Difference polynomials explained

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

p _{n(z) = (z)/(n)} {{z - \beta n -1 } \choose {n -1 }}

where

{z \choose n} is the binomial coefficient. For \beta =0 , the generated polynomials p _{n(z)} are the Newton polynomialsp _{n(z)=} {z \choose n} = (z(z -1 ) … (z - n +1 ))/(n!) .

The case of

\beta =1 generates Selberg's polynomials, and the case of \beta =-1 / 2 generates Stirling's interpolation polynomials.

Moving differences

f(z) , define the moving difference of f asl{L} _{n(f)} = \Delta ^{n} f (\beta n)

where

\Delta is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials asf(z) = \sum ^{inf ty} _{n =0} p _{n(z)} l{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 ^{zt} = \sum ^{inf ty} _{n =0} p _{n(z)} \left[\left(e ^{t-1\right)e} ^{\beta} \right] ^{n.} K(z,w) = A(w)\Psi(zg(w)) = \sum ^{inf ty} _{n =0} p _{n(z)} w ^{n}

by setting

A(w) =1 , \Psi(x) = e ^{x} , g(w) = t and w = (e ^{t-1)e} ^{\beta} .

See also

• Carlson's theorem
• Bernoulli polynomials of the second kind

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.