Difference polynomials
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
- [math]\displaystyle{ p_n(z)=\frac{z}{n} {{z-\beta n -1} \choose {n-1}} }[/math]
where [math]\displaystyle{ {z \choose n} }[/math] is the binomial coefficient. For [math]\displaystyle{ \beta=0 }[/math], the generated polynomials [math]\displaystyle{ p_n(z) }[/math] are the Newton polynomials
- [math]\displaystyle{ p_n(z)= {z \choose n} = \frac{z(z-1)\cdots(z-n+1)}{n!}. }[/math]
The case of [math]\displaystyle{ \beta=1 }[/math] generates Selberg's polynomials, and the case of [math]\displaystyle{ \beta=-1/2 }[/math] generates Stirling's interpolation polynomials.
Moving differences
Given an analytic function [math]\displaystyle{ f(z) }[/math], define the moving difference of f as
- [math]\displaystyle{ \mathcal{L}_n(f) = \Delta^n f (\beta n) }[/math]
where [math]\displaystyle{ \Delta }[/math] is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as
- [math]\displaystyle{ f(z)=\sum_{n=0}^\infty p_n(z) \mathcal{L}_n(f). }[/math]
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
- [math]\displaystyle{ e^{zt}=\sum_{n=0}^\infty p_n(z) \left[\left(e^t-1\right)e^{\beta t}\right]^n. }[/math]
This generating function can be brought into the form of the generalized Appell representation
- [math]\displaystyle{ K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n }[/math]
by setting [math]\displaystyle{ A(w)=1 }[/math], [math]\displaystyle{ \Psi(x)=e^x }[/math], [math]\displaystyle{ g(w)=t }[/math] and [math]\displaystyle{ w=(e^t-1)e^{\beta t} }[/math].
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.
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