Generalized Appell polynomials

In mathematics, a polynomial sequence ${\displaystyle \{p_{n}(z)\}}$ has a generalized Appell representation if the generating function for the polynomials takes on a certain form:

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

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

${\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}\quad }$ with ${\displaystyle a_{0}\neq 0}$

and

${\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}\quad }$ and all ${\displaystyle \Psi _{n}\neq 0}$

and

${\displaystyle g(w)=\sum _{n=1}^{\infty }g_{n}w^{n}\quad }$ with ${\displaystyle g_{1}\neq 0.}$

Given the above, it is not hard to show that ${\displaystyle p_{n}(z)}$ is a polynomial of degree ${\displaystyle n}$.

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