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

$p_n(z)=\frac{z}{n}(\genfrac{}{}{0ex}{}{z-\beta n-1}{n-1})$

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

$p_n(z)=(\genfrac{}{}{0ex}{}{z}{n})=\frac{z(z-1)\cdots (z-n+1)}{n!}.$

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

${ℒ}_n(f)={\Delta }^{n}f(\beta n)$

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

$f(z)=\sum _{n=0}^{\mathrm{\infty }}{p}_n(z){ℒ}_n(f).$

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

$e^{zt}=\sum _{n=0}^{\mathrm{\infty }}{p}_n(z){\left[(e^t-1)e^{\beta t}\right]}^n.$

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

$K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\mathrm{\infty }}{p}_n(z)w^n$

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

See also

• Carlson's theorem
• Bernoulli polynomials of the second kind

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.