Poisson Brackets of Orthogonal Polynomials

TL;DR

This paper computes Poisson brackets for orthogonal polynomials on the real line and unit circle, relating them to symplectic forms, Jacobians, and basic variable changes, enhancing understanding of their algebraic structure.

Contribution

It introduces explicit calculations of Poisson brackets for orthogonal polynomials, connecting them to symplectic forms and variable transformations, which is a novel analytical development.

Findings

Poisson brackets for OPRL and OPUC are explicitly computed.

Connections established between Poisson brackets, symplectic forms, and Jacobians.

Provides new insights into the algebraic structure of orthogonal polynomials.

Abstract

For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable.