Binomial type

:For other concepts using the name "binomial", see binomial (disambiguation).

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

::(x)_n=x(x-1)(x-2)cdotcdotscdot(x-n+1).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:(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.

Related Topics:
Upper factorial - Combinatorialists - Empty product

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Similarly the "upper factorials"

::x^{(n)}=x(x+1)(x+2)cdotcdotscdot(x+n-1)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:are a polynomial sequence of binomial type.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• The Abel polynomials

::p_n(x)=x(x-an)^{n-1} ,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:are a polynomial sequence of binomial type.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• The Touchard polynomials

::p_n(x)=sum_{k=1}^n S(n,k)x^k

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: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) = pn(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".

Related Topics:
Eric Temple Bell - Stirling number - Poisson distribution - Random variable - Bell number

~ ~ ~ ~ ~ ~ ~ ~ ~ ~