Differential equations for discrete Jacobi-Sobolev orthogonal polynomials

TL;DR

This paper investigates the differential properties of discrete Jacobi-Sobolev orthogonal polynomials, constructing them via Casorati determinants and proving their eigenfunction status for certain parameter values.

Contribution

It explicitly constructs the orthogonal polynomials and derives the differential operators they satisfy when Jacobi parameters are nonnegative integers.

Findings

Orthogonal polynomials are eigenfunctions of finite order differential operators.

Explicit construction of these polynomials using Casorati determinants.

Order of the differential operator is computed based on the bilinear form matrices.

Abstract

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters $\alpha$ and $\beta$ are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrices which define the discrete Jacobi-Sobolev bilinear form.