The Ratio Monotonicity of the Boros-Moll Polynomials

Interlacing Log-concavity of the Boros-Moll Polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which i…

转置Boros-Moll序列的扩展反向超对数凹性

The extended reverse ultra log-concavity of transposed Boros-Moll sequences

本文研究转置Boros-Moll序列的对数凹相关性质，证明其扩展反向超对数凹性，给出对应比值的上下界，得出其对数凹性强于原Boros-Moll序列的结论，并提出相关猜想。

A simple proof of higher order Turán inequalities for Boros-Moll sequences

Recently, the higher order Turán inequalities for the Boros-Moll sequences $\{d_\ell(m)\}_{\ell=0}^m$ were obtained by Guo. In this paper, we show a different approach to this result. Our proof is based on a criterion derived by Hou and Li, which need only checking four simple inequalities related to sufficiently sharp bounds for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$. In order to do so, we adopt the upper bound given by Chen and Gu in studying the reverse ultra log-concavity of Boros-Moll polynomials, and establish a desired lower bound for $d_\ell(m)^2/(d_{\ell-1}(m)d_{\ell+1}(m))$ which…

On the positive zeros of generalized Narayana polynomials related to the Boros-Moll polynomials

The generalized Narayana polynomials $N_{n,m}(x)$ arose from the study of infinite log-concavity of the Boros-Moll polynomials. The real-rootedness of $N_{n,m}(x)$ had been proved by Chen, Yang and Zhang. They also showed that when $n\geq m+2$, each of the generalized Narayana polynomials has one and only one positive zero and $m$ negative zeros, where the negative zeros of $N_{n,m}(x)$ and $N_{n+1,m+1}(x)$ have interlacing relations. In this paper, we study the properties of the positive zeros of $N_{n,m}(x)$ for $n\geq m+2$. We first obtain a new recurrence relation for the generalized Naray…

Boros--Moll 序列的无限对数凹性

Infinite log-concavity of the Boros--Moll sequences

该论文证明了 Boros--Moll 序列的无限对数凹性猜想，通过构造多项式 $M_n(x)$ 并证明其零点与 Narayana 多项式零点严格交错，从而解决了该猜想。

A Proof of Moll's Minimum Conjecture

Let $d_i(m)$ denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence $\{i(i+1)(d_i^2(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ attains its minimum with $i=m$. This conjecture is a stronger than the log-concavity conjecture proved by Kausers and Paule. We give a proof of Moll's conjecture by utilizing the spiral property of the sequence $\{d_i(m)\}_{0\leq i \leq m}$, and the log-concavity of the sequence $\{i!d_i(m)\}_{0\leq i \leq m}$.

Boros-Moll序列的Briggs不等式

The Briggs inequality of Boros-Moll sequences

本文证明了Boros-Moll序列及其规范化序列与转置序列均满足Briggs不等式，前两者利用上下界证明，后者通过建立严格对数比值凸性推导，并得出相关序列的严格对数凸性。

Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x+1), which leads to the log-concavity of P(x+c) for any $c\geq 1$ due to Llamas and Mart\'ınez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.

关于Boros-Moll序列的两个猜想

On Two Conjectures Related to the Boros-Moll Sequences

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。

有理函数空间上的进位与映射

Carries and a map on the space of rational functions

本研究将Boros等人关于有理函数空间映射性质与欧拉多项式关联的工作，和进位马尔可夫链建立联系，为其一项结果提供新证明并稍作推广。

组合序列无限对数凹性的统一方法

A unified approach to infinite log-concavity of combinatorial sequences

针对组合序列无限对数凹性，本文建立了统一的充分条件，将其应用于五类组合序列，证明了相关猜想并回答了问题，结合全纯估计与有限计算机辅助验证完成证明。

Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations

Let $P_{n,k}$ be the number of permutations $π$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $π$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. We propose some conjectures on the log-concavity and q-log-convexity of the polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of $\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higher order log-concavity with respect to $\infty$-q-log-convex or $\infty$-q-log…

Infinite log-concavity: developments and conjectures

Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) "infinitely log-concave" if L^i(a_k) is nonnegative for all i >= 1. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the nth row for all n <= 1450. We also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of in…

On multiple and infinite log-concavity

Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if $\mathcal{L}^j (a_n) \geq 0$ for all $j = 0, 1, \ldots, m$. Here, $\mathcal{L}$ is the operator defined by $\mathcal{L} (a_n) = a_n^2 - a_{n - 1} a_{n + 1}$. By a criterion of Craven--Csordas and McNamara--Sagan it is known that a sequence is $\infty$-log-concave if it satisfies the stronger inequality $a_k^2 \geq r a_{k - 1} a_{k + 1}$ for large enough $r$. On the other hand, a recent result of Brändén shows that $\infty$-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this pap…