arXiv subjects

Ana F. Loureiro

Publications and source records attributed to Ana F. Loureiro.

7 recordsLinked to original sources

arXiv2019

Threefold symmetric Hahn-classical multiple orthogonal polynomials

We characterize all the multiple orthogonal threefold 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 $2$-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 $3$-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them $2$-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

math.CA↗

arXiv2018

Volterra-type convolution of classical polynomials

We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $\{P_k(x)\}_{k \geqslant 0}$ with $°P_k(x) = k$. 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.

math.CA↗

arXiv2015

Unique positive solution for an alternative discrete Painlevé I equation

We show that the alternative discrete Painlevé I equation (alt-dP$_{\rm I}$) has a unique solution which remains positive for all $n \geq 0$. Furthermore, we identify this positive solution in terms of a special solution of the second Painlevé equation (P$_{\rm II}$) involving the Airy function $\mathop{\rm Ai}(t)$. The special-function solutions of P$_{\rm II}$ involving only the Airy function $\mathop{\rm Ai}(t)$ therefore have the property that they remain positive for all $n\geq 0$ and all $t \geq 0$, which is a new characterization of these special solutions of P$_{\rm II}$ and alt-dP$_{\rm I}$.

nlin.SI↗

arXiv2015

q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers

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 $q$-differential 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 {\it 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.

math.CA↗

arXiv2012

The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials

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.

math.CA↗

arXiv2012

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.

math.CA↗

arXiv2011

On a nonorthogonal polynomial sequence associated with Bessel operator

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them, its explicit expression, the connection with the Euler numbers, its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality, 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.

math.CA↗