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THE FERMAT CUBIC, ELLIPTIC FUNCTIONS, CONTINUED FRACTIONS, AND A COMBINATORIAL EXCURSION

By Eric Van, Fossen Conrad and Philippe Flajolet

Abstract

Kindly dedicated to Gérard · · ·Xavier Viennot on the occasion of his sixtieth birthday. Abstract. Elliptic functions considered by Dixon in the nineteenth century and related to Fermat’s cubic, x 3 +y 3 = 1, lead to a new set of continued fraction expansions with sextic numerators and cubic denominators. The functions and the fractions are pregnant with interesting combinatorics, including a special Pólya urn, a continuous-time branching process of the Yule type, as well as permutations satisfying various constraints that involve either parity of levels of elements or a repetitive pattern of order three. The combinatorial models are related to but different from models of elliptic functions earlier introduced by Viennot, Flajolet, Dumont, and Françon. In 1978, Apéry announced an amazing discovery: “ζ(3) ≡ ∑ 1/n 3 is irrational”. This represents a great piece of Eulerian mathematics of which van der Poorten has written a particularly vivid account in [58]. At the time of Apéry’s result, nothing was known about the arithmetic nature of the zeta values at odd integers, and not unnaturally his theorem triggered interest in a whole range of problems that are now recognized to relate to much “deep ” mathematics [38, 51]. Apéry’s original irrationality proof crucially depends on a continued fraction representation of ζ(3). To wit: (1) ζ(3)

Year: 2005
OAI identifier: oai:CiteSeerX.psu:10.1.1.237.6876
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