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:

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

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

$A(w)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{w}^n$ with $a_0\ne 0$

and

$\mathrm{\Psi }(t)=\sum _{n=0}^{\mathrm{\infty }}{\mathrm{\Psi }}_n{t}^n$ and all ${\mathrm{\Psi }}_n\ne 0$

and

$g(w)=\sum _{n=1}^{\mathrm{\infty }}{g}_n{w}^n$ 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 $\mathrm{\Psi }(t)=e^t$ 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 $\mathrm{\Psi }(t)=e^t$ gives the Appell sequence of polynomials.

Explicit representation

The generalized Appell polynomials have the explicit representation

$p_n(z)=\sum _{k=0}^n{z}^k{\mathrm{\Psi }}_k{h}_k.$

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 $A(w)\mathrm{\Psi }(zg(w))$ 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

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

and

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

Substituting

$K(z,w)=\sum _{n=0}^{\mathrm{\infty }}{p}_n(z)w^n$

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.