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arXiv · 2602.14568

A combinatorial proof of Jacobi's elliptic identity via alternating permutations

Jean-christophe Pain
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Abstract

We provide a unified combinatorial framework connecting Entringer numbers, Dumont-Viennot snakes, and elliptically weighted continued fractions, which gives a structural interpretation of the Jacobi elliptic identity \begin{equation} \mathrm{sn}'(u)=\mathrm{cn}(u)\,\mathrm{dn}(u), \end{equation} where $\mathrm{sn}$, $\mathrm{cn}$ and $\mathrm{dn}$ are the Jacobi elliptic functions. This framework allows the decomposition of weighted snakes corresponding to the derivative of $\mathrm{sn}$ into canonical $\mathrm{cn}$- and $\mathrm{dn}$-components, bridging classical combinatorics and elliptic function theory.

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Jean-christophe Pain. 2026-02-16. A combinatorial proof of Jacobi's elliptic identity via alternating permutations. https://arxiv.org/abs/2602.14568

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