{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08816",
  "author_total": 1,
  "authors": [
    {
      "name": "Ernest X. W. Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn7hd5edd77nv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08816",
  "fulltext_url": null,
  "id": "bn7hd5edd77nv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08816",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212003532",
  "pdf_url": "https://arxiv.org/pdf/2609.08816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn7hd5edd77nv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The modulo 9 Kanade--Russell identities and their Nahm-sum duals",
  "title_text": "The modulo 9 Kanade--Russell identities and their Nahm-sum duals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/bn7hd5edd77nv",
  "venue": "arXiv",
  "versions": []
}