Generalized Hermite and Laguerre polynomials in multiple symmetric matrix arguments and their applications

Abstract

We investigate properties of the generalized Hermite [ H ( m) κ[ r]; φ ( X [ q] ; B [ r] )] and Laguerre [ L u[ q] κ[ r]; φ ( X [ q] ; B [ r] )] polynomials in q m × m symmetric matrix arguments X 1,…, X q ( = X [ q] ) and r m × m constant matrices B 1,…, B r ( = B [ r] ) and with q parameters u 1,…, u q ( = u [ q] ) ( q ⩽ r), and consider their applications in multivariate distribution theory. The results extend those for the case of one symmetric matrix argument obtained by Chikuse. We derive the generalized multivariate Rodrigues formula (a differential equation) for the polynomial H ( m) κ[ r]; φ , with respect to the associated joint multiple symmetric matrix-variate normal p.d.f., and the generalized multivariate differential operator, i.e., the invariant polynomial in the multiple matrices of differential operators, which plays the role of the ordinary differential. The generalized multivariate Rodrigues formula and the associated polynomials H ( m) κ[ r]; φ are utilized to present the general form of the series (Edgeworth) expansion for a distribution of multiple symmetric random matrices and then give a simple example. Alternative forms of differential equations are derived for the polynomial L u[ q] κ[ r]; φ . Series expansions and recurrence relationships (or some related results) are obtained for the polynomials H ( m) κ[ r];φ and L u[ q] κ[ r];φ .