Fish curve

2013-03-18T03:55:52Z

74.111.243.245: /* Equations */

{{expert-subject|Mathematics|date=November 2009}}
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 polynomial]]s, '''Selberg's polynomials''', and the '''Stirling interpolation polynomials''' as special cases.

==Definition==
The general difference polynomial sequence is given by

:<math>p_n(z)=\frac{z}{n} {{z-\beta n -1} \choose {n-1}}</math>

where <math>{z \choose n}</math> is the [[binomial coefficient]]. For <math>\beta=0</math>, the generated polynomials <math>p_n(z)</math> are the Newton polynomials

:<math>p_n(z)= {z \choose n} = \frac{z(z-1)\cdots(z-n+1)}{n!}.</math>

The case of <math>\beta=1</math> generates Selberg's polynomials, and the case of <math>\beta=-1/2</math> generates Stirling's interpolation polynomials.

==Moving differences==
Given an [[analytic function]] <math>f(z)</math>, define the '''moving difference''' of ''f'' as

:<math>\mathcal{L}_n(f) = \Delta^n f (\beta n)</math>

where <math>\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>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>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>K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n</math>

by setting <math>A(w)=1</math>, <math>\Psi(x)=e^x</math>, <math>g(w)=t</math> and <math>w=(e^t-1)e^{\beta t}</math>.

==See also==
* [[Carlson's theorem]]

==References==
{{reflist}}
* 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.

[[Category:Polynomials]]
[[Category:Finite differences]]
[[Category:Factorial and binomial topics]]

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Fish curve