Asymptotic zero distribution of Jacobi-Pi\~neiro and multiple Laguerre polynomials

We give the asymptotic distribution of the zeros of Jacobi-Pi\~neiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution.Comment: 19 pages, 2 figures. Some minor corrections and four new references adde

An Algebraic Model for the Multiple Meixner Polynomials of the First Kind

An interpretation of the multiple Meixner polynomials of the first kind is provided through an infinite Lie algebra realized in terms of the creation and annihilation operators of a set of independent oscillators. The model is used to derive properties of these orthogonal polynomials

Average Characteristic Polynomials of Determinantal Point Processes

We investigate the average characteristic polynomial $\mathbb E\big[\prod_{i=1}^N(z-x_i)\big]$ where the $x_i$'s are real random variables which form a determinantal point process associated to a bounded projection operator. For a subclass of point processes, which contains Orthogonal Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as $N$ goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from a theorem of Kuijlaars and Van Assche a unified way to describe the almost sure convergence for classical Orthogonal Polynomial Ensembles. As another application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures.Comment: 26 page

Discrete integrable systems generated by Hermite-Pad\'e approximants

We consider Hermite-Pad\'e approximants in the framework of discrete integrable systems defined on the lattice $\mathbb{Z}^2$. We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Pad\'e approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page

On the q-Charlier Multiple Orthogonal Polynomials

We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.The research of J. Arves u was partially supported by the research grant MTM2012-36732-C03-01 (Ministerio de Econom a y Competitividad) of Spain

On 2D discrete Schr\"odinger operators associated with multiple orthogonal polynomials

A class of cross-shaped difference operators on a two dimensional lattice is introduced. The main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials. In other words, this scheme generalizes the classical connection between Jacobi matrices and orthogonal polynomials to the case of operators on lattices. Furthermore we also show how to obtain 2D discrete Schr\"odinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials.Comment: 15 page

Laguerre-Angelesco multiple orthogonal polynomials on an rr-star

We investigate the type I and type II multiple orthogonal polynomials on an $r$-star with weight function $|x|^{\beta}e^{-x^r}$, with $\beta>-1$. Each measure $\mu_j$, for $1\leq j \leq r$, is supported on the semi-infinite interval $[0,\omega^{j-1}\infty)$ with $\omega=e^{2\pi i/r}$. For both the type I and the type II polynomials we give explicit expressions, the coefficients in the recurrence relation, the differential equation and we obtain the asymptotic zero distribution of the polynomials on the diagonal. Also, we give the connection between the Laguerre-Angelesco polynomials and the Jacobi-Angelesco polynomials on an $r$-star.Comment: 33 pages, 4 figures. arXiv admin note: text overlap with arXiv:1804.0751

The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

We show that multiple orthogonal polynomials for r measures $(\mu_1,...,\mu_r)$ satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices $\vec{n}\pm \vec{e}_j$, where $\vec{e}_j$ are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of measures $\mu_j$. We show how the Christoffel-Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials.Comment: 22 page