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

2005/07/13 by Eric van Fossen Conrad, Conrad, Eric van Fossen, Philippe Flajolet +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #math.CO #math.PR #msc:05A15 #msc:30B70 #msc:33C75 #msc:60C05

paper · pdf · doi:10.48550/arxiv.math/0507268

44 pages; submitted to "Seminaire Lotharingien de Combinatoire" (journal), July 2005

arxiv created 2005/07/13 · arxiv updated 2009/12/01

Abstract

Elliptic functions considered by Dixon in the nineteenth century and related to Fermat's cubic, x³+y³=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.

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