Binomial Type - Umbral Composition of Polynomial Sequences

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 { p_n(x) : n = 0, 1, 2, 3, ... } and { q_n(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, and

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).

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 formal composition of formal power series.