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.

图/表 5

图 1

左图是当 $\alpha = -\frac{1}{3}, 0, \frac{3}{2}$ 和 $3$ 时, $\mu_0(t, \alpha)$ 图像; 右图是当 $t = -4, 0, \frac{1}{2}$ 和 $1$ 时, $\mu_0(t, \alpha)$ 图像"

图 1

图 2

当 $\alpha\equiv 1, t = -4, -\frac{1}{2}, 0$ 或 $2$ , $n = 2, 3, 4$ 或 $5$ 时, $P_n(x;t, \alpha)$ 图像"

图 2

图 3

当 $\alpha\equiv 1, t = -4, -\frac{1}{2}, 0$ 或 $2$ 时, $P_n(x;t, \alpha)$ 图像, $n = 2, 3, 4$ 或 $5$ "

图 3

图 4

当 $\alpha\equiv 1, t = -4, -\frac{1}{4}, \frac{1}{2}$ 或 $5$ , $\beta_n(t, \alpha)$ 图像"

图 4

图 5

当 $t\equiv 1, \alpha = -\frac{1}{2}, 0, \frac{1}{2}$ 或 $5$ 时, $\beta_n(t, \alpha)$ 图像"

图 5

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