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

2026/07/13 by Shivam Nalin Patel · 1 citation
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Abstract

Let f(x) = Γ(x)²/(2Γ(2x)) and set λ_α= 4sin²α 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 ∑_k≥1 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 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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