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

Special cases

Explicit representation

The generalized Appell polynomials have the explicit representation The constant is where this sum extends over all compositions of n into k+1 parts; that is, the sum extends over all {j} such that For the Appell polynomials, this becomes the formula

Recursion relation

Equivalently, a necessary and sufficient condition that the kernel K(z,w) can be written as with g_1=1 is that where b(w) and c(w) have the power series and Substituting immediately gives the recursion relation For the special case of the Brenke polynomials, one has g(w)=w and thus all of the b_n=0, simplifying the recursion relation significantly.