Umbral "classical" polynomials

Abstract

We study the umbral "classical" orthogonal polynomials with respect to a generalized derivative operator $\mathcal{D}$ which acts on monomials as $\mathcal{D}{x}^n={\mu }_n{x}^{n-1}$ with some coefficients ${\mu }_n$. Let $P_n(x)$ be a set of orthogonal polynomials. Define the new polynomials $Q_n(x)={\mu }_{n+1}^{-1}\mathcal{D}{P}_{n+1}(x)$. We find necessary and sufficient conditions when the polynomials $Q_n(x)$ will also be orthogonal. Apart from well known examples of the classical orthogonal polynomials we present a new example of umbral classical polynomials expressed in terms of elliptic functions.

24 pages