Differential properties of Jacobi-Sobolev polynomials and electrostatic interpretation

TL;DR

This paper investigates Jacobi-Sobolev orthogonal polynomials, deriving connection formulas, differential equations, and electrostatic interpretations of their zeros, expanding understanding of their properties and applications.

Contribution

It introduces new connection formulas, ladder operators, and electrostatic models for Jacobi-Sobolev polynomials, highlighting their differential properties and zero distributions.

Findings

Derived connection formulas relating Sobolev and Jacobi polynomials

Established ladder differential operators and second-order differential equations

Provided electrostatic interpretations for zeros of Jacobi-Sobolev polynomials

Abstract

We study the sequence of monic polynomials $\{S_n\}_{n\geqslant 0}$, orthogonal with respect to the Jacobi-Sobolev inner {product} \;$$ \langle f,g\rangle_{\mathsf{s}}= \int_{-1}^{1} f(x) g(x)\, d\mu^{\alpha,\beta}(x)+\sum_{j=1}^{N}\sum_{k=0}^{d_j}\lambda_{j,k} f^{(k)}(c_j)g^{(k)}(c_j), $$ \; where $N,d_j \in \ZZ_+$, $\lambda_{j,k}\geqslant 0$, $d\mu^{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta} dx$, $\alpha,\beta>-1$, and $c_j\in\RR\setminus (-1,1)$. A connection formula that relates the Sobolev polynomials $S_n$ with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence $\{S_n\}_{n\geqslant 0}$ and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of $n$ unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.