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.
