## Isabel Cação

## A basic hypercomplex polynomial sequence and its different representations

Hypercomplex analysis (also called Clifford analysis) provides a generalization to higher dimensions of the theory of holomorphic functions of one complex variable by using Clifford algebras. In this framework the analogue of holomorphic functions is obtained as null-solutions of a generalized Cauchy-Riemann system and they are usually called monogenic. An essential difference between this approach and the classical generalization in terms of several complex variables relies on the underlying algebra, that remains commutative in the case of the function theory of several complex

variables and is no longer commutative in the case of Clifford analysis. Moreover, contrary to the complex case, the product of two monogenic functions is not in general monogenic. However, these constraints do not constitute a limitation to the construction of monogenic polynomials and, consequently, to the representation of monogenic functions as series of properly chosen polynomials.

The talk intends to provide an insight into the structure of special homogeneous monogenic polynomials that constitute a basic polynomial sequence with respect to the generalized conjugate Cauchy-Riemann operator. The different representations obtained lead to some new features, such as multiplicative methods for generating monogenic polynomials that compensate the lack of multiplication in the class of monogenic functions.

Joint work with M. I. Falcão (University of Minho, mif@math.uminho.pt) and H.R. Malonek (University of Aveiro, hrmalon@ua.pt).

variables and is no longer commutative in the case of Clifford analysis. Moreover, contrary to the complex case, the product of two monogenic functions is not in general monogenic. However, these constraints do not constitute a limitation to the construction of monogenic polynomials and, consequently, to the representation of monogenic functions as series of properly chosen polynomials.

The talk intends to provide an insight into the structure of special homogeneous monogenic polynomials that constitute a basic polynomial sequence with respect to the generalized conjugate Cauchy-Riemann operator. The different representations obtained lead to some new features, such as multiplicative methods for generating monogenic polynomials that compensate the lack of multiplication in the class of monogenic functions.

Joint work with M. I. Falcão (University of Minho, mif@math.uminho.pt) and H.R. Malonek (University of Aveiro, hrmalon@ua.pt).

João Morais

## An Extension of the Zernike Spherical Polynomials within Quaternionic Analysis

It is truly uncommon that a paper that has been set aside for almost eighty years nds its way back to scientic spotlight. Yet this is exactly what the 1934 paper by F. Zernike's Nobel prize has accomplished in the last decade. As a matter of fact, in the last years considerable attention has been paid to the role played by the Zernike polynomials (ZPs) in many dierent elds of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this development. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In this talk, we introduce the Zernike spherical polynomials within quaternionic analysis (R(Q)ZSPs), which rene and extend the Zernike moments dened through their polynomial counterparts. In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identied, respectively, with R^3 and R^4). We prove that the R(Q)ZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second order homogeneous dierential equation are also discussed. We address all the above and explore some basic facts of the arising quaternionic function theory.

R(Q)ZSPs are new in literature and have some consequences that are now under investigation.

The talk is based on joint work with I. Cação (University of Aveiro, Portugal)

R(Q)ZSPs are new in literature and have some consequences that are now under investigation.

The talk is based on joint work with I. Cação (University of Aveiro, Portugal)