• The choice of ${\displaystyle g(w)=w}$  gives the class of Brenke polynomials.
• The choice of ${\displaystyle \Psi (t)=e^{t}}$  results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.
• The combined choice of ${\displaystyle g(w)=w}$  and ${\displaystyle \Psi (t)=e^{t}}$  gives the Appell sequence of polynomials.

The generalized Appell polynomials have the explicit representation

${\displaystyle p_{n}(z)=\sum _{k=0}^{n}z^{k}\Psi _{k}h_{k}.}$ 

The constant is

${\displaystyle h_{k}=\sum _{P}a_{j_{0}}g_{j_{1}}g_{j_{2}}\cdots g_{j_{k}}}$ 

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

${\displaystyle j_{0}+j_{1}+\cdots +j_{k}=n.\,}$ 

For the Appell polynomials, this becomes the formula

${\displaystyle p_{n}(z)=\sum _{k=0}^{n}{\frac {a_{n-k}z^{k}}{k!}}.}$ 

Equivalently, a necessary and sufficient condition that the kernel ${\displaystyle K(z,w)}$  can be written as ${\displaystyle A(w)\Psi (zg(w))}$  with ${\displaystyle g_{1}=1}$  is that

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

where ${\displaystyle b(w)}$  and ${\displaystyle c(w)}$  have the power series

${\displaystyle b(w)={\frac {w}{g(w)}}{\frac {d}{dw}}g(w)=1+\sum _{n=1}^{\infty }b_{n}w^{n}}$ 

and

${\displaystyle c(w)={\frac {1}{A(w)}}{\frac {d}{dw}}A(w)=\sum _{n=0}^{\infty }c_{n}w^{n}.}$ 

Substituting

${\displaystyle K(z,w)=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}$ 

immediately gives the recursion relation

${\displaystyle z^{n+1}{\frac {d}{dz}}\left[{\frac {p_{n}(z)}{z^{n}}}\right]=-\sum _{k=0}^{n-1}c_{n-k-1}p_{k}(z)-z\sum _{k=1}^{n-1}b_{n-k}{\frac {d}{dz}}p_{k}(z).}$ 

For the special case of the Brenke polynomials, one has ${\displaystyle g(w)=w}$  and thus all of the ${\displaystyle b_{n}=0}$ , simplifying the recursion relation significantly.

• q-difference polynomials

• 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.
• Brenke, William C. (1945). "On generating functions of polynomial systems". American Mathematical Monthly. 52 (6): 297–301. doi:10.2307/2305289.
• Huff, W. N. (1947). "The type of the polynomials generated by f(xt) φ(t)". Duke Mathematical Journal. 14 (4): 1091–1104. doi:10.1215/S0012-7094-47-01483-X.