Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

We find and discuss asymptotic formulas for orthonormal polynomials $P_n(z)$ with recurrence coefficients $a_n,b_n$. Our main goal is to consider the case where off-diagonal elements $a_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_n$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_n,b_n$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_n(z)$, $n\ge -1$, of such equations by a condition for $n\to \mathrm{\infty }$ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schr"odinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions $P_n(z)$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $P_n(z)$ as $n\to \mathrm{\infty }$ in terms of the Wronskian of the solutions $P_n(z)$ and $f_n(z)$. The formulas obtained for $P_n(z)$ generalize the asymptotic formulas for the classical Hermite polynomials where $a_n=\sqrt{(n+1)\mathrm{/}2}$ and $b_n=0$.