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

:$p_n(z)=\frac$

where $$ is the binomial coefficient. For $\beta=0$, the generated polynomials $p_n(z)$ are the Newton polynomials

:$p_n(z)= = \frac.$

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

:$\mathcal_n(f) = \Delta^n f (\beta n)$

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

:$f(z)=\sum_^\infty p_n(z) \mathcal_n(f).$

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^=\sum_^\infty p_n(z)
\left left(e^t-1\right)e^\rightn.

This generating function can be brought into the form of the generalized Appell representation

:$K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n$

by setting $A(w)=1$, $\Psi(x)=e^x$, $g(w)=t$ and $w=(e^t-1)e^$.

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.

Polynomials
Finite differences
Factorial and binomial topics