Wikipedia:Difference polynomials
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In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.
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Definition
The general difference polynomial sequence is given by
where 
is the binomial coefficient. For β = 0, the generated polynomials pn(z) are the Newton polynomials
The case of β = 1 generates Selberg's polynomials, and the case of β = − 1 / 2 generates Stirling's interpolation polynomials.
Moving differences
Given an analytic function f(z), define the moving difference of f as
where Δ is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as
The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than exponential type. Summability conditions are discussed in detail in Boas & Buck.
Generating function
The generating function for the general difference polynomials is given by
![e^{zt}=\sum_{n=0}^\infty p_n(z)
\left[\left(e^t-1\right)e^{\beta t}\right]^n.](http://upload.wikimedia.org/math/b/c/a/bca7633d1a65858d7e2edfa6adf02c00.png)
This generating function can be brought into the form of the generalized Appell representation
by setting A(w) = 1, Ψ(x) = ex, g(w) = t and w = (et − 1)eβt.
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.
