Sequences of Binomial Type with Persistent Roots

We find all sequences of polynomials pn.nG0 with persistent roots i.e., pn x.s cn xyr1. xyr2.??? xyrn..that are of binomial type in Viskov’s generalization of Rota’s umbral calculus to generalized Appell polynomials. We show that such sequences only exist in the classical umbral calculus, the divided difference umbral calculus, and the new ‘‘hyperbolic’’ umbral calculus generated by drdʹx.2. In each of these three umbral calculi, we also find all Sheffer sequences with persistent roots