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

TL;DR

This paper introduces a combinatorial framework linking Entringer numbers, snakes, and elliptic functions, providing a structural proof of Jacobi's elliptic identity through permutation-based interpretations.

Contribution

It offers a novel combinatorial approach to prove Jacobi's elliptic identity by connecting classical combinatorics with elliptic function theory.

Findings

Unified combinatorial framework for elliptic functions and permutations

Structural interpretation of Jacobi's elliptic identity

Decomposition of weighted snakes into canonical components

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.