Combinatorics of orthogonal polynomials on the unit circle

TL;DR

This paper explores the combinatorial structures underlying orthogonal polynomials on the unit circle, providing path interpretations, explicit formulas for moments, and new combinatorial insights into their properties.

Contribution

It introduces three combinatorial path models for OPUC moments, derives explicit formulas for specific polynomial families, and defines generalized linearization coefficients with combinatorial interpretations.

Findings

Path interpretations for OPUC moments: Łukasiewicz, Motzkin, Schröder

Explicit formulas for circular Jacobi and Rogers–Szegő polynomials

Combinatorial interpretations for generalized linearization coefficients

Abstract

Orthogonal polynomials on the unit circle (OPUC for short) are a family of polynomials whose orthogonality is given by integration over the unit circle in the complex plane. There are combinatorial studies on the moments of various types of orthogonal polynomials, including standard orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of type \( R_I \). In this paper, we study the moments of OPUC from a combinatorial perspective. We provide three path interpretations for them: \L{}ukasiewicz paths, gentle Motzkin paths, and Schr\"oder paths. Additionally, using these combinatorial interpretations, we derive explicit formulas for the generalized moments of some examples of OPUC, including the circular Jacobi polynomials and the Rogers--Szeg\H{o} polynomials. Furthermore, we introduce several kinds of generalized linearization coefficients and give combinatorial interpretations for them.