Difference polynomials

(Redirected from Difference series)

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

edit

The general difference polynomial sequence is given by

 

where   is the binomial coefficient. For  , the generated polynomials   are the Newton polynomials

 

The case of   generates Selberg's polynomials, and the case of   generates Stirling's interpolation polynomials.

Moving differences

edit

Given an analytic function  , 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.

Generating function

edit

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  ,  ,   and  .

See also

edit

References

edit
  • 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.