A. Foulquié Moreno

Normal indices in Nikishin systems

We improve the class of indices for which normality takes place in a Nikishin system and apply this in Hermite-Pade approximation of such systems of functions.

Orthogonal polynomial interpretation of Δ-Toda equations

In this paper a discretization of Toda equations is analyzed. The correspondence between these Δ-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. It is shown that the spectral measure of these operators evolve in t like where is a given positive Borel measure. The Lax pair for the Δ-Toda equations is derived and characterized in terms of linear functionals, where orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ appear in a natural way. In order to illustrate the results of the paper we work out two examples of Δ-Toda equations related with Jacobi and Laguerre orthogonal polynomials.

Riemann-Hilbert problem associated with Angelesco systems

Angelesco systems of measures with Jacobi-type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szego-type functions. In the procedure, an approach from the Riemann-Hilbert problem plays a fundamental role.

An extension of Markov’s Theorem

We give a general sufficient condition for the uniform convergence of sequences of type II Hermite-Pade approximants associated with Nikishin systems of functions.

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

ABSTRACT In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.