Macdonald-Koornwinder moments and the two-species exclusion process

Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey-Wilson polynomials, a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey-Wilson polynomials can be viewed as a specialization of the multivariate Macdonald-Koornwinder polynomials (also known as Koornwinder polynomials), which in turn give rise to the Macdonald polynomials associated to any classical root system via a limit or specialization. In light of the fact that Koornwinder polynomials generalize the Askey-Wilson polynomials, it is natural to ask whether one can find a particle model whose partition function is related to Koornwinder polynomials. In this article we answer this question affirmatively, by showing that the "homogeneous" Koornwinder moments at q=t recover the partition function for the two-species exclusion process. We also provide a "hook length" formula for Koornwinder moments when q=t=1.