Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising shift operator

Abstract

We introduce the notion of $$\mathcal {R}_{\mu }$$ -classical orthogonal polynomials, where $$\mathcal {R}_{\mu }$$ is the degree raising shift operator for the sequence of Laguerre polynomials of parameter $$\mu $$ . Then we show that the Laguerre polynomials $$L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0$$ , are the only $$\mathcal {R}_{\mu }$$ -classical orthogonal polynomials.