Dr Ana Loureiro

Senior Lecturer in Mathematics

Ana joined Kent in 2012, after holding a post-doctoral position at the University of Porto in Portugal.She works in the field of orthogonal polynomials and special functions, and her interests include aspects of analytical and algebraic combinatorics as well as integrable systems. Ana’s work also covers applications to spectral methods and numerical linear algebra. Currently Ana is a member of the international Steering Committee for Orthogonal Polynomials and Special Functions.Ana serves on the School's Education Committee.

Research interests

orthogonal polynomials and special functionsalgebraic combinatoricsintegrable systemsapplications to numerical analysis and computational methods

Supervision

Ana is supervising Helder Lima's PhD research

Publications

Article

• Loureiro, A. and Xu, K. (2019). Volterra-type convolution of classical polynomials. Mathematics of Computation [Online]. Available at: https://doi.org/10.1090/mcom/3427.
We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence {Pk(x)}k?0 with degPk(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.
• Xu, K. and Loureiro, A. (2018). Spectral approximation of convolution operator. SIAM Journal on Scientific Computing [Online] 40:A2336-A2355. Available at: https://epubs.siam.org/doi/abs/10.1137/17M1149249.
We develop a unified framework for constructing matrix approximations for 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.
• Loureiro, A. and Zeng, J. (2016). q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers. q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers [Online] 289:693-717. Available at: http://onlinelibrary.wiley.com/wol1/doi/10.1002/mana.201400381/abstract.
We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobiâ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 et al. and by the first author, whilst the combinatorics of this new set of numbers is a
q-version of the JacobiâStirling numbers given by Gelineau and the second author.
• Clarkson, P., Loureiro, A. and Van Assche, W. (2016). Unique positive solution for an alternative discrete PainlevÃ© I equation. Journal of Difference Equations and Applications [Online]. Available at: http://www.tandfonline.com/doi/abs/10.1080/10236198.2015.1127917.
We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second PainlevÃ© equation and the alternative discrete PainlevÃ© I equation.
• Loureiro, A., Maroni, P. and Yakubovich, S. (2014). On a polynomial sequence associated with the Bessel operator. Proceedings of the American Mathematical Society [Online] 142:467-482. Available at: http://dx.doi.org/10.1090/S0002-9939-2013-11658-8.
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.
• Loureiro, A. and Yakubovich, S. (2013). Central factorials under the KontorovichâLebedev transform of polynomials. Integral Transforms and Special Functions [Online] 24. Available at: http://dx.doi.org/10.1080/10652469.2012.672325.
In this paper, we show that slight modifications of the KontorovichâLebedev (KL) 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.
• Loureiro, A. and Yakubovich, S. (2013). The Kontorovich-Lebedev transform as a map between d-orthogonal polynomials. Studies in Applied Mathematics [Online] 131:229-265. Available at: http://dx.doi.org/10.1111/sapm.12009.
A slight modification of the KontorovichâLebedev transform is an auto-morphism on the vector space of polynomials. The action of this inline image-transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the inline image-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose inline image-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.
• Loureiro, A. and Maroni, P. (2012). Polynomial sequences associated with the classical linear functionals. Numerical Algorithms [Online] 60:297-314. Available at: http://dx.doi.org/10.1007/s11075-012-9573-y.
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.
• Cardoso, J. and Loureiro, A. (2011). Iteration functions for pth roots of complex numbers. Numerical Algorithms [Online] 57:329-356. Available at: http://dx.doi.org/10.1007/s11075-010-9431-8.
A novel way of generating higher-order iteration functions for the computation of pth roots of complex numbers is the main contribution of the present work. The behavior of some of these iteration functions will be analyzed and the conditions on the starting values that guarantee the convergence will be stated. The illustration of the basins of attractions of the pth roots will be carried out by some computer generated plots. In order to compare the performance of the iterations some numerical examples will be considered.
