On a family of Jacobi type polynomials as eigenfunctions of $2times 2$ hypergeometric operators: Structural formulas

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2times 2$ introduced by C. Calderon et al. in arXiv:1810.08560, 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 algebra of second order differential operators associated with the weight matrix.