Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

2025/11/27 by Rostyslav Kozhan, Kozhan, Rostyslav, František Štampach +1
Mathematics · #30C15 #42C05 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2511.22507

openalex publication_date 2025/11/27 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients α_n,N when the ratio n/N converges as n,N→∞. First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case of asymptotically constant and asymptotically periodic coefficients α_n,N. Second, we determine the asymptotic zero distribution of paraorthogonal polynomials in the locally constant and locally periodic regimes. Analogous results are obtained for orthogonal polynomials under a mild additional condition on the varying coefficients.

Citations

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Related

• Coefficients of Orthogonal Polynomials on the Unit Circle and Higher Order Szego Theorems
• Asymptotic zero distribution for a class of multiple orthogonal polynomials
• Fine Structure of the Zeros of Orthogonal Polynomials, II. OPUC With Competing Exponential Decay
• Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle
• A universality theorem for ratios of random characteristic polynomials