Asymptotic Properties of Heine-Stieltjes and Van Vleck Polynomials (original) (raw)
Related papers
2012
Investigation of asymptotics of the spectrum for all kinds of spectral problems is a major ingredient in numerous articles in pure and applied mathematics, mathematical and theoretical physics as well as many other areas of natural sciences. At the same time results on the asymptotic behavior of the corresponding sequences of eigenfunctions are incomparably fewer and scattered sporadically in the literature starting with the pioneering Ph.D. thesis of J.D.Birkhoff from 1913. In several recent papers I initiated a further development of this field and discovered its extremely rich connections and plausible applications to several classical branches of mathematics. Goal: The main purpose of this proposal is to pursue a systematic investigation of the asymptotic distributions of the zero loci of solutions and eigenfunctions to linear ODEs with polynomial coefficients depending on parameter(s).
Asymptotics of polynomial solutions of a class of generalized Lamé differential equations
2005
The research of A.M.F. and P.M.G. was partially supported by the Ministry of Science and Technology (MCYT) of Spain through the grant BFM2001-3878-C02-02, and by Junta de Andalucia through Grupo de Investigacion FQM 0229. A.M.F. acknowledges also the support of the European Research Network on Constructive Complex Approximation (NeCCA), INTAS 03-51-6637, and of NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications and Generalizations,” ref. PST.CLG.979738. The research of R.O. was partially supported by grants from Spanish MCYT (Research Project BFM2001-3411) and Gobierno Autonomo de Canarias (Research Project PI2002/136).
Electrostatic models for zeros of polynomials
Journal of Computational and Applied Mathematics, 2007
We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.
Two Approaches to the Asymptotics of the Zeros of a Class of Hypergeometric-Type Polynomials
Russian Academy of Sciences. Sbornik Mathematics, 1995
A class of hypergeometric-type differential equations is considered. It is shown that its polynomial solutions y n exhibit an orthogonality with respect to a "varying measure" (a sequence of measures) on R. From this relation the asymptotic distribution of zeros is obtained by means of a potential theory approach. Moreover, the WKB or semiclassical approximation is used to construct an asymptotically exact sequence of absolutely continuous measures that approximate the zero distribution of y n • Bibliography: 20 titles.
Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
Communications in Mathematical Physics, 2011
We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest.
Asymptotic zero distribution of biorthogonal polynomials
Journal of Approximation Theory, 2014
Let ψ : [0, 1] → R be a strictly increasing continuous function. Let P n be a polynomial of degree n determined by the biorthogonality conditions 1 0 P n (x) ψ (x) j d x = 0, j = 0, 1, . . . , n − 1.
An electrostatics model for zeros of general orthogonal polynomials
Pacific Journal of Mathematics, 2000
We prove that the zeros of general orthogonal polynomials, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an external field determined by the weight function. We compute the total energy of the system in terms of the recursion coefficients of the orthonormal polynomials and study its limiting behavior as the number of particles tends to infinity in the case of Freud exponential weights.
Orthogonal Polynomials, Asymptotics and Heun Equations
The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors", usually dependent on a "time variable" t. From ladder operators [12-14, 30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials P n (x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by P n (x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations (see, for instance, [1, 23, 36]). In this paper we look at three type of weights: the Jacobi type, the Laguerre type and the weights deformed by the indicator function of (a, b) χ (a,b) and the step function θ(x). In particular, we consider the following Jacobi type weights: 1.