This research program consists of three lines of investigation, each connected to multivariate analogues of generalized hypergeometric functions. The first concerns a family of biorthogonal abelian functions associated to elliptic curves that generalize multivariate orthogonal polynomials due to Macdonald and Koornwinder. The second aspect involves certain multivariate analogues of quadratic transformations of hypergeometric functions, which generalize identities from the theory of symmetric spaces; related to this is the study of natural deformations of permutation actions of Coxeter groups to representations of the associated Hecke algebras, some of which have led to proofs of multivariate quadratic transformations. Finally, the investigator will be studying multivariate elliptic hypergeometric (contour) integrals, with a view to understanding the functional equations satisfied by such integrals.
Historically, the study of "special functions" originated in the fact that quite a few functions of interest in applications turned out to be members of a single family, the hypergeometric functions. The new family of functions the investigator is studying is a significant generalization of this family, combining several existing generalizations (each with their own applications and connections to other areas of mathematics) under one roof. In addition to the potential that new applications may arise, this investigation should also lead to a greater understanding of the existing families.
