# Quaternionic Zernike Spherical Polynomials

Presenter: **Isabel Cacao**, University of Aveiro, Portugal

Joint work with **João Morais, CIDMA, University of Aveiro, Portugal**

The so-called Zernike polynomials (ZPs) were introduced by F. Zernike's Nobel prize in 1934 in connection to diffraction theory. They are constituted by the product of radial polynomials by a pair of trigonometric functions (sine and cosine) and they form a complete set of orthogonal polynomials over the unit circle. Since their appearance, a significant number of publications describing various applications of the ZPs have emerged, mainly in
optical engineering. They seem to be well-adapted to corneal
surface modeling and they are commonly used
to describe balanced aberrations. In this talk, we introduce the Zernike spherical
polynomials within quaternionic analysis extending the classical two dimensional ZPs to dimension 3 or 4 in the quaternion algebra's framework. The representation of these
functions in terms of spherical monogenics over the unit sphere are explicitly given,
from which several recurrence formulae for faster computations can be
derived. A summary of their fundamental properties and a second order
differential equation are also discussed.