Umbral "classical" polynomials

Abstract

We study the umbral "classical" orthogonal polynomials with respect to a generalized derivative operator $\cal D$ which acts on monomials as ${\cal D} x^n = \mu_n x^{n-1}$ with some coefficients $\mu_n$. Let $P_n(x)$ be a set of orthogonal polynomials. Define the new polynomials $Q_n(x) =\mu_{n+1}^{-1}{\cal D} P_{n+1}(x)$. 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.