Generalized Jacobi Polynomials as Spherical Functions of the Grassmann Manifold

Abstract

A formula is found for the generalized Jacobi polynomials which are a complete set of orthogonal symmetric polynomials relative to the measure (9.1). The zonal spherical functions of the (non-oriented) Grassmann manifold are a special case. The expansion is in terms of the zonal spherical polynomials of the real positive-definite symmetric matrices, with coefficients each of which factors into a binomial coefficient with partitional elements, a generalized hypergeometric coefficient, and another coefficient which can be calculated by use of a recurrence relation.