Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect towallows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasiorthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems.We also discuss the discrete-to-continuous transition of thesemodels in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.The first author was partially supported by Simons Foundation Collaboration Grants for Mathematicians (grant 710499). He also acknowledges the support of the Spanish Government and the European RegionalDevelopment Fund (ERDF) through grant PID2021-124472NB-I00, Junta deAndalucía (research group FQM-229 and Instituto Interuniversitario Carlos I de Física Teórica y Computacional), and by the University of Almería (Campus de Excelencia Internacional del Mar CEIMAR) in the early stages of this project. The second and third authors were partially supported by Spanish Ministerio de Ciencia, Innovación y Universidades, under grant MTM2015-71352-P. The third author was additionally supported by Junta de Andalucía (research group FQM-384), the University of Granada (Research Project ERDF-UGR A-FQM-246-UGR20), and by the IMAG-Maria de Maeztu grant CEX2020-001105- M/AEI/10.13039/501100011033. Funding for open access publishing: Universidad de Granada/CBU

Some consequences of symmetry in strong Stieltjes distributions

The main purpose of this work is to study a class of strong Stieltjes distributions (t), defined on an interval (a, b) ⊆ (0, ∞), where 0 < < b ≤ ∞ and a = ²/b which satisfy the symmetric property (dψ(t))/t[super]ω=-(dψ(β^2/t))/((β^2/t)[super]ω), tε (a,b), 2ωε We investigate the consequences of this symmetric property on the orthogonal L-polynomials related to distributions ψ(t)and which are the denominators of the two-point Pade approximants for the power series that arise in the moment problem. We examine relations involving the coefficients of the continued fractions that correspond to these power series. We also study the consequences of the symmetry on the associated quadrature formulae

Series Representations and Approximation of some Quantile Functions appearing in Finance

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of a cheap but accurate approximation of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions. As a secondary matter we briefly investigate the problem of approximating the entire quantile function. Indeed with the availability of these new analytic expressions a whole host of possibilities become available. We outline several algorithms and in particular provide a C++ implementation for the variance gamma case. To our knowledge this is the fastest available algorithm of its sort

Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales

El objetivo de esta tesis es dar un algoritmo para decidir si un sistema explicitable de ecuaciones diferenciales kJiferenciales de orden superior sobre las funciones racionales complejas, dado simbólicamente,admite !Soluciones liouvillianas no nulas, calculando una (de laforma dada por un teorema de Singer) en caso !afirmativo. mediante métodos numérico-simbólicos del tipo Introducido por van der Hoeven.donde el uso de álculo numérico no compromete la corrección simbólica. Para ello se Introduce untipo de grupos algebraicos lineales, los grupos euriméricos, y se calcula el cierre eurimérico del grupo de Galois diferencial,mediante una modificación del algoritmo de Derksen y van der Hoeven, dado por los generadores de Ramis.Departamento de Algebra, Análisis Matemático, Geometría y Topologí

An Improved Taylor Algorithm for Computing the Matrix Logarithm

[EN] The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pade approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson-Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.This research was funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad Grant TIN2017-89314-P.Ibáñez González, JJ.; Sastre, J.; Ruíz Martínez, PA.; Alonso Abalos, JM.; Defez Candel, E. (2021). An Improved Taylor Algorithm for Computing the Matrix Logarithm. Mathematics. 9(17):1-19. https://doi.org/10.3390/math9172018S11991

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd