In mathematics, a polynomial sequence
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}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where the generating function or kernel
is composed of the series
with ![{\displaystyle a_{0}\neq 0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
and all ![{\displaystyle \Psi _{n}\neq 0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
with ![{\displaystyle g_{1}\neq 0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Given the above, it is not hard to show that
is a polynomial of degree
.
Boas–Buck polynomials are a slightly more general class of polynomials.
Special cases
Explicit representation
The generalized Appell polynomials have the explicit representation
![{\displaystyle p_{n}(z)=\sum _{k=0}^{n}z^{k}\Psi _{k}h_{k}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The constant is
![{\displaystyle h_{k}=\sum _{P}a_{j_{0}}g_{j_{1}}g_{j_{2}}\cdots g_{j_{k}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where this sum extends over all compositions of
into
parts; that is, the sum extends over all
such that
![{\displaystyle j_{0}+j_{1}+\cdots +j_{k}=n.\,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For the Appell polynomials, this becomes the formula
![{\displaystyle p_{n}(z)=\sum _{k=0}^{n}{\frac {a_{n-k}z^{k}}{k!}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recursion relation
Equivalently, a necessary and sufficient condition that the kernel
can be written as
with
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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
and
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}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
![{\displaystyle c(w)={\frac {1}{A(w)}}{\frac {d}{dw}}A(w)=\sum _{n=0}^{\infty }c_{n}w^{n}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Substituting
![{\displaystyle K(z,w)=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For the special case of the Brenke polynomials, one has
and thus all of the
, simplifying the recursion relation significantly.
See also
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.