Orthogonal Polynomials and Special Functions

正交多项式和特殊函数

• 批准号: 9970865
• 负责人: Mourad Ismail
• 金额: $ 12.28万
• 依托单位: University of South Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970865&HistoricalAwards=false
• 关键词: Orthogonal; Polynomials; Special; Functions

The proposed research deals with discriminants of orthogonal polynomials and energy minimization problems where the equilibrium energy involves a discriminant. We will study q-deformed and quantized analogues of discriminants which are defined as resultants. This involves developing a theory of second order q-difference equations for polynomials orthogonal with respect to a positive measures whose masses are concentrated on the union of at most two geometric progressions. We will also consider certain indeterminate moment problems whose solution involves elliptic functions and their generating functions satisfy a Lame type differential equation. The Lame differential equations are also generating functions for a family of biorthogoal rational functions which are related to R-type continued fractions, a class of continued fractions Masson and I introduced and recently arose in solving generalized eigenvalue problems. We also intend to study spectral theory of second order operator equations involving the Askey-Wilson divided difference operator. We plan to study the equilibrium position of a system of particles in an external field and find explicit formulas for the energy at equilibrium. We also intend to determine how large the energy gets when the number of particles gets large. This will be done in different one dimensional models. Mathematically this leads to a theory of the so called deformed infinite dimensional Lie algebras (named after Sophus Lie). Another focus will be moment problems where the observations (moments) do not necessarily determine the distribution (the physical law) but nevertheless one can find ``best" distributions. We are also concerned with spectral theory of the operators called the Askey-Wilson operators. What we are trying to achieve is to describe their spectrum, which means describe the possible states of the physical system obeying these equations.

本文研究正交多项式的判别式和能量最小化问题，其中平衡能量涉及一个判别式。我们将研究被定义为结果的q-变形和量子化的判别式类似物。这涉及到发展二阶q-差分方程的理论，这些方程与正测度正交，其质量集中在至多两个几何级数的并上。我们还将考虑一类不确定矩问题，其解涉及椭圆函数及其生成函数满足Lame型微分方程。Lame微分方程也为一类双正交有理函数生成函数，这些函数与r型连分式有关，r型连分式是Masson和我介绍的一类连分式，最近在求解广义特征值问题中出现。我们还打算研究涉及Askey-Wilson微分算子的二阶算子方程的谱理论。我们计划研究粒子系统在外场中的平衡位置，并找出平衡能量的显式公式。我们还想确定当粒子数量增加时能量会有多大。这将在不同的一维模型中完成。从数学上讲，这导致了所谓的变形无限维李代数（以Sophus Lie命名）的理论。另一个焦点将是力矩问题，其中观测（力矩）不一定决定分布（物理定律），但仍然可以找到“最佳”分布。我们还关注称为Askey-Wilson算子的谱理论。我们试图实现的是描述它们的光谱，也就是说描述符合这些方程的物理系统的可能状态。