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arXiv · 2105.06991

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

C. Calderón·M. M. Castro
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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.

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C. Calderón, M. M. Castro. 2021-05-14. Structural formulas for matrix-valued orthogonal polynomials related to $2\times 2$ hypergeometric operators. https://doi.org/10.1007/s40840-021-01211-x

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