Author:
Martínez-Finkelshtein Andrei,Orive Ramón,Sánchez-Lara Joaquín
Abstract
AbstractFor 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 to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal 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 these models 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.
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,General Mathematics,Analysis
Reference106 articles.
1. Adler, M., Moser, J.: On a class of polynomials connected with the Korteweg–de Vries equation. Commun. Math. Phys. 61(1), 1–30 (1978)
2. Angelesco, M.A.: Sur deux extensions des fractions continues algébriques. C. R. Acad. Sci. Paris 18, 262–263 (1919)
3. Appell, P.: Sur une suite des polynômes ayant toutes leurs racines réelles. Arch. Math. Phys. 1(3), 69–71 (1901)
4. Aptekarev, A.I.: Strong asymptotics of polynomials of simultaneous orthogonality for Nikishin systems. Mat. Sb. 190(5), 3–44 (1999)
5. Aptekarev, A.I.: Asymptotics of Hermite–Padé approximants for a pair of functions with branch points. Dokl. Akad. Nauk 422(4), 443–445 (2008). English transl. in Dokl. Math. 78:2, 717–719 (2008)