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
where formula_2 is the binomial coefficient. For formula_3, the generated polynomials formula_4 are the Newton polynomials
The case of formula_6 generates Selberg's polynomials, and the case of formula_7 generates Stirling's interpolation polynomials.
Moving differences.
Given an analytic function formula_8, define the moving difference of "f" as
where formula_10 is the forward difference operator. Then, provided that "f" obeys certain summability conditions, then it may be represented in terms of these polynomials as
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
This generating function can be brought into the form of the generalized Appell representation 
by setting formula_14, formula_15, formula_16 and formula_17.