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，其中平衡能量涉及一个判别式。我们将研究被定义为结果式的q变形和量化的判别式的类似。这涉及到发展关于质量集中在至多两个几何级数的并上的正测度正交的多项式的二阶q-差分方程组的理论。我们还将考虑一些不定矩问题，它们的解涉及椭圆函数，并且它们的母函数满足Lame型微分方程。Lame微分方程也是与R型连分式有关的一族双正目标有理函数族的母函数，R型连分式是Masson和我最近在解决广义特征值问题时引入的一类连分式。我们还打算研究涉及Askey-Wilson除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，并找到平衡能量的显式公式。我们还打算确定当粒子数量变大时，能量会变得多大。这将在不同的一维模型中完成。在数学上，这导致了一种所谓的变形无限维李代数的理论(以Sophus Lie命名)。另一个焦点将是矩问题，其中观测值(矩)不一定决定分布(物理定律)，但人们仍然可以找到“最佳”分布。我们还关注称为Askey-Wilson算子的算子的谱理论。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.