Let $(d_i(n))_{i=0}^n$ be the Boros--Moll coefficient sequence. We prove that, for every integer $n\ge1$, the polynomial \[ M_n(x)=\sum_{i=0}^n \bigl(d_i(n)^2-d_{i-1}(n)d_{i+1}(n)\bigr)x^i \] has only simple negative zeros, which strictly interlace those of the Narayana polynomial of the same degree. This proves a conjecture of Chen, Yang, and Zhang and, by Br\"and\'en's preservation theorem, settles the infinite log-concavity conjecture of Boros and Moll. The proof uses an expansion of the reversed and normalized form of $M_n(x)$ in derivatives of the Narayana polynomial, together with estimates for the weights and partial sums of the normalized derivatives.
We establish a sufficient condition for infinite log-concavity and apply it to five families of combinatorial sequences. We give a new proof of infinite log-concavity for the Boros--Moll coefficient sequences $(d_k(n))_{k=0}^n$ for fixed $n$. For the transposed sequences $(d_\ell(\ell+k))_{k\ge0}$, we prove Zhao's conj...
Matthew H. Y. Xie, Candice X. T. Zhang, Philip B. Zhang· 0 citations
Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants govern...
For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{\mu(n)\omega(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\varepsilon\bigl((\log\log X)^{5/3+\varepsilon}/\log X\bigr)...
For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusi...
Omkar Baraskar, Prashant Gokhale, Sarvagya Jain et al.· 1 citation
Let $\nu_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+\nu_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};q^{2^{r+1}r}\big...
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-conca...
Heshan Aravinda· 0 citations
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