On Two Conjectures Related to the Boros-Moll Sequences
The Boros-Moll sequences $\{d_i(m)\}_{0\leq i\leq m}$ are defined as $$d_i(m)=2^{-2m}\sum_{k=i}^m 2^k \binom{2m-2k}{m-k}\binom{m+k}{k}\binom{k}{i}.$$ Consider the ratio sequence $$u_i(m)=\frac{d_{i-1}(m)d_{i+1}(m)}{d_i(m)^2}.$$ Chen and Gu conjectured that $\{u_i(m)\}_{2\leq i \leq m-2}$ is both reverse ultra log-concave and log-concave. In this paper, we prove the reverse ultra log-concavity conjecture using bounds of Chen-Gu and Zhao, and prove the log-concavity conjecture asymptotically by showing that $\{u_i(m)\}_{2\leq i \leq m-2}$ is strictly log-concave for all sufficiently large $m$. The key ingredient in the latter result is a recurrence of Kauers and Paule, which we interpret as a nonlinear discrete dynamical system through a backward map. We construct an approximation to the ratio sequence using its stable limiting fixed point and combine localization and contraction arguments with finite-difference estimates and separate interior and edge analyses to obtain the desired strict log-concavity.