Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

Abstract Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and Hessenberg, respectively, Gauss–Borel factorization of moment matrix is given. It shown a hidden freedom exhibited by system related multiple polynomials. Pearson equations discussed, Laguerre–Freud considered, differential for type I II polynomials, as well corresponding linear forms The Jacobi–Piñeiro used an illustrating case relations presented. permuting Christoffel transformation finding families provides convenient illustration these symmetries, giving shifted permuted parameters. We also present general theory perturbation each weight polynomial or rational function aka called Geronimus transformations. connections formulas forms, vector Stieltjes–Markov functions illustrate findings analyzing special modification even polynomial.