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 $a_{0}\neq 0$

and

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

and

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

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.

See also

• q-difference polynomials

References

• 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.