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

TL;DR

This paper studies matrix valued polynomials generated by Rodrigues' formulas, establishing their properties, recurrence relations, and eigenfunction equations, and proves a conjecture relating weight matrices to scalar types.

Contribution

It provides a comprehensive analysis of matrix valued polynomials from Rodrigues' formulas, including new proofs of conjectures and characterization of commutative classes.

Findings

Established recurrence relations and completeness of these polynomials.

Proved the Duran-Grunbaum conjecture for specific cases and dimensions.

Identified classes of quasi-orthogonal polynomials with properties similar to orthogonal polynomials.

Abstract

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. The conjecture of Duran and Grunbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type is proved for Q(x)=x and Q(x)=x^2-1 in dimension two, and for any dimension under genericity assumptions. Commutative classes of quasi-orthogonal polynomials are found, which satisfy all the properties usually associated to orthogonal polynomials.