5. Properties of General Systems of Orthogonal Polynomials with a Symmetric Matrix Argument

There exists a large literature of the orthogonal polynomials (OPs) with a single variable associated with a univariate distribution. The theory of these OPs is well established and many properties of them are developed. Then, some authors have discussed the OPs with matrix arguments in the past. However, there are many unsolved properties, owing to the complex structures of the OPs with a matrix argument. In this paper, we extend some properties, which are well known for the OPs with a single variable, to those with a matrix argument. We give a brief discussion on the zonal polynomials and the general system of OPs with a symmetric matrix argument, with examples, the Hermite, the Laguerre, and the Jacobi polynomials. We derive the so-called three-term recurrence relations, and then, the Christoffel–Darboux formulas satisfied by the OPs with a symmetric matrix argument as a consequence of the three-term recurrence relations. Also, we present the “\((2k+1)\)-term recurrence relations”, an extension of the three-term recurrence relations, and then an extension of the Christoffel–Darboux formulas as its consequence. Finally we give a brief discussion on the linearization problem and the representation of Hermite polynomials as moments. For the derivations of those results, the theory of zonal and invariant polynomials with matrix arguments is useful.