We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals.
Garza, L. G., Garza, L. E., Marcellán, F., & Pinzón-Cortés, N. C. (2015). A matrix approach for the semiclassical and coherent orthogonal polynomials. Applied Mathematics and Computation, 256, 459–471. https://doi.org/10.1016/j.amc.2015.01.071