Difference polynomials

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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.