Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

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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 one Hessenberg, respectively, and the Gauss-Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre-Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi-Pineiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi-Pineiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes-Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.
CRUE-CSIC (Acuerdos Transformativos 2022) © The Author(s) 2022 AB acknowledges Centro de Matematica da Universidade de Coimbra (CMUC)UID/MAT/00324/2020, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. AFM is partially supported by CIDMA Center for Research and Development inMathematics and Applications (University ofAveiro) and the Portuguese Foundation for Science andTechnology (FCT) within project UID/MAT/04106/2020. MM thanks the financial support from the Spanish "Agencia Estatal de Investigacion" research project [PGC2018-096504-B-C33], Ortogonalidad y Aproximacion: Teoria y Aplicaciones en Fisica Matematica and research project [PID2021-122154NB-I00], Ortogonalidad y aproximacion con aplicaciones en machine learning y teoria de la probabilidad. The authors also acknowledge economical support from ICMAT's Severo Ochoa program mobility B. The authors are grateful for the excellent job of the referees, whose suggestions and remarks improved the final text.
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