Binomial Type - Characterization By Bell Polynomials

Characterization By Bell Polynomials

For any sequence a₁, a₂, a₃, ... of scalars, let

Where B_n,k(a₁, ..., 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.