关于模九 Kanade--Russell 恒等式的十八个猜想恒等式的证明

中文摘要 AI 辅助

Uncu 和 Zudilin 研究了模九 Kanade--Russell 恒等式的有限形式在 $q\mapsto q^{-1}$ 下的反射，从而得出十个猜想性的和-积求值，其中包括 Warnaar 提出的两个。Hickerson 在研究相关参数化二重和式的连续关系时，又猜想出四个恒等式。通过构造 Hickerson 和式的有限形式并研究其反射，Konenkov 提出了两个额外的乘积恒等式，其共轭给出另外两个。在本文中，我们通过将作者建立的 Nahm 和对偶求值与参数化分差恒等式相结合，证明了全部十八个猜想恒等式。对于对称反射，Schur 型分解和有限有理恒等式将所需的求值化简为非奇异二元线性方程组。非对称和分圆情形通过 $q$-Airy--theta 连接公式处理，而 Hickerson 恒等式则由低阶分差导出。

英文摘要

Uncu and Zudilin studied reflections of finite forms of the modulo nine Kanade--Russell identities under $q\mapsto q^{-1}$, leading to ten conjectural sum--product evaluations, including two proposed by Warnaar. Hickerson conjectured four further identities in his study of contiguous relations for the associated parameterized double sums. By constructing finite forms of Hickerson's sums and studying their reflections, Konenkov proposed two additional product identities, whose conjugates give two more. In this paper, we prove all eighteen conjectural identities by combining Nahm-sum dual evaluations established by the author with parameterized divided-difference identities. For the symmetric reflections, Schur-type decompositions and finite rational identities reduce the required evaluations to nonsingular two-by-two linear systems. The asymmetric and cyclotomic cases are treated through $q$-Airy--theta connection formulas, whereas the Hickerson identities are derived from low-order divided differences.

发表机构

• Suzhou University of Science and Technology（苏州科技大学）

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