Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

AbstractThe method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form e−t2T(t)T∗(t), tαe−tT(t)T∗(t), and (1−t)α(1+t)βT(t)T∗(t), with T satisfying T′=(2Bt+A)T, T(0)=I, T′=(A+B/t)T, T(1)=I, and T′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (Pn)n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F2, F1 and F0 (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F0 are simultaneously triangularizable (but without making any commutativity assumption)