Table of contents |
2 Examples 3 A simple characterization 4 Delta operators 5 Umbral composition of polynomial sequences 6 Cumulants and moments 7 Applications 8 References |
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
Definition
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).
It can be shown that a polynomial sequence { pn(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
Examples
A simple characterization
is shift-equivariant and p0(x) = 1 for all x and pn(0) = 0 for n > 0. (The statement that this operator is shift-equivariant 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 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
Delta operators
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 { pn(x) : n = 0, 1, 2, 3, ... } and { qn(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, and
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
Let
The concept of binomial type has applications in combinatorics, probability, statistics, and a variety of other fields.
Umbral composition of polynomial sequences
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.Cumulants and moments
and
These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution.
be the (formal) cumulant-generating function. Then
is the delta operator associated with the polynomial sequence, i.e., we haveApplications
References
As the title suggests, the second of the above is explicit about applications to combinatorial enumeration.