On a class of equilibrium problems in the real axis

Authors:
Ramón Orive;Zebenzuí García
Affiliations:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Canary Islands, Spain;Departamento de Economía Aplicada, Universidad de La Laguna, 38071 La Laguna, Tenerife, Canary Islands, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

Citing 8
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Abstract

In a series of seminal papers, Thomas J. Stieltjes (1856-1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lame equations), the so-called Heine-Stieltjes polynomials. In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x"1,...,x"n@?R are in presence of a finite family of ''attractors'' (i.e., negative charges) z"1,...,z"m@?C@?R, is considered and its connection with certain class of Lame-type equations is shown. In addition, we study the situation when both n-~ and m-~, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.