Ana Loureiro - Academia.edu (original) (raw)

Papers by Ana Loureiro

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

We discuss polynomials orthogonal with respect to a semi-classical generalized higher-order Freud... more We discuss polynomials orthogonal with respect to a semi-classical generalized higher-order Freud weight ω ( x ; t , λ ) = | x | 2 λ + 1 exp ⁡ ( t x 2 − x 2 m ) , x ∈ R , with parameters λ > − 1 , t ∈ R and m = 2 , 3 , … . The sequence of generalized higher-order Freud weights for m = 2 , 3 , … , 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 generalized hypergeometric 1 F 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 → ∞ . We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.

Studies in Applied Mathematics, 2021

A new set of multiple orthogonal polynomials of both type I and type II with respect to two weigh... more 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 has direct applications in the investigation of singular values of products of Ginibre random matrices 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 indices lie on the step‐line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues‐type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2‐orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third‐...

SIAM Journal on Scientific Computing, 2018

We develop a unified framework for constructing matrix approximations to the convolution operator... more 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. Laguerrebased convolution matrices that approximate Volterra convolution operator defined by functions on [0, ∞] are also discussed for the sake of completeness.

Analysis and Applications, 2019

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose seque... more We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [Formula: see text]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [Formula: see text]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [Formula: see text]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

Mathematics of Computation, 2019

We present a general framework for calculating the Volterra-type convolution of polynomials from ... more We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence {P k (x)} k 0 with deg P k (x) = k. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas.

Journal of Difference Equations and Applications, 2016

The version in the Kent Academic Repository may differ from the final published version. Users ar... more The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.

Mathematische Nachrichten, 2015

We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi-... more We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order qdifferential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q-version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q-version of the Jacobi-Stirling numbers given by Gelineau and the second author.

Analytic Number Theory, Approximation Theory, and Special Functions, 2014

After a slight modification, the Kontorovich-Lebedev transform is an automorphism in the vector s... more After a slight modification, the Kontorovich-Lebedev transform is an automorphism in the vector space of polynomials. The action of this transformation over special cases of Boas-Buck-type polynomial sequences is under analysis.

Studies in Applied Mathematics, 2013

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space... more A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this KL α-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the KL α-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose KL α-transform is a d-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.

Proceedings of the American Mathematical Society, 2013

By means of the Bessel operator a polynomial sequence is constructed to which several properties ... more By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them are its explicit expression, the connection with the Euler numbers, and its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality (with respect to an L 2-inner product), it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed.

Numerical Algorithms, 2012

This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, def... more This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, defined by integral powers of certain first order differential operators in deep connection to the classical polynomials of Hermite, Laguerre, Bessel and Jacobi. This connection is streamed from the canonical element of their dual sequences. Meanwhile new Rodrigues-type formulas for the Hermite and Bessel polynomials are achieved.

Applied Mathematics and Computation, 2011

ABSTRACT In this work a condition on the starting values that guarantees the convergence of the S... more ABSTRACT In this work a condition on the starting values that guarantees the convergence of the Schröder iteration functions of any order to a pth root of a complex number is given. Convergence results are derived from the properties of the Taylor series coefficients of a certain function. The theory is illustrated by some computer generated plots of the basins of attraction.

Journal of Approximation Theory, 2011

A Laguerre polynomial sequence of parameter ε/2 was previously characterized in a recent work [An... more A Laguerre polynomial sequence of parameter ε/2 was previously characterized in a recent work [Ana F. Loureiro and P. Maroni (2008) [28]] as an orthogonal F ε-Appell sequence, where F ε represents a lowering (or annihilating) operator depending on the complex parameter ε ̸ = −2n for any integer n ⩾ 0. Here, we proceed to the quadratic decomposition of an F ε-Appell sequence, and we conclude that the four sequences obtained by this approach are also Appell but with respect to another lowering operator consisting of a Fourth-order linear differential operator G ε,µ , where µ is either 1 or −1. Therefore, we introduce and develop the concept of the G ε,µ-Appell sequences and we prove that they cannot be orthogonal. Finally, the quadratic decomposition of the non-symmetric sequence of Laguerre polynomials (with parameter ε/2) is fully accomplished. c