The Laguerre constellation of classical orthogonal Polynomials

TL;DR

This paper introduces the Laguerre constellation, a set of classical orthogonal polynomial families characterized by specific algebraic identities and properties, along with a theorem that classifies these families within the constellation.

Contribution

It presents the Laguerre constellation of classical orthogonal polynomials and provides new algebraic identities and a classification theorem for these families.

Findings

Derived new structure formulas for Laguerre constellation polynomials.

Established orthogonality properties and Rodrigues formulas.

Provided a classification theorem for classical families in the constellation.

Abstract

A linear functional $\bf u$ is classical if there exist polynomials, $\phi$ and $\psi$, with $\deg \phi\le 2$, $\deg \psi=1$, such that ${\mathscr D}\left(\phi(x) {\bf u}\right)=\psi(x){\bf u}$, where ${\mathscr D}$ is a certain differential, or difference, operator. The polynomials orthogonal with respect to the linear functional ${\bf u}$ are called {\sf classical orthogonal polynomials}. In the theory of orthogonal polynomials, a correct characterization of the classical families is of great interest. In this work, on the one hand, we present the Laguerre constellation, which is formed by all the classical families for which $\deg \phi=1$, obtaining for all of them new algebraic identities such as structure formulas, orthogonality properties as well as new Rodrigues formulas; on the other hand, we present a theorem that characterizes the classical families belonging to the Laguerre constellation.