Generalized Zernike polynomials: operational formulae and generating functions

@article{Aharmim2013GeneralizedZP,
  title={Generalized Zernike polynomials: operational formulae and generating functions},
  author={Bouchra Aharmim and El Hamyani Amal and El Wassouli Fouzia and Allal Ghanmi},
  journal={Integral Transforms and Special Functions},
  year={2013},
  volume={26},
  pages={395 - 410}
}
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. 
2 Citations

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting

References

SHOWING 1-10 OF 37 REFERENCES

The Poisson kernel for Heisenberg polynomials on the disk

n = 0 and Jerison [10] which can be used to show (Greiner and Koornwinder [8]) that the functions {O~--~C(,~'~)(ei~ n>0} on [0,rc] span a (weighted sup-norm) dense set in the continuous functions

Generalized Zernike or disc polynomials

Complex Hermite polynomials: Their combinatorics and integral operators

Abstract. We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these

An identity in Hermite polynomials

SUMMARY An extension of the Runge (1914) identity in Hermite polynomials is derived, and a test of the assumption of bivariate normality is developed using the identity.

A class of generalized complex Hermite polynomials

Operational formulae for certain classical polynomials - III

Operational formulae for Jacobi and other polynomials

The Generating Function of Jacobi Polynomials