A. Dibucchianico et De. Loeb, SEQUENCES OF BINOMIAL TYPE WITH PERSISTENT ROOTS, Journal of mathematical analysis and applications, 199(1), 1996, pp. 39-58
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.