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

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

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

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

References

  • 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.
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