Special Functions of Matrix and Single Argument in Statistics

Abstract

This chapter discusses some special functions of matrix and single argument in statistics. The zonal spherical functions are related to certain symmetric spaces and to the representation theory of Lie groups. The spherical functions on the Grassmann manifold of unoriented m subspaces of n space can be the even spherical functions on the symmetric space of oriented subspaces. The spaces have an invariant measure and an invariant metric, the dual of which is an invariant differential operator called the Laplace Beltrami operator. A function on the space belonging to an irreducible representation of the transformation group is called a spherical function, and must be an eigenfunction of the Laplace Beltrami operator. The subgroup of the transformation group, which leaves a point of the space taken as origin fixed, is called the isotropy, stationary, or stability subgroup. Spherical functions invariant under the isotropy group are called zonal spherical functions. The Lie algebra of differential operators, which arise from the Laplace Beltrami operators, supplies the differential operators that occur in the differential equations for the hypergeometric functions.