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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“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)
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