Some conditional and unconditional expectation identities for the multivariate normal with non-zero mean

Abstract

We give formulas for the conditional and unconditional expectations of products of multivariate Hermite and modified Hermite polynomials, each with a multivariate normal argument. A unified approach is given that covers both of these polynomials, each associated with a covariance matrix. This extended Hermite polynomial is associated with a matrix which is the difference between two covariance matrices, in other words, with any symmetric matrix.