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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.

Contents

Definition

The general difference polynomial sequence is given by

p n ( z ) = z n ( z − β n − 1 n − 1 )

where ( z n ) is the binomial coefficient. For β = 0 , the generated polynomials p n ( z ) are the Newton polynomials

p n ( z ) = ( z n ) = z ( z − 1 ) ⋯ ( z − n + 1 ) n ! .

The case of β = 1 generates Selberg's polynomials, and the case of β = − 1 / 2 generates Stirling's interpolation polynomials.

Moving differences

Given an analytic function f ( z ) , define the moving difference of f as

L n ( f ) = Δ n f ( β n )

where Δ is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as

f ( z ) = ∑ n = 0 ∞ p n ( z ) L n ( f ) .

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

e z t = ∑ n = 0 ∞ p n ( z ) [ ( e t − 1 ) e β t ] n .

This generating function can be brought into the form of the generalized Appell representation

K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n

by setting A ( w ) = 1 , Ψ ( x ) = e x , g ( w ) = t and w = ( e t − 1 ) e β t .

References

Difference polynomials Wikipedia
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