Generalized Appell polynomials
where the generating function or kernel is composed of the series
- and all
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 partitions 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
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.
- William C. Brenke, On generating functions of polynomial systems, (1945) American Mathematical Monthly, 52 pp. 297–301.
- W. N. Huff, The type of the polynomials generated by f(xt) φ(t) (1947) Duke Mathematical Journal, 14 pp. 1091–1104.