Binomial Type - Characterization By Generating Functions

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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