Macdonald conjectures for multivariate hypergeometric functions

Hypergeometric functions for symmetric matrices were introduced in the 1950s by Herz, who gave an inductive definition using the Laplace transform. Soon thereafter Constantine obtained an explicit series expansion in terms of the zonal polynomials. These have found considerable applications in multivariate statistics, especially in the theory of non-central distributions.

In the1980s I.G. Macdonald proposed a one-parameter generalization of this theory, replacing Zonal polynomials by Jack polynomials. However many of the results in this general theory are still conjectural; indeed Macdonald's manuscript, though widely circulated, remains unpublished.

One key conjecture relates to the asymptotics of the Laplace kernel, which in the Macdonald setting is given only by a certain power series expansion. The difficulty is somewhat akin to having to prove rapid decay of the function exp(-x) based on its series expansion alone! We will sketch a proof of Macdonald's conjecture using ideas from the theory of Jack polynomials.