Electrostatic models for zeros of Laguerre-Sobolev polynomials

TL;DR

This paper develops electrostatic models for the zeros of Laguerre-Sobolev orthogonal polynomials, deriving formulas, ladder operators, and differential equations, and establishing conditions for the zeros' electrostatic interpretation.

Contribution

It introduces a relation between Laguerre-Sobolev and standard Laguerre polynomials, and provides conditions for electrostatic models of their zeros.

Findings

Derived a formula linking Laguerre-Sobolev and Laguerre polynomials.

Established ladder operators and a second-order differential equation for the polynomials.

Provided a sufficient condition for the zeros to have an electrostatic interpretation.

Abstract

Let {$\{S_n\}_{n\geqslant 0}$} be the sequence of orthogonal polynomials with respect to the Laguerre-Sobolev inner product $$ \langle f,g\rangle_S =\!\int_{0}^{+\infty}\! f(x) g(x)x^{\alpha}e^{-x}dx+\sum_{j=1}^{N}\sum_{k=0}^{d_j}\lambda_{j,k} f^{(k)}(c_j)g^{(k)}(c_j), $$ where $\lambda_{j,k}\geqslant 0$, $\alpha >-1$ and $c_i \in (-\infty, 0)$ for $i=1,2,\dots,N$. We provide a formula that relates the Laguerre-Sobolev polynomials $S_n$ to the standard Laguerre orthogonal polynomials. We find the ladder operators for the polynomial sequence $\{S_n\}_{n\geqslant 0}$ and a second-order differential equation with polynomial coefficients for $\{S_n\}_{n\geqslant 0}$. We establish a sufficient condition for an electrostatic model of the zeros of orthogonal Laguerre-Sobolev polynomials. Some examples are given where this condition is either satisfied or not.