Umbral “classical” polynomials

We consider umbral “classical” orthogonal polynomials with respect to a generalized derivative operator D which acts on monomials as D x n = μ n x n − 1 with some coefficients μ n . Let P n ( x ) be a set of orthogonal polynomials. Define the new polynomials Q n ( x ) = μ n + 1 − 1 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.