English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we use the Riemann-Hilbert problem, with jumps supported onappropriate curves in the complex plane, for matrix biorthogonal polynomialsand apply it to find Sylvester systems of differential equations for theorthogonal polynomials and its second kind functions as well. For this aim,Sylvester type differential Pearson equations for the matrix of weights areshown to be instrumental. Several applications are given, in order ofincreasing complexity. First, a general discussion of non-Abelian Hermitebiorthogonal polynomials in the real line, understood as those whose matrix ofweights is a solution of a Sylvester type Pearson equation with coefficientsfirst order matrix polynomials, is given. All these is applied to thediscussion of possible scenarios leading to eigenvalue problems for secondorder linear differential operators with matrix eigenvalues. Nonlinear matrixdifference equations are discussed next. Firstly, for the general Hermitesituation a general non linear relation (non trivial because the noncommutativity features of the setting) for the recursion coefficients isgotten.  In the next case of higher difficulty, degree two polynomials are allowed inthe Pearson equation, but the discussion is simplified by considering only aleft Pearson equation. In the case, the support of the measure is on anappropriate branch of an hyperbola. The recursion coefficients are shown tofulfill a non-Abelian extension of the alternate discrete Painlev\'e Iequation. Finally, a discussion is given for the case of degree threepolynomials as coefficients in the left Pearson equation characterizing thematrix of weights. However, for simplicity only odd polynomials are allowed. Inthis case, a new and more general matrix extension of the discrete Painlev\'e Iequation is found.

Matrix Biorthogonal Polynomials: eigenvalue problems and non-Abelian discrete Painlev\'e equations