Generalized Appell polynomials

In mathematics, a polynomial sequence formula_1 has a generalized Appell representation if the generating function for the polynomials takes on a certain form:
where the generating function or kernel formula_3 is composed of the series 
and 
and
Given the above, it is not hard to show that formula_10 is a polynomial of degree formula_11.
Boas–Buck polynomials are a slightly more general class of polynomials.
Explicit representation.
The generalized Appell polynomials have the explicit representation
The constant is
where this sum extends over all compositions of formula_11 into formula_19 parts; that is, the sum extends over all formula_20 such that
For the Appell polynomials, this becomes the formula 
Recursion relation.
Equivalently, a necessary and sufficient condition that the kernel formula_3 can be written as formula_24 with formula_25 is that
where formula_27 and formula_28 have the power series
and 
Substituting 
immediately gives the recursion relation
For the special case of the Brenke polynomials, one has formula_12 and thus all of the formula_34, simplifying the recursion relation significantly.