Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations

TL;DR

This paper links bi-orthogonal polynomials on the unit circle with Garnier systems, deriving Hamiltonian formulations and discrete Painlevé equations for multi-variable cases, advancing the understanding of integrable systems.

Contribution

It explicitly constructs Hamiltonian formulations for Garnier systems from bi-orthogonal polynomials and derives canonical forms of discrete Painlevé equations for multiple variables.

Findings

Bi-orthogonal polynomials correspond to classical solutions of Garnier systems.

Hamiltonian formulations for multi-variable Garnier systems are explicitly constructed.

Discrete Painlevé equations characterize the recurrence relations in these systems.

Abstract

We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrence relations of the discrete Painlev\'e type, for example in the case with one free deformation variable the system was found to be characterised by a solution to the discrete fifth Painlev\'e equation. Here we derive the canonical forms of the multi-variable generalisation of the discrete fifth Painlev\'e equation to the Garnier systems, i.e. for arbitrary numbers of deformation variables.