Binomial type

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

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

Related Topics:
Generating function - Power series

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:sum_{n=0}^infty {p_n(x) over n!}t^n=e^{xf(t)}

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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

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:f(t)=sum_{n=1}^infty {p_n,'(0) over n!}t^n.

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The delta operator of the sequence is f−1(D), so that

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:f^{-1}(D)p_n(x)=np_{n-1}(x).

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A way to think about these generating functions

The coefficients in the product of two formal power series

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:sum_{n=0}^infty {a_n over n!}t^n

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and

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:sum_{n=0}^infty {b_n over n!}t^n

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are

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:c_n=sum_{k=0}^n {n choose k} a_k b_{n-k}

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(see also Cauchy product). If we think of x as a parameter indexing a family of such power series, then the binomial identity says in effect that the power series indexed by x + y is the product of those indexed by x and by y. Thus the x is the argument to a function that maps sums to products: an exponential function

Related Topics:
Cauchy product - Exponential function

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:g(t)^x=e^{x f(t)}

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where f(t) has the form given above.

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