Difference polynomials

Last updated August 01, 2020

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}}{{z-\beta n-1} \choose {n-1}}

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

p_{n}(z)={z \choose 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

{\mathcal {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}^{\infty }p_{n}(z){\mathcal {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}^{\infty }p_{n}(z)\left[\left(e^{t}-1\right)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}^{\infty }p_{n}(z)w^{n}

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

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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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