Asymptotic behavior of orthogonal polynomials. Singular critical case

Abstract

Our goal is to find an asymptotic behavior as $n\to \infty$ 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 $a_n\to \infty$, $b_n\to \infty$ in such a way that $\sum a_n^{-1}<\infty$ $($that is, the Carleman condition is violated$)$ and ${\gamma }_n:=2^{-1}{b}_n(a_n{a}_{n-1}{)}^{-1/2}\to \gamma$ as $n\to \infty$. In the case $|\gamma |\ne 1$ asymptotic formulas for $P_n(z)$ are known; they depend crucially on the sign of $|\gamma |-1$. We study the critical case $|\gamma |=1$. The formulas obtained are qualitatively different in the cases $|{\gamma }_n|\to 1-0$ and $|{\gamma }_n|\to 1+0$. 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.

References in corpus (1)

• Asymptotic behavior of orthogonal polynomials without the Carleman condition

Cited by in corpus (1)

• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case