中央研究院數學研究所

Colloquium

Hypergeometric functions of matrix argument

2026/01/15 (Thu.) 15:30~16:30

Date : 2026/01/15 (Thu.) 15:30~16:30
Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
Speaker : Siddhartha Sahi (Rutgers University)
Organizer : Colin McSwiggen (AS)
Abstract :
Hypergeometric functions $\displaystyle {}_pF_q(A)$ of matrix argument were introduced by Herz (1955) for symmetric matrices and by James (1962) for Hermitian matrices. These functions, which depend only on the eigenvalues $x = (x_1, \ldots, x_n)$ of $A$, have many applications in number theory, multivariate statistics, signal processing, and random matrix theory.

In the 1980s, Macdonald introduced a common generalization $\displaystyle {}_pF_q (x;\alpha)$, which for $\alpha=1$ and $\alpha=2$ reduces to the functions of James and Herz. In recent work with Hong Chen we have obtained differential equations that characterize $\displaystyle {}_p F_q (x;\alpha)$, thereby answering a question of Macdonald.

Such equations were previously known only for a small number of cases, all with $p$ and $q$ at most $3$. The main difficulty was that the differential operators involved became very complicated for large $p$ and $q$, a complexity that halted progress for almost 40 years. Our work was made possible by the realization that the operators admit a compact description by means of a generating series.

Refrence: https://arxiv.org/pdf/2510.10875