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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“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)