Umbral “classical” polynomials

Abstract

We consider umbral “classical” orthogonal polynomials with respect to a generalized derivative operator D which acts on monomials as Dxn=μnxn−1 with some coefficients μn. Let Pn(x) be a set of orthogonal polynomials. Define the new polynomials Qn(x)=μn+1−1DPn+1(x). We find necessary and sufficient conditions when the polynomials Qn(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.