Asymptotic behavior of orthogonal polynomials. Singular critical case

Abstract

Our goal is to find an asymptotic behavior as n → ∞ of the orthogonal polynomials P n (z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the caseIn the case |γ| = 1 asymptotic formulas for P n (z) are known; they depend crucially on the sign of |γ| − 1. We study the critical case |γ| = 1. The formulas obtained are qualitatively different in the cases |γ n | → 1 − 0 and |γ n | → 1 + 0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n (z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

“…This case lies on the borderline of our methods, and it is not covered by Corollary 9.1. Let us also mention the recent preprint [58] where the sequences (9.1) were considered for γ ∈ 3 2 , ∞ and proved that σ ess (A) = ∅ provided that A is self-adjoint. This case is also not covered by our results.…”