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.
The general difference polynomial sequence is given by where
are the Newton polynomials The case of
generates Selberg's polynomials, and the case of
generates Stirling's interpolation polynomials.
, define the moving difference of f as where
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.
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