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

Shivam Nalin Patel

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arXiv · 2607.15303

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

Shivam Nalin Patel

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Abstract

Let $f(x) = \Gamma(x)^2/(2\Gamma(2x))$ and set $\lambda_\alpha = 4\sin^2\alpha$ for $0 < \alpha < \pi/2$. 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 $\sum_{k\ge1} \lambda_\alpha^{k-1} f^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $\alpha = \pi/6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value $\mathrm{Gl}_{4,1}(\pi/3)$ occurs. The construction complements general cyclotomic-multiple-zeta methods for inverse-binomial harmonic sums by supplying a continuous master identity, with concrete specializations at $\alpha = \pi/4$ and $\alpha = \pi/3$.

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Shivam Nalin Patel. 2026-07-13. Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations. https://arxiv.org/abs/2607.15303

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