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 $a\_0\; e\; 0$

and :$Psi(t)=\; sum\_\{n=0\}^infty\; Psi\_n\; t^n\; quad$ and all $Psi\_n\; e\; 0$

and:$g(w)=\; sum\_\{n=1\}^infty\; g\_n\; w^n\; quad$ with $g\_1\; e\; 0.$

Given the above, it is not hard to show that $p\_n(z)$ is a polynomial of degree $n$.

pecial 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 partitions 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

:$z^\{n+1\}\; frac\; \{d\}\{dz\}\; left\; [\; frac\{p\_n(z)\}\{z^n\}\; ight]\; =\; -sum\_\{k=0\}^\{n-1\}\; c\_\{n-k-1\}\; p\_k(z)\; -z\; sum\_\{k=1\}^\{n-1\}\; b\_\{n-k\}\; frac\{d\}\{dz\}\; p\_k(z).$

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.

ee 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.

* William C. Brenke, "On generating functions of polynomial systems", (1945) American Mathematical Monthly, 52 pp. 297-301.

* W. N. Huff, "The type of the polynomials generated by f(xt) φ(t)" (1947) Duke Mathematical Journal, 14 pp 1091-1104.