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- W4288889609 abstract "We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}toinfty$ as $ntoinfty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $ngeq -1$, of such equations by a condition for $ntoinfty$ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schrodinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions $P_{n}(z)$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $P_{n}(z)$ as $n toinfty$ in terms of the Wronskian of the solutions $ P_{n} (z) $ and $ f_{n} (z)$. The formulas obtained for $P_{n}(z)$ generalize the asymptotic formulas for the classical Hermite polynomials where $a_{n}=sqrt{(n+1)/2}$ and $b_{n}=0$." @default.
- W4288889609 created "2022-07-31" @default.
- W4288889609 creator A5082940011 @default.
- W4288889609 date "2022-02-04" @default.
- W4288889609 modified "2023-09-24" @default.
- W4288889609 title "Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials" @default.
- W4288889609 doi "https://doi.org/10.48550/arxiv.2202.02087" @default.
- W4288889609 hasPublicationYear "2022" @default.
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