与扎吉尔三阶例子对偶的纳姆和的模性
Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities
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中文总结 AI 辅助
研究与扎吉尔三阶例子对偶的纳姆和，结合已有工作确立多数对偶的模性。对第九个例子部分证明相关恒等式得条件结果，对第十二个例子给出猜想恒等式。证明依赖特定恒等式，还证明了一些新的纳姆和恒等式及模性。
中文摘要 AI 辅助
2007年，扎吉尔识别出十二组三阶模纳姆和，并证明其中三组的模性，其余例子的模性由王确认。本文研究与扎吉尔三阶例子对偶的纳姆和，结合结果与早期工作，除对应第九和第十二个例子的对偶外，确立了所有对偶的模性。对第九个例子，证明了建立乘积表示所需的四组恒等式中的三组，留下一组作为猜想，得到条件模性结果。对第十二个例子，给出暗示其对偶模性猜想的乘积恒等式。证明依赖于将相关纳姆和表示为无穷乘积有限组合的罗杰斯 - 拉马努金型恒等式。过程中，证明了曹、王及本文作者之前猜想的四个四阶蝌蚪纳姆和恒等式，还发现并证明了几个新的三阶纳姆和的模性。
英文摘要
In 2007, Zagier identified twelve families of candidates for rank-three modular Nahm sums and established the modularity of three of them. The remaining cases were subsequently confirmed by Wang. Zagier's duality observation indicates that the Nahm sums associated with the duals of these examples might still be modular. We confirm that this is indeed true. The modularity of the dual of the sixth example was previously established by Milas and Wang, while that of the dual of the eleventh example was partially established by the present authors. We settle all remaining cases, thereby establishing the modularity of every defined dual in Zagier's rank-three list. We achieve this by proving new Rogers--Ramanujan type identities that express the relevant Nahm sums as finite sums of infinite products. Along the way we discover two new families of rank-three Nahm sums. As applications, we prove some Nahm sum identities conjectured by Cao--Wang, Li and the present authors.
