LAUSR

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} 1 $ are also qualitatively different from each other. These results imply, in particular, that the corresponding Jacobi operator has deficiency indices $(1,1)$ in the first case, while it is essentially self-adjoint in the second case.