Properties of Hermite and Laguerre polynomials in matrix argument and their applications

Abstract

This paper is concerned with the Hermite polynomials in symmetric and rectangular matrix argument and the Laguerre polynomials in symmetric matrix argument, which are associated with the symmetric and rectangular matrix-variate normal distributions and the Wishart distribution, respectively. We derive the multivariate Rodrigues formula (a differential equation) for each of the two kinds of Hermite polynomials, with respect to the associating p.d.f., and the mutivariate differential operator, i.e., the zonal polynomial in the matrix of differential operators, which plays the role of the ordinary differential. The multivariate Rodrigues formulae are useful in asymptotic multivariate distribution theory for matrix variates. We consider the general forms of the series (Edgeworth) expansions for distributions of both symmetric and rectangular random matrices, and then give an asymptotic (Edgeworth) expansion for the distribution of a scale mixture matrix, as a simple example. As for the Laguerre polynomials, alternative forms of differential equations are derived. We also present series expansions and recurrence relationships for the Hermite and Laguerre polynomials, and some limit properties of the Laguerre polynomials.