Polynomials of binomial type and compound Poisson processes (Q1101762)

The theory of polynomials of binomial type, i.e. of polynomials \((q_ n)_ 0\) satisfying \(q_ n(x+y)=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)q_ k(x)q_{n-k}(y)\), was developed by \textit{G.-C. Rota}, \textit{D. Kahaner} and \textit{A. Odlyzko} [cf. J. Math. Anal. Appl. 42, 684-760 (1973; Zbl 0267.05004)]. Following a suggestion by Rota et al. the author studies \((q_ n)\) and the associated Sheffer sets \((s_ n)\) in terms of an integer-valued compound Poisson process \(\{Y_ t\); \(t>0\}\). He is especially interested in the asymptotic behaviour for \(n\to \infty\) of the probability generating function \(q_ n(x)/q_ n(1)\) and \(s_ n(x)/s_ n(1).\) Partial results are obtained under conditions on the radius of convergence of a power series related to \((q_ n)\). The problems are difficult and involve a complicated system of notations. The problem has ramifications into several other areas of analysis and probability: Lagrange expansions, renewal theory, subexponential distributions and infinite divisibility. It is yet not very clear where the investigation will lead to.