The unimodality of a polynomial coming from a rational integral. Back to the original proof

Amdeberhan, Tewodros; Dixit, Atul; Guan, Xiao Wei; Jiu, Lin; Moll, Victor H.

doi:10.1016/j.jmaa.2014.06.006

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2015

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Cited by 2 publications

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“…They also established the unimodality of the sequence {d i (m)} m i=0 by taking a different approach [6]. Amdeberhan, Dixit, Guan, Jiu and Moll [1] presented another proof of (8). Amdeberhan, Manna and Moll [2] analyzed properties of the 2-adic valuation of an the electronic journal of combinatorics 22(1) (2015), #P1.8…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Concavity and Convexity of the Boros-Moll Sequences

Xia

2015

Electron. J. Combin.

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In their study of a quartic integral, Boros and Moll discovered a special class of sequences, which is called the Boros–Moll sequences. In this paper, we consider the concavity and convexity of the Boros–Moll sequences $\{d_i(m)\}_{i=0}^m$. We show that for any integer $m\geq 6$, there exist two positive integers $t_0(m)$ and $t_1(m)$ such that $d_i(m)+d_{i+2}(m)>2d_{i+1}(m)$ for $i\in [0,t_0(m)]\bigcup[t_1(m),m-2]$ and $d_i(m)+d_{i+2}(m) < 2d_{i+1}(m)$ for $i\in [t_0(m)+1,t_1(m)-1]$. When $m$ is a square, we find $t_0(m)=\frac{m-\sqrt{m}-4}{2}$ and $t_1(m) =\frac{m+\sqrt{m}-2}{2}$. As a corollary of our results, we show that \[\lim_{m\rightarrow +\infty }\frac{{\rm card}\{i|d_i(m)+d_{i+2}(m)< 2d_{i+1}(m), 0\leq i \leq m-2\}}{\sqrt{m}}=1. \]

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Section: Introduction and Main Resultsmentioning

confidence: 99%