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HYPERGEOMETRIC FUNCTIONS OF MATRIX ARGUMENT
Speaker: Siddhartha Sahi, Rutgers University

Hypergeometric functions pFq(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=(x1,…,xn)
of A, have many applications in number theory, multivariate statistics, signal processing, and random matrix theory.

In the 1980s, Macdonald introduced a common generalization pFq(x;α)
, which for α=1 and α=2 reduces to the functions of James and Herz. In recent work with Hong Chen, we have obtained differential equations that characterize pFq(x;α), 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.