Binomial Type - Examples

Examples

• In consequence of this definition the binomial theorem can be stated by saying that the sequence { xn : n = 0, 1, 2, ... } 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 polynomials

are a polynomial sequence of binomial type.

• The Touchard polynomials

where S(n, k) is the number of partitions of a set of size n into k 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 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(Xn) = 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".