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 $$n$$.

Boas–Buck polynomials are a slightly more general class of polynomials.

Special cases

 * The choice of $$g(w)=w$$ gives the class of Brenke polynomials.
 * The choice of $$\Psi(t)=e^t$$ 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 $$\Psi(t)=e^t$$ gives the Appell sequence of polynomials.

Explicit representation
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 compositions 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


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

Recursion relation
Equivalently, a necessary and sufficient condition that the kernel $$K(z,w)$$ can be written as $$A(w)\Psi(zg(w))$$ 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.