Metraipolis › archive › dc_320b85344c9a

The modulo 9 Kanade--Russell identities and their Nahm-sum duals — arXiv:2609.08816

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.08816
License (as archived): CC0-1.0
Archived: 2026-10-16T14:49:29.784Z
SHA-256 of the archived text: d4001b877dd6fe2cf168fd6b88f11a4e568993b6c002b7ca55d3a7e93ad9acb1
Stable URL of this page: https://metraipolis.com/documento/dc_320b85344c9a

How to cite

The modulo 9 Kanade--Russell identities and their Nahm-sum duals — arXiv:2609.08816 — arXiv (en), https://arxiv.org/abs/2609.08816 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_320b85344c9a (Metraipolis archive dc_320b85344c9a, sha256 d4001b877dd6fe2cf168fd6b88f11a4e568993b6c002b7ca55d3a7e93ad9acb1)

BibTeX

@misc{dc320b85344c9a,
  title = {The modulo 9 Kanade--Russell identities and their Nahm-sum duals — arXiv:2609.08816},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.08816},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_320b85344c9a; sha256 d4001b877dd6fe2cf168fd6b88f11a4e568993b6c002b7ca55d3a7e93ad9acb1},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.08816. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Kanade and Russell initiated a family of conjectural Rogers--Ramanujan type identities of moduli $9$ and $12$, which ultimately comprised five modulo $9$ identities and eleven modulo $12$ identities. The eleven modulo $12$ conjectures were subsequently settled through the work of Bringmann, Jennings-Shaffer, and Mahlburg and of Rosengren. In this paper, we prove all five modulo $9$ Kanade--Russell sum-product identities, four individual generalized Nahm-sum dual identities, and a product formula for the natural dual companion of the fifth Kanade--Russell identity, which is expressed as a linear combination of two negative-mixed-term generalized Nahm sums. The first three individual dual identities settle Conjecture~3.6 of Wang and Wang, while the fourth proves the corresponding conjecture of Li and Wang. Our results also connect directly with the recent Dynkin-diagram framework of Sun and Wang for generalized Nahm sums. They identified the rank-two pairs $(T_1,G_2)$ and $(G_2,T_1)$ as unresolved cases whose modularity would follow, respectively, from the first modulo $9$ Kanade--Russell identity and its Wang-Wang dual. The present results prove precisely these two required identities and hence establish the corresponding modularity statements unconditionally.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.CO).
Categorie arXiv: Number Theory (math.NT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.08816 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:32:29.951Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).