A q-generalization of the Toda equations for the q-Laguerre/Hermite orthogonal polynomials

TL;DR

This paper introduces a q-deformed version of the Toda equations tailored for q-Laguerre and Hermite orthogonal polynomial ensembles, extending the classical Toda lattice correspondence to a quantum setting.

Contribution

It develops a novel q-generalization of the Toda equations that aligns with the deformation theory of specific orthogonal polynomials.

Findings

Derived a q-deformed Toda equation consistent with quadratic relations

Extended Toda lattice correspondence to q-Laguerre and Hermite ensembles

Confirmed compatibility with polynomial deformation theory

Abstract

Based on the motivation of generalizing the correspondence between the Lax equation for the Toda lattice and the deformation theory of the orthogonal polynomials, we derive a q-deformed version of the Toda equations for both q-Laguerre/Hermite ensembles, and check the compatibility with the quadratic relation.