Appell Sequence - Recursion Formula

Recursion Formula

Suppose

where the last equality is taken to define the linear operator S on the space of polynomials in x. Let

be the inverse operator, the coefficients a_k being those of the usual reciprocal of a formal power series, so that

In the conventions of the umbral calculus, one often treats this formal power series T as representing the Appell sequence {p_n}. One can define

by using the usual power series expansion of the log(1 + x) and the usual definition of composition of formal power series. Then we have

(This formal differentiation of a power series in the differential operator D is an instance of Pincherle differentiation.)

In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.