Algebraic and Enumerative combinatorics seminar - Siddhartha Sahi-Hypergeometric functions of matrix argument

Abstract:In a widely circulated manuscript from the 1990s, I.G. Macdonald introduced certain higher-rank analogs of the classical hypergeometric functions $_pF_q$, which are expressed as explicit series in Jack and Macdonald polynomials in one and two sets of variables. For special choices of parameters, these series reduce to the hypergeometric functions of matrix argument introduced earlier by C. Herz and A.T. James, which have numerous applications in number theory, multivariate statistics, signal processing, and random matrix theory.

The classical hypergeometric functions are solutions to the hypergeometric differential equation. Macdonald raised the problem of providing an analogous characterization for higher-rank functions by means of differential equations. Over the years, this problem was solved for a small number of cases where p and q are at most 3. However, as the operators become increasingly complicated, the general problem remained open for 40 years. In this talk, we will present a complete solution. This is joint work with Hong Chen.

There will be a pre-seminar at 1:30pm in MC 6029 in a flipped classroom format based on Macdonald’s manuscript on hypergeometric functions (https://arxiv.org/abs/1309.4568). Participants are expected to read the manuscript in advance, and the session will focus on questions and discussion led by the speaker.