A miraculously commuting family of orthogonal matrix polynomials satisfying second order differential equations

We find structural formulas for a family (P-n)(n) of matrix polynomials of arbitrary size orthogonal with respect to the weight matrix e(-t2) e(At) e(A)*(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 P-n, n >= 0, form a commuting family. This commuting property is a genuine and miraculous matrix setting because, in general, the coefficients of P-n do not commute with those of P-m, n not equal m. (C) 2011 Elsevier Inc. All rights reserved.