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 $J$. 
@article{FAA_2021_55_2_a6,
     author = {D. R. Yafaev},
     title = {Universal relations in asymptotic formulas for orthogonal polynomials},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {77--99},
     publisher = {mathdoc},
     volume = {55},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a6/}
}
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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/