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Concept
Binomial type
Summary
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers 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.
In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type.
The sequence of "lower factorials" is defined by(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.
The Abel polynomialsare a polynomial sequence of binomial type.
The Touchard polynomialswhere is the number of partitions of a set of size into disjoint non-empty 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 are "Stirling numbers of the second kind". This sequence has a curious connection with the Poisson distribution: If is a random variable with a Poisson distribution with expected value then . In particular, when , we see that the th moment of the Poisson distribution with expected value is the number of partitions of a set of size , called the th Bell number. This fact about the th moment of that particular Poisson distribution is "Dobinski's formula".
It can be shown that a polynomial sequence { pn(x) : n = 0, 1, 2, ...
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