Combinatorics of Orthogonal Polynomials

Orthogonal polynomials are classical objects arising from the study of continued fractions. Due to the long history of orthogonal polynomials, they have now become important objects of study in many areas: classical analysis and PDE, mathematical physics, probability, random matrix theory, and combinatorics. The combinatorial study of orthogonal polynomials was pioneered by Flajolet and Viennot in the 1980s. In this lecture series, we study fascinating combinatorial properties of orthogonal polynomials.

We first study basic properties of univariate orthogonal polynomials including Viennot's combinatorial theory. Some of these properties can be naturally generalized to orthogonal polynomials of type R₁ and R₂. We will show that moments of multivariate little q-Jacobi polynomials are generating functions for lecture hall tableaux, which are 2-dimensional generalizations of lecture hall partitions. These moments are closely related to q-Selberg integrals. We will also show that such an approach can be generalized to all orthogonal polynomials in the q-Askey scheme.

This is based on several joint papers with Sylvie Corteel, Bhargavi Jonnadula, Jon Keating, Minho Song, and Dennis Stanton.