An introduction to the quaternionic Zernike spherical polynomials

In continuation of [4], this paper discusses the quaternionic Zernike spherical polynomials (QZSPs), which refine and extend the Zernike polynomials or radial polynomials introduced in the early thirties by F. Zernike's Nobel prize. In particular, the underlying polynomials are of three real variables and take on values in the quaternions (identified with ${\mathbb{R}}^4$). QZSPs are complete and orthonormal in the unit ball. The representation of these functions are explicitly given, and a summary of their fundamental properties is also discussed. To the best of our knowledge, this does not appear to have been done in literature before.