A combinatorial proof of Jacobi's elliptic identity via alternating permutations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911450164887552 |
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| author | Pain, Jean-christophe |
| author_facet | Pain, Jean-christophe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14568 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A combinatorial proof of Jacobi's elliptic identity via alternating permutations Pain, Jean-christophe Combinatorics 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. |
| title | A combinatorial proof of Jacobi's elliptic identity via alternating permutations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.14568 |