Generalized Appell polynomials

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:

$K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n$

where the generating function or kernel $K (z, w)$ is composed of the series

$A(w)= \sum_{n=0}^\infty a_n w^n \quad$ with $a_0 \ne 0$

and

$\Psi(t)= \sum_{n=0}^\infty \Psi_n t^n \quad$ and all $\Psi_n \ne 0$

and

$g(w)= \sum_{n=1}^\infty g_n w^n \quad$ with $g_1 \ne 0.$

Given the above, it is not hard to show that $p n (z)$ is a polynomial of degree $n$ .

[edit] Special cases

The choice of $Ψ(t) = e t$ results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.

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

The generalized Appell polynomials have the explicit representation

$p_n(z) = \sum_{k=0}^n z^k \Psi_k h_k.$

The constant is

$h_k=\sum_{P} a_{j_0} g_{j_1} g_{j_2} \cdots g_{j_k}$

where this sum extends over all partitions of $n$ into $k + 1$ parts; that is, the sum extends over all ${j}$ such that

$j_0+j_1+ \cdots +j_k = n.\,$

For the Appell polynomials, this becomes the formula

$p_n(z) = \sum_{k=0}^n \frac {a_{n-k} z^k} {k!}.$

Equivalently, a necessary and sufficient condition that the kernel $K (z, w)$ can be written as $A (w)Ψ(z g (w))$ with $g 1 = 1$ is that

$\frac{\partial K(z,w)}{\partial w} = c(w) K(z,w)+\frac{zb(w)}{w} \frac{\partial K(z,w)}{\partial z}$

where $b (w)$ and $c (w)$ have the power series

$b(w) = \frac{w}{g(w)} \frac {d}{dw} g(w) = 1 + \sum_{n=1}^\infty b_n w^n$

and

$c(w) = \frac{1}{A(w)} \frac {d}{dw} A(w) = \sum_{n=0}^\infty c_n w^n.$

Substituting

$K(z,w)= \sum_{n=0}^\infty p_n(z) w^n$

immediately gives the recursion relation

$z^{n+1} \frac {d}{dz} \left[ \frac{p_n(z)}{z^n} \right]= -\sum_{k=0}^{n-1} c_{n-k-1} p_k(z) -z \sum_{k=1}^{n-1} b_{n-k} \frac{d}{dz} p_k(z).$

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.

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.