Zeros of Difference Polynomials
with R. Evans

Journal of Approximation Theory, 69 (1992), 1-14

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These papers are based on experiments with over 10,000 roots of various difference polynomials. The polynomials in question are the difference polynomials Dm(xn) where D(f)(x)=f(x+1)-f(x). After normalization and change of variables the study is reduced to the study of a family of polynomials Cn,m of degree c = int((n-m)/2) which have positive real roots.

The coefficients arise as differences of very large numbers - so roundoff error occurs very quickly when ordinary floating point arithmetic is used. A special purpose system was constructed to access and analyze the system of roots. We found several other representations of the polynomials - one of which allowed the computation of all the needed roots to high accuracy.

A special purpose software system was created to provide facilities for the generation and analysis of the roots.  The project saw an interesting interplay between theory and software development. The paper presents theorems and conjectures about the difference polynomials and their roots.