Macdonald polynomials and basic hypergeometric series

The main objective of this project is to study Macdonald polynomials (which are a family of symmetric multivariable orthogonal polynomials introduced by 1. G. Macdonald in the 1980`s), with a particular emphasis an their connection to (multivariate) basic hypergeometric series. More generally, this project shall involve the study of selected topics in multivariable q-orthogonal polynomials, and their extensions, such as multivariable basic or elliptic biorthogonal rational functions, using tools from algebra and classical analysis. Among the different (possible) multivariate theories, this project particularly emphasizes q- orthogonal polynomials associated to root systems, or, equivalently, to Lie algebras. A substantial amount of theory for multivariable q-orthogonal polynomials associated to root systems already exists and has been developed only very recently over the last 15-20 years. These development include the Macdonald polynomials (of type A; in more generality, associated to reduced root systems) and the Koomwinder-Macdonald polynomials (of type BC, a non- reduced root system). The lauer caass of polynomials of type BC includes the former classes associated with classical (i.e., non-exceptional) root systems as special cases. Macdonald polynomials of type A are indexed by partitions, i.e. finite weakly decreasing sequences of positive integers. These polynomials form a basis of the algebra of symmetric functions with rational coefficients in two parameters q, t. They generalize many classical bases of this algebra, including monomial, elementary, Schur, Hall-Littlewood, and Jack symmetric functions. These particular cases correspond to various specializations of the indeterminates q and t. In terms of basic hypergeometric series, the Macdonald polynomials correspond to a multivariable generalization of the q- ultraspherical polynomials. A general goal of this project is to (further) develop the theory of multivariable q-orthogonal polynomials, analogous to the classical one-variable theory. For instance, it is likely that some classes of q-orthogonal polynomials which have been extended to one particular root system (such as BC) may also have corresponding extensions to other root systems. One of the further goals of this project is to provide, wherever possible, explicit "analytic" expressions for the respective multivariable q-orthogonal polynomials. In view of prospective applications of multivariable (q-)orthogonal polynomials (e.g., in harmonic analysis or in mathematical physics), it is certainly desirable to have explicit formulae at hand.