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.
The generalized Appell polynomials have the explicit representation The constant is where this sum extends over all compositions of
parts; that is, the sum extends over all
such that For the Appell polynomials, this becomes the formula Equivalently, a necessary and sufficient condition that the kernel
have the power series and Substituting immediately gives the recursion relation For the special case of the Brenke polynomials, one has
, simplifying the recursion relation significantly.
This polynomial-related article is a stub.