Binomial type

In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers formula_1 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 under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus.
Characterization by delta operators.
It can be shown that a polynomial sequence { "p""n"(x) : "n" = 0, 1, 2, … } is of binomial type if and only if all three of the following conditions hold:
Delta operators.
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 (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.
Characterization by Bell polynomials.
For any sequence "a"1, "a"2, "a"3, … of scalars, let
where "B""n","k"("a"1, …, "a""n"−"k"+1) 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 (see "References" below) states that every polynomial sequence { "p""n"("x") }"n" of binomial type is determined by the sequence { "p""n"′(0) }"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.
Characterization by a convolution identity.
For sequences "a""n", "b""n", "n" = 0, 1, 2, …, define a sort of convolution by
Let formula_17 be the "n"th term of the sequence
Then for any sequence "a""i", "i" = 0, 1, 2, ..., with "a"0 = 0, the sequence defined by "p"0("x") = 1 and
for "n" ≥ 1, is of binomial type, and every sequence of binomial type is of this form.
Characterization by generating functions.
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. 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("D"), so that
A way to think about these generating functions.
The coefficients in the product of two formal power series
and
are
(see also Cauchy product). 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"("t") has the form given above.
Umbral composition of polynomial sequences.
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 "n"th term is
(the subscript "n" appears in "p""n", since this is the "n" term of that sequence, but not in "q", since this refers to the sequence as a whole rather than one of its terms).
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.
Cumulants and moments.
The sequence κ"n" 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 (formal) 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.
References.
As the title suggests, the second of the above is explicitly about applications to combinatorial enumeration.