SEQUENCES OF BINOMIAL TYPE WITH PERSISTENT ROOTS

We find all sequences of polynomials (p(n))(n greater than or equal to 0) with persistent roots (i.e., p(n)(x) = C-n(x - r(1))(x - r(2))...( x - r(n))) that are of binomial type in Viskov's generalization of Rot a's umbral calculus to generalized Appell polynomials. We show that su ch sequences only exist in the classical umbral calculus, the divided difference umbral calculus, and the new ''hyperbolic'' umbral calculus generated by (d/d root x)(2). In each of these three umbral calculi, we also find all Sheffer sequences with persistent roots. (C) 1996 Aca demic Press, Inc.