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 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.

Abstract. Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner productwhere E is a rectifiabl Jordan curve or arc in the complex planeA is an M × M Hermitian matrix, Ml 1 + · · · + l m + m, |dξ | denotes the arc length measure, ρ is a nonnegative function on E, and z i ∈ , i = 1, 2, . . . , m, where is the exterior region to E.

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In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if ( 0 , 1 ) is a coherent pair of measures on the unit circle, then 0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with 1 is a specific rational transformation of the linear functional corresponding to 0 . Some examples are given.

Laguerre-Hahn families on the real line are characterized in terms of second order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.