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 is composed of the series with and and all and with 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 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. 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 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.