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

${\displaystyle p_{n}(z)={\frac {z}{n}}{{z-\beta n-1} \choose {n-1}}}$

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

${\displaystyle p_{n}(z)={z \choose n}={\frac {z(z-1)\cdots (z-n+1)}{n!}}.}$

The case of ${\displaystyle \beta =1}$ generates Selberg's polynomials, and the case of ${\displaystyle \beta =-1/2}$ generates Stirling's interpolation polynomials.

Moving differences

Given an analytic function ${\displaystyle f(z)}$, define the moving difference of f as

${\displaystyle {\mathcal {L}}_{n}(f)=\Delta ^{n}f(\beta n)}$

where ${\displaystyle \Delta }$ is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as

${\displaystyle f(z)=\sum _{n=0}^{\infty }p_{n}(z){\mathcal {L}}_{n}(f).}$

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

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

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

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

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

See also

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