论文标题
正交多项式的渐近行为。奇异的关键案例
Asymptotic behavior of orthogonal polynomials. Singular critical case
论文作者
论文摘要
我们的目标是找到正交多项式的$ n \ to \ infty $ $ p_ {n}（z）$由jacobi复发系数$ a_ {n} $（off-diagonal项）和$ b_ {n} $（diagonal项）。我们考虑$ a_ {n} \ to \ infty $，$ b_ {n} \ to \ infty $，以$ \ sum a_ {n}^{ - 1} <\ infty $ $（$ a_ a _ {n}^{ - 1} （a_ {n} a_ {n-1}）^{ - 1/2} \toγ$ as $ n \ to \ infty $。在情况下$ |γ| \ neq 1 $渐近公式为$ p_ {n}（z）$是已知的；它们取决于$ |的标志。 γ| -1 $。我们研究关键案例$ | γ| = 1 $。在$ |γ_{n} |的情况下，获得的公式在质量上有所不同。 \至1-0 $和$ |γ_{n} | \至1+0 $。本文的另一个目的是提倡一种基于jacobi差异方程式和schrödinger类型的微分方程的$ p_ {n}（z）$的渐近行为的方法。
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 $γ_{n}:=2^{-1}b_{n} (a_{n}a_{n-1})^{-1/2} \to γ$ as $n\to\infty$. In the case $|γ| \neq 1$ asymptotic formulas for $P_{n}(z)$ are known; they depend crucially on the sign of $| γ|-1$. We study the critical case $| γ|=1$. The formulas obtained are qualitatively different in the cases $|γ_{n}| \to 1-0$ and $|γ_{n}| \to 1+0$. 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.
