Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is givenpublishe

Generating new classes of orthogonal polynomials

Given a sequence of monic orthogonal polynomials (MOPS), {Pn}, with respect to a quasi-definite linear functional u, we find necessary and sufficient conditions on the parameters an and bn for the sequence Pn(x)+anPn−1(x)+bnPn−2(x),   n≥1P0(x)=1,P−1(x)=0 to be orthogonal. In particular, we can find explicitly the linear functional v such that the new sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite and Tchebychev orthogonal polynomials of second kind are obtained

Quadratic decomposition of bivariate orthogonal polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Backlund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.publishe

Riemann–Hilbert problem and matrix biorthogonal polynomials

Recently the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, has been presented for matrix biorthogonal polynomials, in particular non-Abelian Hermite matrix biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coe cients first order matrix polynomials. We will explore this discussion, present some achievements and consider some new examples of weights for matrix biorthogonal polynomials.publishe

Matrix Jacobi Biorthogonal Polynomials via Riemann-Hilbert problem

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlev\'e d-PIV equations are obtained for the three term recurrence relation coefficients

Matrix Toda and Volterra lattices

We consider matrix Toda and Volterra lattice equations and their relation with matrix biorthogonal polynomials. From that relation, we give a method for constructing a new solution of these systems from another given one. An illustrative example is presented.publishe