• Loureiro, A. and Maroni, P. (2011). Around q-Appell polynomial sequences. Ramanujan Journal [Online] 26:311-321. Available at: http://dx.doi.org/10.1007/s11139-011-9336-8.
First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator Mq;q-eq;q?, where ? represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of MÃq;q-e/2q;q?2 (the MÃq;q-e/2q;q?2 -Appell), only the Wall q-polynomials with parameter q ?/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.
• Loureiro, A. and Maroni, P. (2011). Quadratic decomposition of Laguerre polynomials via lowering operators. Journal of Approximation Theory [Online] 163:888-903. Available at: http://dx.doi.org/10.1016/j.jat.2010.07.009.
A Laguerre polynomial sequence of parameter ?/2 was previously characterized in a recent work [Ana F. Loureiro and P. Maroni (2008) [28]] as an orthogonal F?-Appell sequence, where F? represents a lowering (or annihilating) operator depending on the complex parameter ???2n for any integer n?0. Here, we proceed to the quadratic decomposition of an F?-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?,?, where ? is either 1 or ?1. Therefore, we introduce and develop the concept of the G?,?-Appell sequences and we prove that they cannot be orthogonal. Finally, the quadratic decomposition of the non-symmetric sequence of Laguerre polynomials (with parameter ?/2) is fully accomplished.
• Loureiro, A. and Cardoso, J. (2011). On the convergence of SchrÃ¶der iteration functions for pth roots of complex numbers. Applied Mathematics and Computation [Online] 217:8833-8839. Available at: http://dx.doi.org/10.1016/j.amc.2011.03.047.
In this work a condition on the starting values that guarantees the convergence of the SchrÃ¶der 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.
• Loureiro, A. (2010). New results on the Bochner condition about classical orthogonal polynomials. Journal of Mathematical Analysis and Applications [Online] 364:307-323. Available at: http://dx.doi.org/10.1016/j.jmaa.2009.12.003.
The classical polynomials (Hermite, Laguerre, Bessel and Jacobi) are the only orthogonal polynomial sequences (OPS) whose elements are eigenfunctions of the Bochner second-order differential operator F (Bochner, 1929 [3]). In Loureiro, Maroni and da Rocha (2006) [18] these polynomials were described as eigenfunctions of an even order differential operator Fk with polynomial coefficients defined by a recursive relation. Here, an explicit expression of Fk for any positive integer k is given. The main aim of this work is to explicitly establish sums relating any power of F with Fk, k?1, in other words, to bring a pair of inverse relations between these two operators. This goal is accomplished with the introduction of a new sequence of numbers: the so-called A-modified Stirling numbers, which could be also called as Bessel or JacobiâStirling numbers, depending on the context and the values of the complex parameter A.
• Loureiro, A. and Maroni, P. (2008). Quadratic decomposition of Appell sequences. Expositiones Mathematicae [Online] 26:177-186. Available at: http://dx.doi.org/10.1016/j.exmath.2007.10.002.
We proceed to the quadratic decomposition of Appell sequences and we characterise the four derived sequences obtained by this approach. We prove that the two monic polynomial sequences associated to such quadratic decomposition are also Appell sequences with respect to another (lowering) operator, which we call as F?, where either ?=1 or -1. Thus, we introduce and develop the concept of the Appell polynomial sequences with respect to the operator F? (where, ? is a parameter belonging to the field of the complex numbers): the F?-Appell sequences. The orthogonal polynomial sequences that are also F?-Appell correspond to the Laguerre sequences with parameter ?/2. Indeed, this brings an entirely new characterisation of the Laguerre sequences.
• Loureiro, A., Maroni, P. and Rocha, Z. (2006). The generalised Bochner condition about classical orthogonal polynomials revisited. Journal of Mathematical Analysis and Applications [Online] 322:645-667. Available at: http://dx.doi.org/10.1016/j.jmaa.2005.09.026.
We bring a new proof for showing that an orthogonal polynomial sequence is classical if and only if any of its polynomial fulfils a certain differential equation of order 2k, for some k?1. So, we build those differential equations explicitly. If k=1, we get the Bochner's characterization of classical polynomials. With help of the formal computations made in Mathematica, we explicitly give those differential equations for k=1,2 and 3 for each family of the classical polynomials. Higher order differential equations can be obtained similarly.

