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

Abstract

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calderón 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.

27 pages

References in corpus (7)

• Infinite J-matrices and a matrix moment problem
• Some examples of orthogonal matrix polynomials satisfying odd order differential equations
• Bispectrality and Time-Band-Limiting: Matrix valued polynomials
• Time and band limiting for matrix valued functions: an integral and a commuting differential operator
• The Matrix Bochner Problem
• $2\times 2$ Hypergeometric operators with diagonal eigenvalues
• A further look at time-and-band limiting for matrix orthogonal polynomials