Mathematical Sciences: Research in Generalized Special Functions

本项目涉及广义特殊 函数，定义在矩阵空间和领域的积极性。 该项目的一部分将是开发的精细结构的 纯量值和算子值广义超几何函数 定义在积极性的一般领域。 的另一部分 该项目将开发通用特殊 完全正性理论和数学 统计 这项研究涉及的一般基础， 矩阵空间上的超几何函数 这项工作 经典不变理论在组合数学中的应用， 多项式恒等式的发展，在物理学中的应用 在超几何函数中，在量子的矩阵论元中 场论及其在统计学中的应用 概率“自然指数族”的推论 分布。

This project concerns research on generalized special functions, defined on matrix spaces and on domains of positivity. Part of the project will be to develop the fine structure of the scalar- and operator-valued generalized hypergeometric functions defined on general domains of positivity. The other part of the project will be to develop applications of the generalized special functions to the theory of total positivity and to mathematical statistics. This research deals with the general foundations of the hypergeometric functions on the space of matrices. The work has applications to combinatorics in the classical invariant theory and the development of polynomial identities, applications to physics in the hypergeometric functions in the matrix arguments in quantum field theory, and applications to statistics in statistical inference for the 'natural exponential family' of probability distributions.