Matrix orthogonal polynomials satisfying second-order differential equations: Coping without help from group representation theory

AbstractThe method developed in Duran and Grünbaum [Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004) 461–484] led us to consider 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 tα(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/t+B/(1-t))T, T(1/2)=I, respectively. Here A and B are in general two non-commuting matrices. To proceed further and find situations where these polynomials satisfied second-order differential equations, we needed to impose commutativity assumptions on the pair of matrices A,B. In fact, we only dealt with the case when one of the matrices vanishes.The only exception to this arose as a gift from group representation theory: one automatically gets a situation where A and B do not commute, see Grünbaum et al. [Matrix valued orthogonal polynomials of the Jacobi type: the role of group representation theory, Ann. Inst. Fourier Grenoble 55 (6) (2005) 2051–2068]. This corresponds to the last of the three cases mentioned above.The purpose of this paper is to consider the other two situations and since now we do not get any assistance from representation theory we make a direct attack on certain differential equations in Duran and Grünbaum [Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004) 461–484]. By solving these equations we get the appropriate weight matrices W(t), where the matrices A,B give rise to a solvable Lie algebra