Asymptotic behavior of orthogonal polynomials without the Carleman condition

TL;DR

This paper investigates the asymptotic behavior of orthogonal polynomials defined by Jacobi recurrence coefficients when the Carleman condition is violated, revealing distinct formulas and conditions for self-adjointness.

Contribution

It provides new asymptotic formulas for orthogonal polynomials without the Carleman condition and characterizes self-adjointness of the associated Jacobi operator.

Findings

Asymptotic formulas differ significantly from the Carleman condition case.

Phase factors in formulas are independent of the spectral parameter when _n^{-1} converges.

Necessary and sufficient conditions for self-adjointness are established.

Abstract

Our goal is to find an asymptotic behavior as $n\to\infty$ of orthogonal polynomials $P_{n}(z)$ defined by the Jacobi recurrence coefficients $a_{n}, b_{n}$. We suppose that the off-diagonal coefficients $a_{n}$ grow so rapidly that the series $\sum a_{n}^{-1}$ converges, that is, the Carleman condition is violated. With respect to diagonal coefficients $b_{n}$ we assume that $-b_{n} (a_{n}a_{n-1})^{-1/2}\to 2\beta_{\infty}$ for some $\beta_{\infty}\neq \pm 1$. The asymptotic formulas obtained for $P_{n}(z)$ are quite different from the case $\sum a_{n}^{-1}=\infty$ when the Carleman condition is satisfied. In particular, if $\sum a_{n}^{-1}<\infty$, then the phase factors in these formulas do not depend on the spectral parameter $z\in{\Bbb C}$. The asymptotic formulas obtained in the cases $|\beta_{\infty}|<1 $ and $|\beta_{\infty}|>1 $ are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.