The zonal polynomials are symmetric polynomial functions each of which is an eigenfunction of a kind of Laplace–Beltrami operator. We give a simplification of the Laplace–Beltrami operator. We realize a symbolic algorithm for obtaining the coefficients of zonal polynomials.

This article shows how Harmonic Analysis on matrix groups can solve specifc problems of Multivariate Statistical Analysis. In particular are studied the properties of zonal polynomials, seen as spherical functions of a classical Gelfand pair. A duality formula allows to write the zonal...