关于Boros-Moll序列的两个猜想
On Two Conjectures Related to the Boros-Moll Sequences
AI总结：
本文利用Chen-Gu和Zhao的界证明Boros-Moll序列比率序列的反向超对数凹猜想，并通过动力系统方法渐近证明其对数凹猜想。
AI中文摘要：
Boros-Moll序列$\{d_i(m)\}_{0\leq i\leq m}$定义为$$d_i(m)=2^{-2m}\sum_{k=i}^m 2^k \binom{2m-2k}{m-k}\binom{m+k}{k}\binom{k}{i}.$$考虑比率序列$$u_i(m)=\frac{d_{i-1}(m)d_{i+1}(m)}{d_i(m)^2}.$$Chen和Gu猜想$\{u_i(m)\}_{2\leq i \leq m-2}$既是反向超对数凹的又是对数凹的。在本文中，我们利用Chen-Gu和Zhao的界证明了反向超对数凹猜想，并通过证明对所有充分大的$m$，$\{u_i(m)\}_{2\leq i \leq m-2}$严格对数凹，渐近地证明了对数凹猜想。后一结果的关键要素是Kauers和Paule的一个递推关系，我们将其解释为通过一个后向映射的非线性离散动力系统。我们利用其稳定极限不动点构造比率序列的近似，并结合局部化和压缩论证与有限差分估计，以及分别分析内部和边缘情况，获得所需的严格对数凹性。
英文摘要：
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.
