发表机构
Suzhou University of Science and Technology（苏州科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了全部五个模9 Kanade--Russell恒等式及其四个广义Nahm和对偶，解决了Wang-Wang和Li-Wang的猜想，并建立了Sun-Wang框架中两个未决情形的模性。
AI 中文摘要
Kanade和Russell发起了一族模数$9$和$12$的猜想型Rogers--Ramanujan类型恒等式，最终包含五个模$9$恒等式和十一个模$12$恒等式。这十一个模$12$猜想随后通过Bringmann、Jennings-Shaffer和Mahlburg以及Rosengren的工作得到解决。在本文中，我们证明了全部五个模$9$ Kanade--Russell和积恒等式、四个独立的广义Nahm和对偶恒等式，以及第五个Kanade--Russell恒等式的自然对偶伴随体的乘积公式，该对偶伴随体表示为两个负混合项广义Nahm和的线性组合。前三个独立对偶恒等式解决了Wang和Wang的猜想3.6，而第四个证明了Li和Wang的相应猜想。我们的结果还直接与Sun和Wang关于广义Nahm和的近期Dynkin图框架相关联。他们确定了秩二对$(T_1,G_2)$和$(G_2,T_1)$为未解决情形，其模性将分别由第一个模$9$ Kanade--Russell恒等式及其Wang--Wang对偶推出。本文结果恰好证明了这两个所需恒等式，从而无条件地建立了相应的模性陈述。
英文摘要
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.
