Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and study the corresponding second order difference equation. We suggest an Ansatz for its solutions playing the role of the semiclassical Green-Liouville Ansatz for solutions of the Schr\"odinger equation. The formulas obtained for $P_{n}(z)$ as $n\to\infty$ generalize the classical Bernstein-Szeg\"o asymptotic formulas.
Universal relations in asymptotic formulas for orthogonal polynomials
Asymptotic behavior of orthogonal polynomials without the Carleman condition
Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials
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