Genčev–Rucki conjecture

For every integer $r>1$ , the multiple Apéry-like series $\mathcal{A}_r$ introduced by Genčev and Rucki, involving central binomial coefficients and finite multiple harmonic star sums, satisfies $\mathcal{A}_r=2\left(1-4^{-r}\right)\zeta(2r+1)$ . The supplied sources do not reproduce the explicit definition of $\mathcal{A}_r$ .

References

Primary source

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Additional references

On the proof of the Genčev–Rucki conjecture for multiple Apéry-like series — Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas — Ce Xu

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A 2025 preprint, later published in a mathematical journal, claims to prove the conjecture, but the proof has not been independently checked in the available record.

Genčev and Rucki proposed the conjecture in 2025: for every integer $r>1$ , their multiple Apéry-like series $4\mathcal{A}_r$ should equal $2(1-4^{-r})\zeta(2r+1)$ .

October 10, 2025 claimed proof

Ce Xu’s arXiv preprint states that two hypergeometric methods prove the identity, with the result given as Theorem 2.2. A later journal record reports the same claimed resolution, and subsequent arXiv papers cite Xu’s theorem, but no independent verification, referee report, correction, withdrawal, or retraction appears in the retrieved sources.

Current status (as of September 2026): Xu’s proof claim is published and subsequently cited, but independent verification is not recorded, so the conjecture remains unconfirmed.

Sources

No solutions have been posted yet.