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: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ... Jump to: navigation, search In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

where the generating function or kernel K(z,w) is composed of the series The word kernel has a a variety of meanings in a several fields. ...

with

and

and all

and

with

Given the above, it is not hard to show that p_n(z) is a polynomial of degree n. This article is about the term degree as used in mathematics. ...

Special cases

• The choice of g(w) = w gives the class of Brenke polynomials.

• The choice of Ψ(t) = e^t results in the Sheffer sequence of polynomials.

• The combined choice of g(w) = w and Ψ(t) = e^t gives the Appell sequence of polynomials.

In mathematics, a polynomial sequence, i. ... In mathematics, a polynomial sequence, i. ...

Explicit representation

The generalized Appell polynomials have the explicit representation

The constant is

where this sum extends over all partitions of n into k + 1 parts; that is, the sum extends over all {j} such that . In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. ...

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 A(w)Ψ(zg(w)) with g₁ = 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.

References

• 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.