Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations

Let $f(x)=\mathrm{\Gamma }(x{)}^2\mathrm{/}(2\mathrm{\Gamma }(2x))$ and set ${\lambda }_{\alpha }=4{\sin }^2\alpha$ for $0<\alpha <\pi \mathrm{/}2$. 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 ${\sum }_{k\ge 1}{\lambda }_{\alpha }^{k-1}{f}^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $\alpha =\pi \mathrm{/}6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value ${\mathrm{G}\mathrm{l}}_{4,1}(\pi \mathrm{/}3)$ occurs. The construction complements general cyclotomic-multiple-zeta methods for inverse-binomial harmonic sums by supplying a continuous master identity, with concrete specializations at $\alpha =\pi \mathrm{/}4$ and $\alpha =\pi \mathrm{/}3$.