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The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion

Eric van Fossen Conrad·Philippe Flajolet
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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 Fran{ç}on.

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Eric van Fossen Conrad, Philippe Flajolet. 2005-07-13. The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion. https://arxiv.org/abs/math/0507268

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