Book section

• Loureiro, A. and Yakubovich, S. (2014). On especial cases of Boas-Buck type polynomial sequences. in: Milovanovi?, G. and Rassias, M. eds. Analytic Number Theory, Approximation Theory, and Special Functions. Springer New York, pp. 705-720. Available at: http://dx.doi.org/10.1007/978-1-4939-0258-3_26.
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.

Edited journal

• Loureiro, A.F. ed. (2012). Opuscula Mathematica - a special journal issue. Opuscula Mathematica [Online] 32. Available at: http://www.opuscula.agh.edu.pl/om-vol32iss4.

Thesis

• Loureiro, A. (2008). Hahn's generalized problem and corresponding Appell sequences.
This thesis is devoted to some aspects of the theory of orthogonal polynomials, paying a special attention to the classical ones (Hermite, Laguerre, Bessel and Jacobi). The elements of a classical sequence are eigenfunctions of a second order linear differential operator with polynomial coefficients $\mathcal{L}$ known as the Bochner's operator. In an algebraic manner, a classical sequence is also caracterised through the so-called Hahn's property, which states that an orthogonal polynomial sequence is classical if and only if the sequence of its (normalised) derivatives is also orthogonal.

In the present work we show that an orthogonal polynomial sequence is classical if and only if any of its polynomials fulfils a certain differential equation of order $2k$, for some positive integer $k$. We thoroughly reveal the structure of such differential equation and, for each classical family, we explicitly present the corresponding $2k$-order differential operator $\mathcal{L}_{k}$. When we consider $k=1$, we recover the Bochner's differential operator: $\mathcal{L}_{1} = \mathcal{L}$. On the other hand, as a consequence of Bochner's result, any element of a classical sequence must be an eigenfunction of a polynomial with constant coefficients in powers of $\mathcal{L}$. As a result of the introduction of the so-called $A$-modified Stirling numbers (where $A$ indicates a complex parameter), we are able to establish inverse relations between the powers of the Bochner operator $\mathcal{L}$ and $\mathcal{L}_{k}$.

Afterwards, we proceed to the quadratic decomposition of an Appell sequence. The four polynomial sequences obtained by this approach are also Appell sequences but with respect to another lowering differential operator, denoted $\mathcal{F}_{\varepsilon}$, where $\varepsilon$ is either 1 or -1. Thus, we introduce and develop the concept of Appell sequences with respect to the operator $\mathcal{F}_{\varepsilon}$ (where, more generally, $\varepsilon$ denotes a complex parameter): the $\mathcal{F}_{\varepsilon}$-Appell sequences. Subsequently, we seek to find all the orthogonal polynomial sequences that are also $\mathcal{F}_{\varepsilon}$-Appell, which are, indeed, the $\mathcal{F}_{\varepsilon}$-Appell sequences that satisfy Hahn's property respect to $\mathcal{F}_{\varepsilon}$. This latter consists of the Laguerre sequences of parameter $\varepsilon/2$, up to a linear change of variable. Inspired by this problem, we characterise all the $\mathcal{F}_{\varepsilon}$-classical sequences.
While ferreting out the all $\mathcal{F}_{\varepsilon}$-classical sequences, apart from the Laguerre sequence, we find certain Jacobi sequences (with two parameters).
The quadratic decomposition of Appell sequences with respect to other lowering operators is also considered and the results obtained are akin to the aforementioned ones attained in the analogous problem.

Forthcoming

• Clarkson, P. et al. eds. (2018). Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) - Special Issue on Orthogonal Polynomials, Special Functions and Applications. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) - Special Issue on Orthogonal Polynomials, Special Functions and Applications [Online]. Available at: https://www.emis.de/journals/SIGMA/OPSFA2017.html.
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