Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials

TL;DR

This paper investigates the equilibrium configurations of discrete integrable systems related to classical root systems, revealing deformed orthogonal polynomials that extend Hermite, Laguerre, and Jacobi polynomials with preserved orthogonality and recurrence properties.

Contribution

It introduces deformed orthogonal polynomials arising from Ruijsenaars-Schneider systems, extending classical polynomials with new functional equations and recurrence relations.

Findings

Deformed polynomials describe equilibrium positions in discrete integrable systems.

Orthogonality of classical polynomials is preserved in deformations.

Deformed polynomials satisfy three-term recurrence and functional equations.

Abstract

The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of C-M systems. We determine the polynomials describing the equilibrium positions of the rational and trigonometric Ruijsenaars-Schneider systems based on classical root systems. These are deformation of the classical orthogonal polynomials, the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium positions of the corresponding Calogero and Sutherland systems. The orthogonality of the original polynomials is inherited by the deformed ones which satisfy three-term recurrence and certain functional equations. The latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials.