For integers m≥ 4 and 0≤ i≤ m, define the Boros–Moll numbers by d_i(m)=2^-2m∑_k=i^m2^k binom2m-2km-kbinomm+kkbinomki, and define u_i(m)=dfracd_i-1(m)d_i+1(m)d_i(m)^2. Chen and Gu conjectured that, for every m, the sequence u_i(m)_2≤ i≤ m-2 is log-concave, namely u_i(m)^2≥ u_i-1(m)u_i+1(m) for 3≤ i≤ m-3, and reverse ultra-log-concave, namely the normalized sequence dfracu_i(m)binomm-4i-2_2≤ i≤ m-2 is log-convex, equivalently (dfracu_i(m)binomm-4i-2)^2≤dfracu_i-1(m)binomm-4i-3dfracu_i+1(m)binomm-4i-1 for 3≤ i≤ m-3.

Chen–Gu conjectures on Boros–Moll sequences

For integers $m\geq 4$ and $0\leq i\leq m$ , define the Boros–Moll numbers by $d_i(m)=2^{-2m}\sum_{k=i}^{m}2^k\binom{2m-2k}{m-k}\binom{m+k}{k}\binom{k}{i}$ , and define $u_i(m)=\dfrac{d_{i-1}(m)d_{i+1}(m)}{d_i(m)^2}$ . Chen and Gu conjectured that, for every $m$ , the sequence $\{u_i(m)\}_{2\leq i\leq m-2}$ is log-concave, namely $u_i(m)^2\geq u_{i-1}(m)u_{i+1}(m)$ for $3\leq i\leq m-3$ , and reverse ultra-log-concave, namely the normalized sequence $\left\{\dfrac{u_i(m)}{\binom{m-4}{i-2}}\right\}_{2\leq i\leq m-2}$ is log-convex, equivalently $\left(\dfrac{u_i(m)}{\binom{m-4}{i-2}}\right)^2\leq\dfrac{u_{i-1}(m)}{\binom{m-4}{i-3}}\dfrac{u_{i+1}(m)}{\binom{m-4}{i-1}}$ for $3\leq i\leq m-3$ .

References

Primary source

arXiv

Additional references

On Two Conjectures Related to the Boros-Moll Sequences — arXiv

combinatorics discrete-dynamical-systems

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A September 2026 preprint claims to settle one conjecture and prove the other for all sufficiently large cases, but the claim has not been independently verified and some small cases remain open.

Chen and Gu posed two conjectures for the ratio sequence $\left\{d_{i+1}(m)d_{i-1}(m)/d_i(m)^2\right\}_{2\leq i\leq m-2}$ : ordinary log-concavity and reverse ultra-log-concavity. The original paper presented both as conjectures, not theorems.

Known results

Chen and Gu established upper and lower bounds for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$ .
A 2024 paper proved strict extended reverse ultra-log-concavity for the transposed sequences $\{d_\ell(m)\}_{m\geq\ell}$ , but addressed different sequences rather than proving the original conjectures.

September 2026 claimed resolution

On September 9, 2026, a report linked the preprint On Two Conjectures Related to the Boros–Moll Sequences, claiming the reverse ultra-log-concavity conjecture in full and ordinary log-concavity asymptotically. The finite ordinary-log-concavity cases outside that range remain open, and the claimed resolution is unverified.

Current status (as of September 2026): Reverse ultra-log-concavity is claimed in full but unverified; ordinary log-concavity is claimed asymptotically, while small indices remain open.

Sources

No solutions have been posted yet.