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.

• 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. JSTOR 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.