arXiv subjects

Ana Loureiro

Publications and source records attributed to Ana Loureiro.

6 recordsLinked to original sources

arXiv2026

Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials

We investigate the ratio asymptotics and the asymptotic zero distribution of a sequence of polynomials that satisfy a recurrence relation of order $r+1$ with all recurrence coefficients, except the last one, equal to zero. Such a sequence is part of a system of multiple orthogonal polynomials and it satisfies the symmetry property $P_n(\omega_{r+1} z) = \omega_{r+1}^n P_n(z)$, where $\omega_{r+1}$ is the primitive $(r+1)$th root of unity. We consider the unbounded regime in which the recurrence coefficients exhibit algebraic growth and, after division by $n^{\gamma}$ become asymptotically periodic and bounded. After the appropriate scaling, we establish ratio asymptotics and characterize the limiting ratio as the distinguished solution of an algebraic equation. We then determine the limiting zero distribution through its Stieltjes transform and investigate the associated $\St$-transform, which in several cases yields connections with hypergeometric polynomial sequences and distributions arising in free probability. The recurrence is represented by a two-diagonal non-self-adjoint Hessenberg operator, so that the limiting zero measure also admits a natural interpretation as a limiting empirical spectral distribution of its rescaled finite sections. Our analysis is based solely on the positivity and asymptotic behavior of the recurrence coefficients and requires no explicit knowledge of the underlying orthogonality measures.

math.CA↗

arXiv2025

Symmetric Sextic Freud Weight

This paper investigates properties of the sequence of coefficients $(β_n)_{n\geq0}$ in the recurrence relation satisfied by the sequence of monic symmetric polynomials, orthogonal with respect to the symmetric sextic Freud weight $$ω(x;τ, t) = \exp(-x^6 +τx^4 + t x^2), \qquad x \in \mathbb{R},$$ with real parameters $τ$ and $t$. It is known that the recurrence coefficients $β_n$ satisfy a fourth-order nonlinear discrete equation, which is a special case of the second member of the discrete Painlevé I hierarchy, often known as the ''string equation''. The recurrence coefficients have been studied in the context of Hermitian one-matrix models and random symmetric matrix ensembles with researchers in the 1990s observing ''chaotic, pseudo-oscillatory'' behaviour. More recently, this ''chaotic phase'' was described as a dispersive shockwave in a hydrodynamic chain. Our emphasis is a comprehensive study of the behaviour of the recurrence coefficients as the parameters $τ$ and $t$ vary. Extensive computational analysis is carried out, using Maple, for critical parameter ranges, and graphical plots are presented to illustrate the behaviour of the recurrence coefficients as well as the complexity of the associated Volterra lattice hierarchy. The corresponding symmetric sextic Freud polynomials are shown to satisfy a second-order differential equation with rational coefficients. The moments of the weight are examined in detail, including their integral representations, differential equations, and recursive structure. Closed-form expressions for moments are obtained in several special cases in terms of generalised hypergeometric functions and modified Bessel functions. The results highlight the rich algebraic and analytic structures underlying the Freud weight and its connections to integrable systems.

nlin.SI↗

arXiv2023

Generalised higher-order Freud weights

We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},\] with parameters $λ> -1$, $t\in\mathbb{R}$ and $m=2,3,\dots$\ . The sequence of generalised higher order Freud weights for $m=2,3,\dots$, forms a hierarchy of weights, with associated hierarchies for the first moment and the recurrence coefficient. We prove that the first moment can be written as a finite partition sum of generalised hypergeometric $_1F_m$ functions and show that the recurrence coefficients satisfy difference equations which are members of the first discrete Painlevé hierarchy. We analyse the asymptotic behaviour of the recurrence coefficients and the limiting distribution of the zeros as $n \to \infty$. We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.

math.CA↗

arXiv2020

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval $(0,1)$ is studied. This type of polynomials have direct applications in the investigation of singular values of products of Ginibre matrices, in the analysis of rational solutions to Painlevé equations and are connected with branched continued fractions and total positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indexes lie on the step line, for which it is shown that differentiation on the variable gives a shift on the parameters, therefore satisfying Hahn's property. We obtain a Rodrigues-type formula for type I, while a more detailed characterisation is given for the type II polynomials (aka $2$-orthogonal polynomials) which include: an explicit expression as a terminating hypergeometric series, a third-order differential equation, and a third-order recurrence relation. The asymptotic behaviour of their recurrence coefficients mimics those of Jacobi-Piñeiro polynomials, based on which, their zero asymptotic distribution and a Mehler-Heine asymptotic formula near the origin are given. Particular choices on the parameters degenerate in some known systems such as special cases of the Jacobi-Piñeiro polynomials, Jacobi-type $2$-orthogonal polynomials, and components of the cubic decomposition of threefold symmetric Hahn-classical polynomials. Equally considered are confluence relations to other known polynomial sets, such as multiple orthogonal polynomials with respect to Tricomi functions.

math.CA↗

arXiv2020

Multiple orthogonal polynomials associated with confluent hypergeometric functions

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues formula. We characterise the type II polynomials via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the type II polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

math.CA↗

arXiv2018

Spectral approximation of convolution operator

We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on $[-1, 1]$. The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerre-based convolution matrices that approximate Volterra convolution operator defined by functions on $[0, \infty]$ are also discussed for the sake of completeness.

math.NA↗