d-dimensional orthogonal polynomials: Commutator theorem and other results

We give new and simplified proofs of three basic theorems in the theory of orthogonal polynomials associated to a classical, [Formula: see text]-valued random variable [Formula: see text] with all moments, namely: (1) The characterization of [Formula: see text] in terms of commutators among the creation–annihilation–preservation (CAP) operators in its quantum decomposition. (2) The characterization, in terms of the same objects, of the fact that the distribution of [Formula: see text] is a product measure. (3) The equivalence of the symmetry of [Formula: see text] with the vanishing of the associated preservation operator. Our new formulation of these results allows one to obtain a stronger form of the above statements.