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-Pi\~neiro 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-Pi\~neiro 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.
