Structural formulas for matrix-valued orthogonal polynomials related to $2\times 2$ hypergeometric operators

TL;DR

This paper develops explicit structural formulas for 2x2 matrix-valued orthogonal polynomials related to hypergeometric operators, including Rodrigues formulas, recurrence relations, and differential operators, expanding understanding of their algebraic properties.

Contribution

It provides explicit Rodrigues formulas, recurrence relations, and differential operators for matrix-valued orthogonal polynomials of size 2x2, linked to hypergeometric operators, which were not previously detailed.

Findings

Explicit Rodrigues formula in terms of Jacobi polynomials

Three-term recurrence relation for orthonormal polynomials

Identification of differential operators and algebraic structures

Abstract

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calder\'on et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel-Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.