A Characterization of Exponential and Ordinary Generating Functions

It is common to represent a sequence a=(a0, a1, …) of complex numbers with a generating function. G.C. Rota once remarked that among all the possible generating functions that might be used to represent a, the ordinary and exponential generating functions are the most ubiquitous. It is unclear what, if anything, makes these two particular representations special. We show here that the ordinary and exponential representations uniquely possess the property that the determinants of the Hankel matrices of certain convolutional polynomials in a are independent of a1. Hankel matrices are closely associated with the problem of moments and the problem of moment preserving maps and hence the independence of a1 has some curious implications. For example determining if a is a sequence of cumulants for some distribution is necessarily independent of the value for the mean a1. We explore this and other applications of the generating function characterization property within probability and combinatorics.