The zonal polynomials of the positive definite real symmetric matrices, which appear in the expansions of the functions occurring in many multivariate non null distributions and moment formulae are eigenfunctions of the Laplace Beltrami operator. The resulting differential equation gives a recurrence relation between the coefficients from which they can be calculated.
A. T. James. "Calculation of Zonal Polynomial Coefficients by Use of the Laplace-Beltrami Operator." Ann. Math. Statist. 39 (5) 1711 - 1718, October, 1968. https://doi.org/10.1214/aoms/1177698153