Umbral composition

Properties

The set of all Sheffer sequences is a group under the operation of umbral composition of polynomial sequences, defined as follows. Suppose ( p_n(x) : n = 0, 1, 2, 3, ... ) and ( q_n(x) : n = 0, 1, 2, 3, ... ) are polynomial sequences, given by $${\displaystyle p_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k}\ {\mbox{and}}\ q_{n}(x)=\sum _{k=0}^{n}b_{n,k}x^{k}~.}$$