Spherical functions on the Grassmann manifold and generalized Jacobi polynomials — Part 2

Abstract

Part 1 of this paper presented an explicit formula for generalized Jacobi polynomials of matrix argument. These polynomials constitute a complete system of orthogonal symmetric polynomials with respect to a multivariate beta measure. Zonal spherical functions on the Grassmann manifold may be expressed in terms of generalized Jacobi polynomials, and it is shown in Part 2 that they have an integral representation which generalizes the well-known integrals for Legendre and Gegenbauer polynomials of even order. In particular cases, this integral representation may be used to construct the zonal and associated spherical functions in terms of univariate special functions.