Orthogonal Polynomials and Special Functions

点击翻译按钮获取中文摘要

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.