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Asymptotic behavior of orthogonal polynomials. Singular critical case. / Yafaev, D.R. .
In: Journal of Approximation Theory, Vol. 262, 105506, 02.2021.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Asymptotic behavior of orthogonal polynomials. Singular critical case
AU - Yafaev, D.R.
N1 - Publisher Copyright: © 2020
PY - 2021/2
Y1 - 2021/2
N2 - Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P n(z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n→∞, b n→∞ in such a way that ∑a n −1<∞ (that is, the Carleman condition is violated) and γ n:=2 −1b n(a na n−1) −1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for P n(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γ n|→1−0 and |γ n|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.
AB - Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P n(z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n→∞, b n→∞ in such a way that ∑a n −1<∞ (that is, the Carleman condition is violated) and γ n:=2 −1b n(a na n−1) −1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for P n(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γ n|→1−0 and |γ n|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.
KW - Increasing Jacobi coefficients
KW - Carleman condition
KW - Difference equations
KW - Jost solutions
UR - https://www.sciencedirect.com/science/article/abs/pii/S0021904520301428#!
UR - http://www.scopus.com/inward/record.url?scp=85095679362&partnerID=8YFLogxK
U2 - 10.1016/j.jat.2020.105506
DO - 10.1016/j.jat.2020.105506
M3 - Article
VL - 262
JO - Journal of Approximation Theory
JF - Journal of Approximation Theory
SN - 0021-9045
M1 - 105506
ER -
ID: 71379289