Difference polynomials
title: "Difference polynomials" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polynomials", "finite-differences", "factorial-and-binomial-topics"] topic_path: "arts/film" source: "https://en.wikipedia.org/wiki/Difference_polynomials" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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.
Definition
The general difference polynomial sequence is given by
:p_n(z)=\frac{z}{n}
where {z \choose n} is the binomial coefficient. For \beta=0, the generated polynomials p_n(z) are the Newton polynomials
:p_n(z)= {z \choose n} = \frac{z(z-1)\cdots(z-n+1)}{n!}.
The case of \beta=1 generates Selberg's polynomials, and the case of \beta=-1/2 generates Stirling's interpolation polynomials.
Moving differences
Given an analytic function f(z), define the moving difference of f as
:\mathcal{L}_n(f) = \Delta^n f (\beta n)
where \Delta is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as
:f(z)=\sum_{n=0}^\infty p_n(z) \mathcal{L}_n(f).
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.
This generating function can be brought into the form of the generalized Appell representation
:K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n
by setting A(w)=1, \Psi(x)=e^x, g(w)=t and w=(e^t-1)e^{\beta t}.
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.
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::