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A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss\u2013Borel factorization of the moment matrix is given. It is shown a <jats:italic>hidden<\/jats:italic> freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre\u2013Freud 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\u2013Pi\u00f1eiro multiple orthogonal polynomials of type\u00a0I and type\u00a0II are used as an illustrating case and the corresponding differential relations are\u00a0presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi\u2013Pi\u00f1eiro 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\u2013Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.\n<\/jats:p>","DOI":"10.1007\/s13324-022-00734-1","type":"journal-article","created":{"date-parts":[[2022,10,3]],"date-time":"2022-10-03T22:02:41Z","timestamp":1664834561000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Multiple orthogonal polynomials: Pearson equations and Christoffel formulas"],"prefix":"10.1007","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4685-1583","authenticated-orcid":false,"given":"Am\u00edlcar","family":"Branquinho","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5097-950X","authenticated-orcid":false,"given":"Ana","family":"Foulqui\u00e9-Moreno","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3764-5737","authenticated-orcid":false,"given":"Manuel","family":"Ma\u00f1as","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,3]]},"reference":[{"key":"734_CR1","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1016\/j.aim.2011.03.008","volume":"227","author":"C \u00c1lvarez-Fern\u00e1ndez","year":"2011","unstructured":"\u00c1lvarez-Fern\u00e1ndez, C., Prieto, U.F., Ma\u00f1as, M.: Multiple orthogonal polynomials of mixed type: Gauss\u2013Borel factorization and the multi-component 2D Toda hierarchy. 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