Derivative Sums of Balanced Gamma Quotients and Multiple Zeta Values: Five Conjectures of Zhi-Wei Sun

Abstract

We introduce a uniform reduction for derivative sums of balanced gamma quotients. For exponent data $(a_i,e_i)$ satisfying ${\sum }_i{e}_i{a}_i=0$, the translation-dependent gamma prefactor is governed by the characteristic power sums ${\chi }_m={\sum }_i{e}_i{a}_i^m$ through $\log C(u)={\sum }_{m\ge 2}(-1{)}^m{\chi }_m\zeta (m)u^m/m$. This separates the universal gamma contribution from a hypergeometric coefficient-extraction problem and organizes three exponential families. For ${\chi }_m=2-2^m$, diagonal and symmetric specializations of a four-parameter Wilf--Zeilberger identity prove Conjectures 4.2 and 4.3 of Zhi-Wei Sun. For ${\chi }_m=3-3^m$, exact span certificates in the coefficient spaces of Au's Example IV prove corrected forms of Conjectures 4.4 and 4.5. For ${\chi }_m=4^m-10\cdot 2^m+16$, a half-integer specialization of Au's first $1/{\pi }^4$ construction, combined with the diagonal transformation in his Example VI, proves Conjecture 4.6 through weight eleven. The transformed sides reduce to ordinary multiple zeta values, and every computer-assisted acceptance test is exact: explicit rational WZ certificates and separately implemented MZV certificate checkers use no numerical recognition, PSLQ, or conjectural MZV dimensions. We also identify four errors in the printed statements of Conjectures 4.2--4.5.

36 pages, 7 tables. v2 adds the proof of Conjecture 4.6, restores its exact ancillary certificate branch, updates the title and abstract from four to five conjectures, and corrects the companion-paper citation