Generalized Appell polynomials

In mathematics, a polynomial sequence has a generalized Appell representation if the generating function for the polynomials takes on a certain form:
where the generating function or kernel is composed of the series
and
and
Given the above, it is not hard to show that is a polynomial of degree.
Boas–Buck polynomials are a slightly more general class of polynomials.

Special cases

The choice of gives the class of Brenke polynomials.
The choice of results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.
The combined choice of and gives the Appell sequence of polynomials.

Explicit representation

The generalized Appell polynomials have the explicit representation
The constant is
where this sum extends over all compositions of into parts; that is, the sum extends over all such that
For the Appell polynomials, this becomes the formula

Recursion relation

Equivalently, a necessary and sufficient condition that the kernel can be written as with is that
where and have the power series
and
Substituting
immediately gives the recursion relation
For the special case of the Brenke polynomials, one has and thus all of the, simplifying the recursion relation significantly.