That linear transformation is clearly a delta operator, i.e., a shift-equivariant linear transformation on the space of polynomials in x that reduces degrees of polynomials by 1. The most obvious examples of delta operators are difference operators and differentiation. It can be shown that every delta operator can be written as a power series of the form where D is differentiation. Each delta operator Q has a unique sequence of "basic polynomials", i.e., a polynomial sequence satisfying It was shown in 1973 by Rota, Kahaner, and Odlyzko, that a polynomial sequence is of binomial type if and only if it is the sequence of basic polynomials of some delta operator. Therefore, this paragraph amounts to a recipe for generating as many polynomial sequences of binomial type as one may wish.
Characterization by Bell polynomials
For any sequence a1, a2, a3,... of scalars, let where Bn,k is the Bell polynomial. Then this polynomial sequence is of binomial type. 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 states that every polynomial sequence n of binomial type is determined by the sequence n, but those sources do not mention Bell polynomials. This sequence of scalars is also related to the delta operator. Let Then is the delta operator of this sequence.
For sequences an, bn, n = 0, 1, 2,..., define a sort of convolution by Let be the nth term of the sequence Then for any sequence ai, i = 0, 1, 2,..., with a0 = 0, the sequence defined by p0 = 1 and for n ≥ 1, is of binomial type, and every sequence of binomial type is of this form.
Polynomial sequences of binomial type are precisely those whose generating functions are formal power series of the form where f is a formal power series whose constant term is zero and whose first-degree term is not zero. 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 f−1, so that
A way to think about these generating functions
The coefficients in the product of two formal power series and are . If we think of x as a parameter indexing a family of such power series, then the binomial identity says in effect that the power series indexed by x + y is the product of those indexed by x and by y. Thus the x is the argument to a function that maps sums to products: an exponential function where f has the form given above.
The set of all polynomial sequences of binomial type is a group in which the group operation is "umbral composition" of polynomial sequences. That operation is defined as follows. Suppose and are polynomial sequences, and Then the umbral composition p o q is the polynomial sequence whose nth term is . 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.
The sequence κn of coefficients of the first-degree terms in a polynomial sequence of binomial type may be termed the cumulants of the polynomial sequence. It can be shown that the whole polynomial sequence of binomial type is determined by its cumulants, in a way discussed in the article titled cumulant. Thus and These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution. Let be the cumulant-generating function. Then is the delta operator associated with the polynomial sequence, i.e., we have
Applications
The concept of binomial type has applications in combinatorics, probability, statistics, and a variety of other fields.
