关于Han和Xiong关于分数高斯二项式系数的一个猜想
On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients
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中文总结 AI 辅助
本文针对Han和Xiong提出的分数高斯二项式系数猜想，通过证明支持度优势定理解决了大部分情形，将猜想简化为有限单位分数情形，结合计算机计算和Lean中的自动验证，最终验证了所有正有理数$r$及$k\neq200$时猜想成立。
中文摘要 AI 辅助
Han和Xiong近期将高斯二项式系数$\binom{r+k}{k}_q$推广到正有理数$r$，并猜想其整数迹（所得幂级数的整数指数部分）在$r=1/2$时系数最大。我们证明了一个支持度优势定理，用于在显式整除条件下比较有理参数，该定理对所有$r\neq1/2$解决了此猜想，并将完整猜想简化为单位分数$r=\frac{1}{2m}$的情形，对每个固定$k$而言，其中仅有有限个非平凡情形。随后通过计算机计算验证了所有正有理数$r$及$k\neq200$时该猜想成立。理论结果由AxiomProver在Lean中自动生成并验证。
英文摘要
Han and Xiong recently extended the Gaussian binomial coefficient $\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}$ to positive rational $r$ and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at $r=1/2$. We prove a support-dominance theorem comparing rational parameters under an explicit divisibility condition. It settles the conjecture for every $r\geq 1/2$ and reduces the full conjecture to the unit fractions $r=\frac{1}{2m}$, only finitely many of which are nontrivial for each fixed $k$. A computer computation then verifies the conjecture for every positive rational $r$ and every $k\leq 200$. The theoretical results were autonomously produced and verified in Lean by AxiomProver.
