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arXiv · 1811.09254

Semiclassical asymptotic behavior of orthogonal polynomials

D. R. Yafaev
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Abstract

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.

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BibTeXRIS

D. R. Yafaev. 2018-11-22. Semiclassical asymptotic behavior of orthogonal polynomials. https://arxiv.org/abs/1811.09254

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