Generalized Appell polynomials

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

where the generating function or kernel is composed of the series

with

and

and all

and

with

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

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Explicit representation

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The generalized Appell polynomials have the explicit representation

 

The constant is

 

where this sum extends over all compositions of   into   parts; that is, the sum extends over all   such that

 

For the Appell polynomials, this becomes the formula

 

Recursion relation

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Equivalently, a necessary and sufficient condition that the kernel   can be written as   with   is that

 

where   and   have the power series

 

and

 

Substituting

 

immediately gives the recursion relation

 

For the special case of the Brenke polynomials, one has   and thus all of the  , simplifying the recursion relation significantly.

See also

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References

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  • 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.