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Title: Multiple orthogonal polynomials: pearson equations and Christoffel formulas
Author: Branquinho, Amílcar
Foulquié-Moreno, Ana
Mañas, Manuel
Keywords: Multiple orthogonal polynomials
Banded tetradiagonal recursion matrices
Christoffel transformations
Geronimus transformations
Pearson equation
Issue Date: 3-Oct-2022
Publisher: Birkhauser
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 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–Piñeiro 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–Piñeiro 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 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.
Peer review: yes
DOI: 10.1007/s13324-022-00734-1
ISSN: 1664-2368
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Appears in Collections:CIDMA - Artigos
CHAG - Artigos

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