Generalized Zernike polynomials: operational formulae and generating functions

Abstract

We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish recurrence relations with respect to the argument and the integer indices, as well as Nielsen identities and Runge addition formula. In addition, various new generating functions for these disk polynomials are proved.

Keywords:

• complex disk polynomials
• operational formulae
• Nielsen's identities
• Runge's additionformula
• generating functions

AMS Subject Classification :

• 33C45
• 42C05
• 58C40

Acknowledgments

The authors are indebted to the anonymous referee for providing insightful comments, remarks and suggestions. We would like to thank Prof. Abdellah Sebbar for the effort to spend time to read the paper and make useful corrections. The assistance of the members of the seminars ‘Partial differential equations and spectral geometry’ is gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author.

Funding

A.G. is partially supported by the Hassan II Academy of Sciences and Technology.