Binomial type

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

Umbral composition of polynomial sequences

The set of all polynomial sequences of binomial type is a group in which the group operation is "umbral composition" of polynomial sequences. That operation is defined as follows. Suppose { pn(x) : n = 0, 1, 2, 3, ... } and { qn(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, and

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:p_n(x)=sum_{k=0}^n a_{n,k}, x^k.

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Then the umbral composition p o q is the polynomial sequence whose nth term is

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:(p_ncirc q)(x)=sum_{k=0}^n a_{n,k}, q_k(x).

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With the delta operator defined by a power series in D as above, the natural bijection between delta operators and polynomial sequences of binomial type, also defined above, is a group isomorphism, in which the group operation on power series is (perhaps surprisingly) formal composition of formal power series.

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