Derivative sums of balanced gamma quotients and multiple zeta values: four conjectures of Zhi-Wei Sun

2026/07/14 by Shivam Nalin Patel
Mathematics · #math.GM #msc:11M32 #msc:33B15 #msc:11B65 #msc:33C20 #msc:33F10

paper · pdf

29 pages, 6 tables. Ancillary files contain exact rational certificates, Python verification scripts for the inverse-4 branch, Wolfram Language verification scripts and saved exact outputs for the inverse-27 branch, and Supplement A

arxiv created 2026/07/14 · arxiv updated 2026/07/31

Abstract

We introduce a uniform reduction for derivative sums of balanced gamma quotients. For exponent data (a_i,e_i) satisfying ∑_i e_i a_i=0, the entire translation-dependent gamma prefactor is governed by the characteristic power sums χ_m=∑_i e_i a_i^m through the identity log C(u)=∑_m≥2(-1)^mχ_mζ(m)u^m/m. This separates the analytic gamma contribution from a hypergeometric coefficient-extraction problem and explains, in a single framework, the inverse-4 and inverse-27 families occurring in four conjectures of Zhi-Wei Sun. For the branch χ_m=2-2^m, diagonal and symmetric specializations of a four-parameter Wilf--Zeilberger identity prove Conjectures 4.2 and 4.3. For the branch χ_m=3-3^m, exact span certificates in the coefficient spaces of Au's Example IV prove corrected forms of Conjectures 4.4 and 4.5. The transformed sides reduce to ordinary multiple zeta values of weights at most nine. Every computer-assisted step is exact: the two branches use independent rational-certificate toolchains, and no numerical recognition or PSLQ is used. We also identify four errors in the printed conjectures and show that the same balanced-quotient reduction applies to Conjecture 4.6, isolating the missing WZ input.

Citations

• Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations
• Discovering and Proving Infinite Binomial Sums Identities
• Mixed Tate motives over \Z
• Relations of multiple zeta values and their algebraic expression
• Central Binomial Sums, Multiple Clausen Values and Zeta Values
• Quasi-shuffle products

Related

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