Multivariate generalisations of classical hypergeometric polynomials from Lie theory

Abstract

In this thesis, we will be studying Lie groups and their connection to certain orthogonal polynomials. We will look into the classical Krawtchouk, Meixner and Laguerre polynomials, and the multivariate Krawtchouk and Meixner polynomials as defined by Iliev. Using representations of the Lie groups SU(2) and SU(1,1), it will be shown that the three classical polynomials can be described as matrix coefficients of the representations. Using this connection of the polynomials to Lie groups, we derive various properties of the polynomials from the unitarity of the representation and the associated Lie algebra representation. Next, the representations are generalised to higher dimensional spaces. Doing so, a new connection is shown between the Lie group SU(d+1) and the multivariate Krawtchouk polynomials, extending the known theory for the univariate polynomials. Another new result that will be established is the connection between the multivariate Meixner polynomials and Lie theory. This will be done by defining a representation of SU(1,d) in the Bergman space of the d-dimensional unit ball. Similar as for the univariate polynomials, we will derive the orthogonality, recurrence relations and difference equations from the associated Lie theory.