On some gateways between sum rules

TL;DR

This paper explores classical measure mappings linking measures on intervals and the unit circle, revealing new sum rule formulations and connections between orthogonal polynomials, recursion coefficients, and equilibrium measures in random matrix theory.

Contribution

It introduces new correspondences between measures and sum rules, expanding understanding of orthogonal polynomials and equilibrium measures in random matrix ensembles.

Findings

New measure correspondences linking interval and circle measures

Formulation of novel sum rules for classical models

Connections established between orthogonal polynomials and equilibrium measures

Abstract

We present correspondences induced by some classical mappings between measures on an interval and measures on the unit circle. More precisely, we link their sequences of orthogonal polynomial and their recursion coefficients. We also deduce some correspondences between particular equilibrium measures of random matrix ensembles. Additionally, we show that these mappings open up gateways between the sum rules associated with some classical models, leading to new formulations of several sum rules.