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 ak 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 {pn}. 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.
Read more about this topic: Appell Sequence
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