BCn-symmetric abelian functions

Abstract

We construct a family of ${\mathrm{BC}}_n$-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials (see [10]) and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials (see [13]). We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson's summation and Bailey's transformation