Even-weight alternating effective-sum identities

Establish, for even weight \(w\ge 4\), the four conjectured identities relating the effective finite Euler sums \(Z_A^{(\bar a,b)}\), \(Z_A^{(a,\bar b)}\), and \(Z_A^{(\bar a,\bar b)}\), including the two sum formulas, the stuffle relation, and the weighted combined sum formula stated in the conjecture.

Background

After constructing several families of effective double Euler sums, the paper identifies further relations that are expected to hold in even weight. The authors explain that proving these identities requires a triple Euler-sum analogue of a cited result, which they have only checked numerically. The displayed conjecture records the unresolved relations.

References

It is natural to expect the following relations to hold. To prove these, however, one first needs the triple Euler sum analog of Cor.~5.9, which we have verified numerically only.

— Effective Euler Sums  (2609.09260 - Zhao, 8 Sep 2026) in Conjecture 5.4, subsection “Effective Euler sums $Z_A^{(a),(b)}$”

Suppose $a,b\inN$ with $a+b$ even. Then $Z_A({a},b)=Z_A{a},b$ and $Z_A(a,{b})=Z_Aa,{b}$.

— Effective Euler Sums  (2609.09260 - Zhao, 8 Sep 2026) in Conjecture A.2, Appendix A, subsection “Effective double Euler sums of types $(\bar a,b)$ and $(a,\bar b)$”