An Algebra Model for the Higher Order Sum Rules

TL;DR

This paper develops an algebraic framework to analyze higher order sum rules for orthogonal polynomials on the unit circle, connecting algebraic structures with sum rule expressions and proving new cases of conjectures.

Contribution

It introduces an algebra model that relates sum rules to Hall-Littlewood type polynomials, recovering known results and proving a new case of the Lukic conjecture.

Findings

Recovered Golinskii and Zlatoš's result

Proved half of the Lukic conjecture for a single critical point

Established an algebraic expression linking sum rules and polynomials

Abstract

We introduce an algebra model to study higher order sum rules for orthogonal polynomials on the unit circle. We build the relation between the algebra model and sum rules, and prove an equivalent expression on the algebra side for the sum rules, involving a Hall-Littlewood type polynomial. By this expression, we recover an earlier result by Golinskii and Zlat\v{o}s, and prove a new case - half of the Lukic conjecture in the case of a single critical point with arbitrary order.