Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by { 0, 1, 2, 3, ... } in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities
Many such sequences exist. The set of all such sequences forms a
Lie group in a natural way explained below. Every sequence of binomial type is a
Sheffer sequence (but most Sheffer sequences are not of binomial type).
(In the theory of special functions, this same notation denotes upper factorials, but this present usage is universal among combinatorialists.) The product is understood to be 1 if n = 0, since it is in that case an empty product. This polynomial sequence is of binomial type.
 Similarly the "upper factorials"
are a polynomial sequence of binomial type.
are a polynomial sequence of binomial type.
where S(n, k) is the number of partitions of a set of size n into k disjoint nonempty subsets, is a polynomial sequence of binomial type. Eric Temple Bell called these the "exponential polynomials" and that term is also sometimes seen in the literature. The coefficients S(n, k ) are "Stirling numbers of the second kind". This sequence has a curious connection with the Poisson distribution: If X is a random variable with a Poisson distribution with expected value λ then E(X^{n}) = p_{n}(λ). In particular, when λ = 1, we see that the nth moment of the Poisson distribution with expected value 1 is the number of partitions of a set of size n, called the nth Bell number. This fact about the nth moment of that particular Poisson distribution is "Dobinski's formula".
A simple characterization
It can be shown that a polynomial sequence { p_{n}(x) : n = 0, 1, 2, ... } is of binomial type if and only if the linear transformation on the space of polynomials in x that is characterized by

is shiftequivariant and
p_{0}(
x) = 1 for all
x and
p_{n}(0) = 0 for
n > 0. (The statement that this operator is shiftequivariant is the same as saying that the polynomial sequence is a
Sheffer sequence; the set of sequences of binomial type is properly included within the set of Sheffer sequences.)
That linear transformation is clearly a delta operator, i.e., a shiftequivariant 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 (note that the lower bound of summation is 1). 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.
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 { p_{n}(x) : n = 0, 1, 2, 3, ... } and { q_{n}(x) : n = 0, 1, 2, 3, ... } 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 (perhaps surprisingly) formal composition of formal power series.
The sequence κ_{n} of coefficients of the firstdegree 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 (formal) cumulantgenerating 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.
 G.C. Rota, D. Kahaner, and A. Odlyzko, "Finite Operator Calculus," Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.
 R. Mullin and G.C. Rota, "On the Foundations of Combinatorial Theory III: Theory of Binomial Enumeration," in Graph Theory and Its Applications, edited by Bernard Harris, Academic Press, New York, 1970.
As the title suggests, the second of the above is explicit about applications to
combinatorial enumeration.