Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

The properties of matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators.A new, direct proof is given to the conjecture of Durán and Grünbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type.Commutative classes of orthogonal polynomials (corresponding to weights that are self-adjoint but not positive semidefinite) are found, which satisfy all the properties usually associated to orthogonal polynomials, and are not of scalar type.