Geodesic
Universal relations in asymptotic formulas for orthogonal polynomials
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 2, pp. 77-99.

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Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as $n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on the coefficients of the operator $J$, the amplitude and phase factors in asymptotic formulas for $P_{n}(\lambda)$ are linked by certain universal relations found in the paper. Our proofs rely on the study of a time-dependent evolution generated by suitable functions of the operator $J$. 
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D. R. Yafaev. Universal relations in asymptotic formulas for orthogonal polynomials. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 2, pp. 77-99. http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a6/

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