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 Zlatoš and prove a new case - half of the Lukic conjecture in the case of a single critical point with arbitrary order.

中文翻译：

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高阶求和规则的代数模型

我们引入了一个代数模型来研究单位圆上正交多项式的高阶求和规则。我们建立了代数模型和求和规则之间的关系，并证明了涉及霍尔-利特伍德型多项式的求和规则在代数方面的等价表达式。通过这个表达式，我们恢复了 Golinskii 和 Zlatoš 较早的结果，并证明了一个新案例——在具有任意顺序的单个临界点的情况下，Lukic 猜想的一半。