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.

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

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

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.

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.
{{cite journal|first1=William C.|last1= Brenke|title=On generating functions of polynomial systems|year= 1945|journal=American Mathematical Monthly|volume = 52|number=6|pages=297–301|doi=10.2307/2305289|jstor= 2305289}}
{{cite journal|first1=W. N.|last1= Huff|title=The type of the polynomials generated by f(xt) φ(t)|year=1947|journal=Duke Mathematical Journal|volume=14|number=4|pages=1091–1104|doi=10.1215/S0012-7094-47-01483-X}}