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


 * $$p_n(z)=\frac{z}{n} {{z-\beta n -1} \choose {n-1}}$$

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 $$w=(e^t-1)e^{\beta t}$$.