On differential relations of 2-orthogonal polynomials

TL;DR

This paper investigates differential relations of 2-orthogonal polynomials, introducing a new identity for differential operators and applying it to find polynomial eigenfunctions within this class.

Contribution

It develops a systematic approach using a new differential operator identity to analyze 2-orthogonal polynomial sequences and their eigenfunctions.

Findings

Proves an identity involving differential operators for 2-orthogonal polynomials.

Analyzes a third order differential operator that preserves polynomial degree.

Provides a method to find polynomial eigenfunctions in 2-orthogonal sequences.

Abstract

A generic differential operator on the vectorial space of polynomial functions was presented in a recent work and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Using the techniques therein developed, we prove an identity fulfilled by different differential operators and apply it in a systematic approach to the problem of finding polynomial eigenfunctions, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. In particular, we analyse a third order differential operator that does not increase the degree of polynomials.