Abstract

Many orthogonal polynomials \(u(n,z)\) (\(n\) is the number of the polynomial, \(z\) is its argument), for example, the Chebyshev, Hermite, Laguerre, Legendre, and other polynomials, are determined by recurrence relations (or finite-difference equations) of the second order. For large numbers \(n\), they are approximated by exponential, trigonometric, or special functions of a compound argument. For example, Hermite polynomials are approximated by the Plancherel–Rotach formulas, in which the special function is \({\rm Ai}\), the Airy function, the Legendre polynomials are approximated by the zero-order Bessel function, etc. In the paper, an approach is developed for finding asymptotics of this type, which are uniform in this case (and unified) with respect to the variable \(z\). The approach is based on the passage from discrete equations to continuous pseudodifferential equations and the subsequent application of the semiclassical approximation to these equations with complex phases. This is a further development of the considerations proposed in the papers of A.I. Aptekarev, S.Yu. Dobrokhotov, D.N. Tulyakov, and A.V. Tsvetkova devoted to asymptotics of the Plancherel–Rotach type for Hermite polynomials and a subclass of Hermite type orthogonal polynomials with multiple indices.

中文翻译：

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求递归关系给定多项式渐近线的一种方法

摘要

许多正交多项式\(u(n,z)\)（\(n\)是多项式的个数，\(z\)是它的参数），例如Chebyshev、Hermite、Laguerre、Legendre等多项式，由二阶递推关系（或有限差分方程）确定。对于大数\(n\)，它们通过指数函数、三角函数或复合参数的特殊函数来近似。例如， Hermite 多项式通过 Plancherel-Rotach 公式近似，其中特殊函数为\({\rm Ai}\)，艾里函数，勒让德多项式由零阶贝塞尔函数等逼近。 在本文中，开发了一种方法来寻找这种类型的渐近线，它们在这种情况下是一致的（并且是统一的）关于变量\(z\)。该方法基于从离散方程到连续伪微分方程的过渡，以及随后将半经典近似应用于这些具有复杂相位的方程。这是对 AI Aptekarev, S.Yu 的论文中提出的考虑的进一步发展。Dobrokhotov、DN Tulyakov 和 AV Tsvetkova 致力于研究 Hermite 多项式的 Plancherel-Rotach 型渐近性和具有多个索引的 Hermite 型正交多项式的子类。