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

By Eric Conrad, Philippe Flajolet. 2005

46 citations recorded by the source corpus.

Elliptic functions considered by Dixon in the nineteenth century and re- lated 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 Polya 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 dierent from models of elliptic functions earlier introduced by Viennot, Flajolet, Dumont, and Francon. In 1978, Apery announced an amazing discovery: " (3) P 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 (59). At the time of Apery'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). Apery's original irrationality proof crucially depends on a continued fraction representation of (3). To wit:

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/253d0c407b14fbda0fc755856c2e97b4a6fd60c6

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In Algebra and Number Theory

  1. Philippe Flajolet

In Discrete Mathematics and Combinatorics

  1. Robert Sedgewick
  2. Philippe Flajolet
  3. Sergi Elizalde
  4. Richard P. Stanley
  5. R. Stephenson
  6. François Charton
  7. Guillaume Lample
  8. Sheng Xu

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