Orthogonal Polynomials in Mathematical Physics

TL;DR

This review explores ($q$-)hypergeometric orthogonal polynomials, their connections to quantum groups, matrix models, integrable systems, and knot theory, highlighting their broad applications in mathematical physics.

Contribution

It provides a comprehensive framework linking orthogonal polynomials with ($q$-)hypergeometric functions and various areas of mathematical physics, including representation theory and conformal field theory.

Findings

Unified framework for ($q$-)hypergeometric orthogonal polynomials

Connections established with quantum groups and knot theory

Integral representations relate to Virasoro algebra conformal blocks

Abstract

This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ($q$-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra.