The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- July 2005
- DOI:
- 10.48550/arXiv.math/0507268
- arXiv:
- arXiv:math/0507268
- Bibcode:
- 2005math......7268V
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Probability;
- 05A15;
- 30B70;
- 33C75;
- 60C05
- E-Print:
- 44 pages