Generalized Appell Representation

In mathematics, a polynomial sequence $\$ has a generalized Appell representation if the generating function for the polynomials takes on a certain form:

:$K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n$
where the generating function or kernel $K(z,w)$ is composed of the series

:$A(w)= \sum_^\infty a_n w^n \quad$ with $a_0 \ne 0$

and
:$\Psi(t)= \sum_^\infty \Psi_n t^n \quad$ and all $\Psi_n \ne 0$

and
:$g(w)= \sum_^\infty g_n w^n \quad$ 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 $\Psi(t)=e^t$ 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

:$p_n(z) = \sum_^n z^k \Psi_k h_k.$

The constant is

:$h_k=\sum_ a_ g_ g_ \cdots g_$

where this sum extends over all compositions of $n$ into $k+1$ parts; that is, the sum extends over all $\$ such that

:$j_0+j_1+ \cdots +j_k = n.\,$

For the Appell polynomials, this becomes the formula

:$p_n(z) = \sum_^n \frac .$

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

:$\frac = c(w) K(z,w)+\frac \frac$

where $b(w)$ and $c(w)$ have the power series

:$b(w) = \frac \frac g(w) = 1 + \sum_^\infty b_n w^n$

and

:$c(w) = \frac \frac A(w) = \sum_^\infty c_n w^n.$

Substituting

:$K(z,w)= \sum_^\infty p_n(z) w^n$

immediately gives the recursion relation

: z^ \frac \left \frac \right
-\sum_^ c_ p_k(z)
-z \sum_^ b_ \frac 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.
*
* {{cite journal, first1=W. N., last1= Huff, title=The type of the polynomials generated by f(xt) φ(t), year=1947, journal=Duke Mathematical Journal, volume=14, number=4, pages=1091–1104, doi=10.1215/S0012-7094-47-01483-X

Polynomials