Binomial type

In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers ${\textstyle \left\{0,1,2,3,\ldots \right\}}$ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities

${\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}\,p_{k}(x)\,p_{n-k}(y).}$

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 may be expressed in terms of the Bell polynomials. 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.