Full length articleA miraculously commuting family of orthogonal matrix polynomials satisfying second order differential equations

We find structural formulas for a family (Pn)n of matrix polynomials of arbitrary size orthogonal with respect to the weight matrix e−t2eAteA∗t, where A is certain nilpotent matrix. It turns out that this family is a paradigmatic example of the many new phenomena that show the big differences between scalar and matrix orthogonality. Surprisingly, the polynomials Pn, n≥0, form a commuting family. This commuting property is a genuine and miraculous matrix setting because, in general, the coefficients of Pn do not commute with those of Pm, n≠m.