Orthogonal polynomials with respect of a class of Fisher-Hartwig symbols

TL;DR

This paper derives asymptotic formulas for the coefficients of orthogonal polynomials on the unit circle with respect to a Fisher-Hartwig type weight, expanding understanding of their behavior in complex analysis and mathematical physics.

Contribution

It provides new asymptotic results for orthogonal polynomials associated with Fisher-Hartwig symbols, a class not fully analyzed before.

Findings

Asymptotic formulas for polynomial coefficients derived

Extension of Fisher-Hartwig analysis to orthogonal polynomials on the circle

Results applicable to complex analysis and mathematical physics

Abstract

In this paper we give an asymptotic of the coefficients of the orthogonal polynomials on the unit circle, with respect of a weight of type $\displaystyle{ f : \theta \mapsto \prod_{1\le j \le M} \vert 1 - e^{i(\theta_{j}-\theta)}\vert ^{2\alpha_{j}} c}$ with $\theta_{j}\in ]-\pi,\pi]$, $-\frac{1}{2} < \alpha_{j}<\frac{1}{2}$ and $c$ a sufficiently smooth function.