Eventual positivity conjecture for generalized Gregory coefficients

Let $G_n^{(\ell)}(\mathbf{k})$ denote the generalized Gregory coefficients associated with a multiple polylogarithm of index $\mathbf{k}$ and positive integer order $\ell$ , as defined by the Laurent expansion of the reciprocal of the corresponding $\ell$ -th power. The conjecture asserts that for every admissible index $\mathbf{k}$ and every $\ell\in\mathbb{N}$ , there exists $N=N(\mathbf{k},\ell)$ such that $G_n^{(\ell)}(\mathbf{k})>0$ for all $n\geq N$ .

References

Primary source

arXiv

Additional references

Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers — arXiv — Ce Xu, Jianqiang Zhao

number-theory special-functions

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new paper confirms the conjecture for two important families, but the general question remains open.

The problem asks whether generalized Gregory coefficients eventually have the conjectured positive sign. The general composition and order cases remain unresolved.

September 2026 special-case confirmation

Ce Xu and Jianqiang Zhao report that the eventual-sign conjecture is confirmed for polylogarithm and double-polylogarithm cases. This is a substantial partial advance, not a resolution of the general conjecture; the reported result is unverified in this scan.

Current status (as of September 2026): The polylogarithm and double-polylogarithm cases are claimed to be settled, while the general composition and order cases remain open.

Sources

No solutions have been posted yet.