The search for differential equations for orthogonal polynomials by using computers

TL;DR

This paper develops a computer-aided method to find spectral differential equations for specific classes of orthogonal polynomials, extending previous work on Jacobi and Sobolev-Laguerre polynomials.

Contribution

It introduces a novel computer algebra approach to identify differential equations for orthogonal polynomials, enhancing analytical techniques in this area.

Findings

Preliminary differential equations for generalized Jacobi polynomials identified.

Differential equations for Sobolev-Laguerre polynomials explored.

Method demonstrates potential for automating spectral equation discovery.

Abstract

We look for spectral type differential equations for the generalized Jacobi polynomials found by T.H. Koornwinder in 1984 and for the Sobolev-Laguerre polynomials. We introduce a method which makes use of computeralgebra packages like Maple and Mathematica and we will give some preliminary results.