伽马商的加权导数和：孙的猜想与分圆特化
Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations
AI总结：
研究伽马商\(f(x)\)的加权导数和，通过建立基本参数恒等式推导出显式公式，其系数满足递推关系，证明了孙志伟的猜想4.1，还给出了\(\alpha = \pi/4\)和\(\alpha = \pi/3\)处的具体特化，补充了分圆多重zeta方法。
AI中文摘要：
设\(f(x)=\Gamma(x)^2/(2\Gamma(2x))\)，对于\(0\lt\alpha\lt\pi/2\)，令\(\lambda_\alpha = 4\sin^2\alpha\)。我们为\(f\)的加权平移建立了一个基本参数恒等式，并推导出一个在每个导数阶都有效的显式公式，用于相关的加权和\(\sum_{k\ge1} \lambda_\alpha^{k - 1} f^{(r)}(k)\)。系数满足关于普通zeta值的有效递推关系。唯一的未加权特化\(\alpha = \pi/6\)证明了孙志伟的猜想4.1；在四阶时出现了一个深度为二的值\(\mathrm{Gl}_{4,1}(\pi/3)\)。该构造通过提供一个连续的主恒等式，补充了用于逆二项调和和的一般分圆多重zeta方法，并在\(\alpha = \pi/4\)和\(\alpha = \pi/3\)处有具体特化。
英文摘要：
Let $f(x) = Γ(x)^2/(2Γ(2x))$ and set $λ_α= 4\sin^2α$ for $0 < α< π/2$. We establish an elementary parameter identity for a weighted translate of $f$ and derive an explicit formula, valid at every derivative order, for the associated weighted sums $\sum_{k\ge1} λ_α^{k-1} f^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $α= π/6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value $\mathrm{Gl}_{4,1}(π/3)$ occurs. The construction complements general cyclotomic-multiple-zeta methods for inverse-binomial harmonic sums by supplying a continuous master identity, with concrete specializations at $α= π/4$ and $α= π/3$.
