Spherical functions on the Grassmann manifold and generalized Jacobi polynomials — part 1

Abstract

Generalized Jacobi polynomials constitute a complete system of orthogonal symmetric polynomials with respect to a multivariate beta measure. This paper gives an explicit formula for these polynomials in terms of the author's invariant polynomials in two matrix arguments, and derives the orthogonality property. Explicit formulas are also given for certain constants which appear in the theory of zonal polynomials, in cases where the partitional parameters have not more than two nonzero parts. Zonal spherical functions on the Grassmann manifold may be expressed in terms of generalized Jacobi polynomials. In Part 2 of this paper, it will be shown that these zonal spherical functions 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.