One must always generalize. (Man muß immer generalisieren)
-- Carl Jacobi quoted in P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981. (Thanks: David Marjanovic of Austria for notes on orthography and translation.)
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.
-- Jacques Hadamard quoted in H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Certain elliptic functions, considered in an 1890 paper by Alfred Cardew Dixon and related to Fermat's cubic equation (x3+y3=1), lead to a new set of continued fraction expansions with sextic numerators and cubic denominators. The functions and the continued fractions are pregnant with interesting combinatorics, including a special Pólya urn, a continuous-time branching process, as well as permutations satisfying constraints that involve either parity of levels of elements or a repetitive pattern of order three.Trivia: Because of this paper (and thanks to Philippe Flajolet), my Erdös number is 3. See http://algo.inria.fr/flajolet/numbers.html.
In a 1907 paper, L. Rogers used two methods to obtain continued fractions for certain Laplace transforms of Jacobi elliptic functions. His first method employed repeated integration by parts, while his second method recalled an 1889 technique of T. Stieltjes. In 1996, S. Milne used these expansions and others obtained by modular transformations to derive results about sums of squares and triangular numbers. Working independently in the 1820's, C. Jacobi and N. Abel both introduced elliptic functions to advance the study of elliptic integrals. In 1981, D. Dumont introduced symmetric variants of the elliptic functions of Jacobi and Abel to facilitate the study of certain combinatorial problems related to coefficients in Maclaurin expansions of Jacobi elliptic functions. In this thesis, we use Dumont's elliptic functions to rederive the continued fraction expansions of Rogers. In the classical approach used by Rogers and Milne, four families of continued fractions are obtained. In our approach, members of the same four families are derived directly by specializing parameters instead of employing modular transformations. To these four families, we add a new set of continued fractions based on certain elliptic functions that were studied in an 1890 paper by A. Dixon. These new continued fractions were discovered in 1999 in the course of work with D. Dumont.
Eliza: Hello. I am ELIZA. How can I help you?At this point the cybertherapist became unintelligible.
Me: By eating porridge.
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Eliza: Oh, I []