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.

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Related concepts (3)

Appell sequence

In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence satisfying the identity and in which is a non-zero constant. Among the most notable Appell sequences besides the trivial example are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.

Sheffer sequence

In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence (pn(x) : n = 0, 1, 2, 3, ...) of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer. Fix a polynomial sequence (pn). Define a linear operator Q on polynomials in x by This determines Q on all polynomials. The polynomial sequence pn is a Sheffer sequence if the linear operator Q just defined is shift-equivariant; such a Q is then a delta operator.

Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.