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.

中文翻译：

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无卡尔曼条件的正交多项式的渐近行为

我们的目标是找到由雅可比递推系数 $a_{n}, b_{n}$ 定义的正交多项式 $P_{n}(z)$ 的渐近行为 $n\to\infty$。我们假设非对角线系数 $a_{n}$ 增长得如此之快，以至于系列 $\sum a_{n}^{-1}$ 收敛，即违反了卡尔曼条件。关于对角线系数 $b_{n}$ 我们假设 $-b_{n} (a_{n}a_{n-1})^{-1/2}\to 2\beta_{\infty}$一些 $\beta_{\infty}\neq \pm 1$。$P_{n}(z)$得到的渐近公式与满足Carleman条件时$\sum a_{n}^{-1}=\infty$的情况大不相同。特别是，如果 $\sum a_{n}^{-1} 1 $ 也有质的不同。这些结果特别意味着，在第一种情况下，相应的雅可比算子具有缺陷指数 $(1,1)$，