Characterization By Generating Functions
Polynomial sequences of binomial type are precisely those whose generating functions are formal (not necessarily convergent) power series of the form
where f(t) is a formal power series whose constant term is zero and whose first-degree term is not zero. It can be shown by the use of the power-series version of Faà di Bruno's formula that
The delta operator of the sequence is f−1(D), so that
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