Proof of Moll’s minimum conjecture

Chen, William Y. C.; Xia, Ernest X. W.

doi:10.1016/j.ejc.2012.11.002

Cited by 7 publications

(7 citation statements)

References 10 publications

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“…Recently, Chen and Xia [13] showed that the sequence {d i (m)} 0≤i≤m satisfies the strongly ratio monotone property which implies the logconcavity and the spiral property. They [14] also confirmed a conjecture of Moll which says that…”

Section: Introductionsupporting

confidence: 71%

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2-Log-Concavity of the Boros–Moll Polynomials

Chen¹,

Xia²

2013

Proceedings of the Edinburgh Mathematical Society

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The Boros-Moll polynomials P m (a) arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that P m (a) is 2-log-concave for any m ≥ 2. Let d i (m) be the coefficient of a i in P m (a). We also show that the sequence {i (i+1)This leads another proof of Moll's minimum conjecture.As will be seen, the 2-log-concavity of P m (a) implies the log-concavity of a sequence considered by Moll [18,21].Since log-concavity implies unimodality, the above property leads to another proof of Moll's minimum conjecture [21] for the sequence {i(i+1)(d 2 i (m)−d i−1 (m)d i+1 (m)} 1≤i≤m . By comparing the first entry with the last entry, we deduce that this sequence attains its minimum at i = m which equals 2 −2m m(m+1) 2m m 2 . This conjecture was confirmed by Chen and Xia [14] by using a result of Chen and Gu [9] and the spiral property of the Boros-Moll sequences [13].

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Section: Introductionsupporting

confidence: 71%

“…By comparing the first entry with the last entry, we deduce that this sequence attains its minimum at i = m, which equals 2 −2m m(m + 1) 2m m 2 . This conjecture was confirmed by Chen and Xia [12] by using a result of Chen and Gu [10] and the spiral property of the Boros-Moll sequences [11].…”

Section: Theorem 14 the Boros-moll Sequences Arementioning

confidence: 55%

2-Log-Concavity of the Boros–Moll Polynomials

Chen¹,

Xia²

2013

Proceedings of the Edinburgh Mathematical Society

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“…Chen and Xia [13] proved that the Boros-Moll sequences satisfy the ratio monotone property which implies the log-concavity and the spiral property. They [14] also confirmed a conjecture given by Moll in [19]. By constructing an intermediate function, Chen and Xia [15] proved the 2-log-concavity of the Boros-Moll sequences.…”

Section: Introduction and Main Resultssupporting

confidence: 79%

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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“…Recently, Chen and Xia [ 4 ] showed that the sequence satisfies the strongly ratio monotone property which implies the log-concavity and the spiral property. They [ 5 ] also confirmed a conjecture of Moll which says that attains its minimum at . Chen et al [ 6 ] proved that the Boros-Moll sequences are interlacing log-concave.…”

Section: Introduction and Main Resultssupporting

confidence: 53%