关于若干Kanade--Russell模12猜想的证明
Proofs of Some Kanade--Russell Mod 12 Conjectures
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中文总结 AI 辅助
本文用线性递推、q-差分方程、q-级数求和及围道积分与留数演算等方法，证明了Kanade--Russell模12猜想中剩余的六个恒等式，包括两个三重和与四个四重和恒等式。
中文摘要 AI 辅助
Kanade和Russell猜想出了十七个模12的Rogers--Ramanujan型恒等式。其中涉及三重和的十一个恒等式已由Bringmann--Jennings-Shaffer--Mahlburg和Rosengren证明。受这些工作的启发并采用类似方法，我们解决了所有剩余的六个猜想，包括源自Russell论文的两个三重和恒等式（标记为$I_{5a}$和$I_{6a}$）以及四个四重和恒等式（标记为7、7a、8和8a）。我们对三重和恒等式的证明结合了线性递推、$q$-差分方程和$q$-级数求和公式。对于四重和恒等式，我们将和表示为被积函数为无穷乘积的围道积分，并通过留数演算求值。所得的留数化简为单和三次基本超几何级数，我们能够将其表示为无穷乘积。
英文摘要
Kanade and Russell conjectured seventeen Rogers--Ramanujan type identities of modulus 12. Eleven of these identities involving triple sums were proved by Bringmann--Jennings-Shaffer--Mahlburg and by Rosengren. Motivated by these works and using similar methods, we settle all the six remaining conjectures including two triple sum identities labeled $I_{5a}$ and $I_{6a}$ originated from Russell's thesis and four quadruple sum identities labeled 7, 7a, 8 and 8a. Our proof of the triple sum identities combines linear recurrences, $q$-difference equations, and $q$-series summation formulas. For the quadruple sum identities, we represent the sums as contour integrals whose integrands are infinite products and evaluate them by residue calculus. The resulting residues reduce to single sum cubic basic hypergeometric series, and we are able to express them as infinite products.
发表机构
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
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