Binomial type

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

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:For related mathematical concepts, see list of factorial and binomial topics.

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In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by { 0, 1, 2, 3, ... } in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities

Related Topics:
Mathematics - Polynomial sequence - Polynomial

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:p_n(x+y)=sum_{k=0}^n{n choose k}, p_k(x), p_{n-k}(y).

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Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus.

Related Topics:
Lie group - Sheffer sequence - Umbral calculus

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