Asymptotics of orthogonal polynomials with asymptotic Freud-like weights
Asymptotics of orthogonal polynomials with asymptotic Freud-like weights
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具有渐近弗洛伊德权重的正交多项式的渐近
DOI：
10.1111/sapm.12291
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发表时间：
2019-12-06
影响因子：
2.7
通讯作者：
Wang，Xiang-Sheng
中科院分区：
文献类型：
--
作者：
Long，Wen-Gao;Dai，Dan;Wang，Xiang-Sheng
我们导出了正交多项式的一致渐近展开式，这些多项式是实解析的，在无穷远处表现为渐近的弗洛伊德权重函数。虽然极限零分布与弗洛伊德情况相同，但渐近展开式是不同的，因为权重函数在虚轴上可能有有限或无限个零。为了解决由这些零点引起的奇异性，在黎曼-希尔伯特分析中引入了一个辅助函数。在覆盖整个复平面的若干区域上建立了渐近公式。我们以连续对偶Hahn多项式为例来说明我们的主要结果。并给出了数值验证。
We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.