In recent years, my primary research interests have revolved around the interplay of classical analysis and approximation theory, computational complexity, number theory and symbolic computation. Analytic problems whose attack and proof lend themselves to extensive computational experimentation have attracted me most.
Substantial symbolic and numeric calculation has led to the discovery of some rather beautiful analytic objects (series, iterations, etc.), and in many cases, such as the derivation of Ramanujan type series, considerably aids the proofs. The kind of questions that arise impinge on issues in approximation theory, number theory, and computational complexity (not to mention the obvious computational issues). This research has also led to some of the most efficient known algorithms for various elementary functions and constants (a number of the recent record calculations of π have used one of our algorithms). A pleasant by-product has the been the detection of various phenomena of a kind probably not visible without substantial interactive computation.
- Expansion problems of a number theoretic variety. The relationship between orthogonal expansions, Padé approximants, and irrationality proofs (such as that for ζ(3)).
- Growth problems concerning polynomials with integer coefficients associated with conjectures of Erdös and Szekeres and conjectures of Littlewood.
- Iterations and expansions related to special functions, emphasizing low complexity.
- Analytic inequalities and expansions for rational functions and polynomials, including lacunary polynomials.
- Approximation issues concerning Chebyshev systems, their Chebyshev polynomials, Müntz systems and incomplete rationals.
- Existence of planar functions.