Orthogonal matrix polynomials satisfying second-order differential equations

We develop a general method that allows us to introduce families of orthogonal matrix polynomials of size N × N satisfying second-order differential equations. The presence of this extra property should make these orthogonal polynomials into useful tools in several areas of mathematics and its applications. Historically, this has certainly been the case for their scalar-valued versions. The subtlety of the noncommutative algebra of matrices can be exploited to yield many different such families, almost dwarfing the scalar situation by comparison. All these families form a nice and rich hierarchy starting from the classical Jacobi, Hermite, and Laguerre families, N=1, and increasing in number and variety as the size N increases. We illustrate the use of our method by giving large classes of generic examples of arbitrary size N.