Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Abstract

We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices, 2004 : 10 (2004), 461–484) and [15] (Matrix valued orthogonal polynomials of the Jacobi type, Indag. Math. 14 nrs. 3, 4 (2003), 353–366). While we restrict ourselves to considering only first order operators, we do not make any assumption as to their symmetry. For simplicity we restrict to the case N = 2. We draw a few lessons from these examples; many of them serve to illustrate the fundamental difference between the scalar and the matrix valued case.