Binomial type

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

Characterization by Bell polynomials

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

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:p_n(x)=sum_{k=1}^n B_{n,k}(a_1,dots,a_{n-k+1}) x^k.

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Where Bn,k(a1, ..., an−k+1) is the Bell polynomial. Then this polynomial sequence is of binomial type. Note that for each n ≥ 1,

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:p_n'(0)=a_n.,

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Here is the main result of this section:

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Theorem: All polynomial sequences of binomial type are of this form.

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A result in Mullin and Rota, repeated in Rota, Kahaner, and Odlyzko (see References below) states that every polynomial sequence { pn(x) }n of binomial type is determined by the sequence { pn′(0) }n, but those sources do not mention Bell polynomials.

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This sequence of scalars is also related to the delta operator. Let

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:P(t)=sum_{n=1}^infty {a_n over n!} t^n.

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Then

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:P^{-1}left({d over dx} ight),

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is the delta operator of this sequence.

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