Mathematical Sciences: Research in Generalized Special Functions

本项目涉及定义在矩阵空间和正定义域上的广义特殊函数的研究。该项目的一部分将是开发定义在一般正性域上的标量值和算子值广义超几何函数的精细结构。该项目的另一部分将是发展广义特殊函数在全正性理论和数理统计中的应用。本文研究了矩阵空间上超几何函数的一般基础。这项工作应用于经典不变量理论中的组合学和多项式恒等式的发展，应用于量子场论中矩阵变元中超几何函数的物理应用，以及应用于概率分布的统计推断中的统计应用。

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.