一类广义的十次Freud-型权函数

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 921-935.

一类广义的十次Freud-型权函数

王丹¹(),朱孟坤^2,*(),YangChen¹(),王晓丽²()

¹ 澳门大学科技学院数学系澳门 999078
² 齐鲁工业大学(山东省科学院)数学与统计学院济南 250353

收稿日期:2020-08-04 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 朱孟坤 E-mail:bohewan@126.com;zmk@qlu.edu.cn;yayangchen@um.edu.mo;wxlspu@qlu.edu.cn
作者简介:王丹, E-mail: bohewan@126.com|Yang Chen, E-mail: yayangchen@um.edu.mo|王晓丽, E-mail: wxlspu@qlu.edu.cn
基金资助:
澳门科学技术发展基金(FDCT023/2017/A1);澳门大学基金(MYRG2018-00125FST);国家自然科学基金(11801292);广东省自然科学基金(2021A1515010361)

A Generalised Decic Freud-Type Weight

Dan Wang¹(),Mengkun Zhu^2,*(),Chen Yang¹(),Xiaoli Wang²()

¹ Department of Mathematics, Faculty of Science and Technology, University of Macau, Avenida da Universidade, Macau Taipa 999078
² School of Mathematics and Statistics, Qilu University of Technology(Shangdong Academy of Sciences), Jinan 250353

Received:2020-08-04 Online:2021-08-26 Published:2021-08-09
Contact: Mengkun Zhu E-mail:bohewan@126.com;zmk@qlu.edu.cn;yayangchen@um.edu.mo;wxlspu@qlu.edu.cn
Supported by:
the Macau Science and Technology Development Fund(FDCT023/2017/A1);the Universidade de Macau(MYRG2018-00125FST);the NSFC(11801292);the NSF of Guangdong Province(2021A1515010361)

摘要/Abstract

摘要：

该文构造了一类广义的十次Freud-型权函数 $ w （x；t，\alpha）=|x|^{\alpha}{\rm e}^{-x^{10}+tx^2}，x，t\in\mathbb{R}，\alpha>-1. $ 研究了正交于该权函数的多项式的性质，得到其相关的循环系数$\beta_n$所满足的一系列差分-微分方程，并给出$\beta_n$的渐近行为.讨论并推导出关于该权函数的Hankel行列式在高维时（$n\rightarrow\infty$）满足的特性.最后，考虑了特殊情形下，即$t=\alpha=0$时，由该权函数生成的Hankel矩阵的最小特征值问题.

关键词: 正交多项式, 循环系数, Hankel行列式, 最小特征值

Abstract:

In this paper, the authors focus on a generalised decic Freud-type weight function $w(x;t, \alpha)=|x|^{\alpha}{\rm e}^{-x^{10}+tx^2}, t, x\in\mathbb{R}, \alpha>-1, $ and study the properties of the orthogonal polynomials with respect to this weight. The difference-differential equations of their associated recurrence coefficients are derived; meanwhile, the authors also find the asymptotic behavior of recurrence coefficients via above mentioned equations. What's more, the authors discuss the Hankel determinant in regard to this weight as $n\rightarrow\infty$, and calculate the smallest eigenvalues of large Hankel matrices generated by this weight when $\alpha=t=0$.

Key words: Orthogonal Polynomials, Recurrence coefficients, Hankel determinant, Smallest eigenvalue

中图分类号:

O175.7

王丹,朱孟坤,YangChen,王晓丽. 一类广义的十次Freud-型权函数[J]. 数学物理学报, 2021, 41(4): 921-935.

Dan Wang,Mengkun Zhu,Chen Yang,Xiaoli Wang. A Generalised Decic Freud-Type Weight[J]. Acta mathematica scientia,Series A, 2021, 41(4): 921-935.

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