KNOWPIA
WELCOME TO KNOWPIA

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

- 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.

- Ralph P. Boas, Jr. and R. Creighton Buck,
*Polynomial Expansions of Analytic Functions (Second Printing Corrected)*, (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. - Brenke, William C. (1945). "On generating functions of polynomial systems".
*American Mathematical Monthly*.**52**(6): 297–301. doi:10.2307/2305289. - Huff, W. N. (1947). "The type of the polynomials generated by f(xt) φ(t)".
*Duke Mathematical Journal*.**14**(4): 1091–1104. doi:10.1215/S0012-7094-47-01483-X.