Orthogonal Polynomials and Special Functions

所提出的研究涉及正交多项式的判别式和能量最小化问题的平衡能量涉及判别式。我们将研究q-变形和量化的类似物的判别式被定义为结式。这涉及到发展一个理论的二阶q-差分方程的多项式正交关于一个积极的措施，其群众集中在工会的最多两个几何级数。我们还将考虑某些不定矩问题，其解涉及椭圆函数，其生成函数满足Lame型微分方程。拉梅微分方程也产生功能的家庭biorthogal合理的职能，这是有关R型连续分数，一类连续分数马松和我介绍，最近出现在解决广义特征值问题。我们还打算研究含有Askey-Wilson分差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，并找到平衡时能量的显式公式。我们还打算确定当粒子数变大时，能量会变大。这将在不同的一维模型中完成。在数学上，这导致了所谓的变形无限维李代数理论（以Sophus Lie命名）。另一个焦点将是矩问题，其中观测（矩）不一定决定分布（物理定律），但仍然可以找到“最佳”分布。我们还关心的频谱理论的运营商称为阿斯基威尔逊运营商。我们试图实现的是描述它们的谱，这意味着描述服从这些方程的物理系统的可能状态。

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.