On Radon hypergeometric functions on the Grassmannian manifold

TL;DR

This paper introduces Radon hypergeometric functions on Grassmannian manifolds, generalizing classical functions and deriving differential equations, with applications to matrix integrals and special functions.

Contribution

It defines Radon hypergeometric functions on Grassmannians, linking them to classical functions and deriving their differential equations.

Findings

Radon HGF generalizes Gelfand HGF on Grassmannians.

Differential equations for Radon HGF are established.

Connections to classical special functions via matrix integrals.

Abstract

We give a definition of Radon hypergeometric function (Radon HGF) of confluent and nonconfluent type, which is a function on the Grassmannian Gr(m,nr) obtained as a Radon transform of a character of the universal covering group of H_{\lambda}\subset GL(nr) specified by a partition \lambda of n, where H_{(1,\dots,1)}\simeq(GL(r))^{n}. When r=1, the Radon HGF reduces to the Gelfand HGF on the Grassmannian. We give a system of differential equations satisfied by the Radon HGF and show that the Hermitian matrix integral analogues of Gauss HGF and its confluent family: Kummer, Bessel, Hermite-Weber and Airy function, are obtained in a unified manner as the Radon HGF on Gr(2r,4r) corresponding to the partitions (1,1,1,1), (2,1,1), (2,2), (3,1) and (4), respectively.