with complex spherical harmonics also will be studied. Moreover, we exhibit an inductive method to construct bases of complex spherical harmonics via our disk functions.

Keywords:

• disk function
• disk polynomial
• Laplace integral representation
• Zernike circle functions

Mathematics Subject Classification:

• 32A17
• 32A25
• 32A50
• 43A90
• 42B35
• 33C55

Acknowledgements

The author completed this work at the Centro de Matemática e Aplicações da Universidade de Lisboa. We are grateful to the professors Jorge Buescu and A. C. Paixão that made this visit possible. The financial support from CAPES-Brasil [grant number 10884/13-0] is gratefully recognized.