The higher-order differential operator for the generalized Jacobi polynomials - new representation and symmetry

TL;DR

This paper introduces a new representation of the higher-order differential operator for generalized Jacobi polynomials, demonstrating its symmetry and orthogonality properties, which enhances understanding of their eigenfunctions.

Contribution

It provides a novel, accessible form of the differential operator for generalized Jacobi polynomials and proves its symmetry and orthogonality, advancing theoretical understanding.

Findings

Differential operator expressed as a combination of four elementary components

Operator shown to be symmetric with respect to the scalar product

Orthogonality of eigenfunctions verified

Abstract

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder's generalized Jacobi polynomials with four parameters $\alpha,\beta\in\mathbb{N}_{0}$ and $M,N \ge 0$ determining the orthogonality measure on the interval $-1 \le x \le 1$. The corresponding differential equation of order $2\alpha+2\beta+6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely accessible for applications. In particular, we show that this operator is symmetric with respect to the underlying scalar product and thus verify the orthogonality of the eigenfunctions.