Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits
J. Borrego-Morell, Mirta M. Castro, Antonio J. Durán · Integral Transforms and Special Functions · 2011
We introduce a family of weight matrices W of the form T(t)T*(t), , where is a certain nilpotent matrix and is a diagonal matrix with negative real entries. The weight matrices W have arbitrary size N×N and depend on N parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second-order differential equation with differential coefficients that are matrix polynomials F 2, F 1 and F 0 (independent of n) of degrees not bigger than 2, 1 and 0, respectively. For size 2×2, we find an explicit expression for a sequence of orthonormal polynomials with respect to W. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.