The Laguerre constellation of classical orthogonal Polynomials

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.