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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 $f(x) = Γ(x)^2/(2Γ(2x))$ 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 $α= π/3$.

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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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