Mathematical Sciences: Symmetry and Orthogonal Polynomials

在这个项目中，首席研究员将继续研究正交多项式和与Coxeter群和根系统相关的积分，方法是利用微分-差分算子的参数化交换代数。特别地，他将研究将这些代数与不同参数值联系起来的缠结算子。该算子被推测为一类由群代数子集支持的分数阶积分算子。与泊松核的关系可能导致显式积分形式。主要研究者正在分析与Coxeter群及其根系统相关的正交多项式，以研究涉及微分算子、积分算子的核以及与拉普拉斯算子相关的特殊函数的问题。正交多项式在数值分析、近似理论和微分方程等许多不同领域中都有重要的应用。在这个项目中，首席研究员将使用与这些多项式相关的代数结构来扩展我们对多项式对称性的认识。

In this project the principal investigator will continue his work on orthogonal polynomials and integrals associated to Coxeter groups and root systems by means of a parametrized commutative algebra of differential-difference operators. In particular, he will look at the intertwining operator which relates these algebras for different parameter values. This operator is conjectured to be a new type of fractional integral operator supported by a subset of the group algebra. There is a relationship to Poisson kernels that could lead to explicit integral forms. The principal investigator is analyzing orthogonal polynomials that are associated to Coxeter groups and their root systems, in order to study questions involving differential- difference operators, kernels of integral operators, and special functions associated with the Laplace operator. Orthogonal polynomials arise in an essential way in a number of different areas, such as numerical analysis, approximation theory and differential equations. In this project the principal investigator will use the algebraic structure associated with these polynomials to extend our knowledge of the symmetry properties of the polynomials.