The set of all such sequences forms a Lie group under the operation of umbral composition, explained below.
Every sequence of binomial type may be expressed in terms of the Bell polynomials.
Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus.
It can be shown that every delta operator can be written as a power series of the form where D is differentiation (note that the lower bound of summation is 1).
Therefore, this paragraph amounts to a recipe for generating as many polynomial sequences of binomial type as one may wish.
Note that for each n ≥ 1, Here is the main result of this section: Theorem: All polynomial sequences of binomial type are of this form.
A result in Mullin and Rota, repeated in Rota, Kahaner, and Odlyzko (see References below) states that every polynomial sequence { pn(x) }n of binomial type is determined by the sequence { pn′(0) }n, but those sources do not mention Bell polynomials.
Polynomial sequences of binomial type are precisely those whose generating functions are formal (not necessarily convergent) power series of the form where f(t) is a formal power series whose constant term is zero and whose first-degree term is not zero.
[2] It can be shown by the use of the power-series version of Faà di Bruno's formula that The delta operator of the sequence is the compositional inverse
With the delta operator defined by a power series in D as above, the natural bijection between delta operators and polynomial sequences of binomial type, also defined above, is a group isomorphism, in which the group operation on power series is formal composition of formal power series.
Then is the delta operator associated with the polynomial sequence, i.e., we have The concept of binomial type has applications in combinatorics, probability, statistics, and a variety of other fields.