Appell Sequence - Subgroup of The Sheffer Polynomials

Subgroup of The Sheffer Polynomials

The set of all Appell sequences is closed 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

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

(the subscript n appears in p_n, since this is the n term of that sequence, but not in q, since this refers to the sequence as a whole rather than one of its terms).

Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup. That it is abelian can be seen by considering the fact that every Appell sequence is of the form

and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator D.