Electrostatic interpretation for the zeros of certain polynomials and the Darboux process

摘要

The purpose of this paper is to extend the electrostatic interpretation of Stieltjes to the zeros of a larger class of important polynomials. To keep things simple I consider polynomials that are obtained from those considered by Stieltjes by means of the so-called Darboux process applied to the difference operator featuring in their three term recursion relation. It will turn out that some of the beauty in the results of Stieltjes is preserved in this process. It would be interesting to interpret the Darboux process at the electrostatic level and this paper gives “experimental data” that might help in this direction.