Binomial type

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

Characterization by a convolution identity

For sequences an, bn, n = 0, 1, 2, ..., define a sort of convolution by

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:(a diamondsuit b)_n = sum_{j=0}^n {n choose j} a_j b_{n-j}.

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Let a_n^{kdiamondsuit}, be the nth term of the sequence

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:underbrace{adiamondsuitcdotsdiamondsuit a}_{k mathrm{factors}}.,

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Then for any sequence ai, i = 0, 1, 2, ..., with a0 = 0, the sequence defined by p0(x) = 1 and

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:p_n(x) = sum_{k=1}^n {a_{n}^{kdiamondsuit} x^k over k!},

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for n ≥ 1, is of binomial type, and every sequence of binomial type is of this form.

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