For each integer n≥ 1, define the Boros--Moll coefficients by P_n(x)=2^-2n∑_k=0^n2^k binom2n+12k+1(x+1)^k=∑_i=0^nd_i(n)x^i. Extend the finite sequence by setting d_i(n)=0 for i notin0,…,n. Define the log-concavity operator L by L(a)_i=a_i^2-a_i-1a_i+1. The Boros--Moll conjecture asserts that, for every n≥ 1, every integer r≥ 0, and every integer i, L^r((d_j(n))_j∈Z)_i≥ 0; equivalently, the Boros--Moll coefficient sequence is infinitely log-concave.

Boros–Moll infinite log-concavity conjecture

For each integer $n\ge 1$ , define the Boros--Moll coefficients by $P_n(x)=2^{-2n}\sum_{k=0}^{n}2^k\binom{2n+1}{2k+1}(x+1)^k=\sum_{i=0}^{n}d_i(n)x^i$ . Extend the finite sequence by setting $d_i(n)=0$ for $i\notin\{0,\ldots,n\}$ . Define the log-concavity operator $\mathcal{L}$ by $\mathcal{L}(a)_i=a_i^2-a_{i-1}a_{i+1}$ . The Boros--Moll conjecture asserts that, for every $n\ge 1$ , every integer $r\ge 0$ , and every integer $i$ , $\mathcal{L}^{r}((d_j(n))_{j\in\mathbb{Z}})_i\ge 0$ ; equivalently, the Boros--Moll coefficient sequence is infinitely log-concave.

References

Primary source

arXiv

Additional references

Infinite log-concavity of the Boros--Moll sequences — arXiv — Matthew H. Y. Xie, Philip B. Zhang

combinatorics real-rooted-polynomials

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new unrefereed preprint claims to settle the conjecture, but the proof has not yet been independently verified.

Boros and Moll conjectured that the coefficient sequence of their special-function expansion is infinitely log-concave. Earlier work established only finite levels of log-concavity and related polynomial properties.

Known results

Chen, Dou, and Yang confirmed auxiliary real-rootedness conjectures, implying $2$ - and $3$ -log-concavity.
The 2013 literature established $2$ -log-concavity, while the full $$-log-concavity conjecture remained open.

September 17, 2026 claimed solution

Matthew H. Y. Xie and Philip B. Zhang’s new preprint is reported to prove a stronger real-rootedness and interlacing theorem, then apply Brändén’s preservation theorem to obtain infinite log-concavity. This is a claimed complete resolution, but the preprint is unrefereed and independent verification is absent.

Current status (as of September 2026): Infinite log-concavity is claimed solved by Xie and Zhang, but the new proof remains unverified; the previously established results cover only finite levels.

Sources

No solutions have been posted yet.