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>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 * Bernoulli polynomials of the second kind
==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