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

The 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)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)(1−t)α(1+t)βT(t)T∗(t), with TT satisfying T′=(2Bt+A)TT′=(2Bt+A)T, T(0)=IT(0)=I, T′=(A+B/t)TT′=(A+B/t)T, T(1)=IT(1)=I, and T′(t)=(−A/(1−t)+B/(1+t))TT′(t)=(−A/(1−t)+B/(1+t))T, T(0)=IT(0)=I, respectively. Here AA and BB are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (Pn)n(Pn)n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F2F2, F1F1 and F0F0 (independent of nn) 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 AA or BB vanishes.The purpose of this paper is to show a method which allows us to deal with the case when AA, BB and F0F0 are simultaneously triangularizable (but without making any commutativity assumption).