Using $\D$-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

TL;DR

This paper introduces $\\\ extbf{\D}$-operators as a new tool to generate orthogonal polynomials that are eigenfunctions of higher order difference or differential operators, expanding classical families like Charlier, Meixner, Krawtchouk, Hahn, Laguerre, and Jacobi.

Contribution

It develops the concept of $\D$-operators to construct new orthogonal polynomial families satisfying higher order equations, generalizing classical polynomials.

Findings

Constructed new polynomial sequences $q_n$ as linear combinations of classical polynomials.

Identified conditions on $\beta_n$ for $q_n$ to be eigenfunctions of higher order operators.

Generated orthogonal polynomials with respect to specific measures.

Abstract

We introduce the concept of $\D$-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the concept of $\D$-operator, we determine the structure of the sequence $(\beta_n)_n$ in order that the polynomials $(q_n)_n$ are common eigenfunctions of a higher order difference operator. In addition, we generate sequences $(\beta_n)_n$ for which the polynomials $(q_n)_n$ are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.