The Concavity and Convexity of the Boros-Moll Sequences

Abstract

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
 \[
\lim_{m\rightarrow +\infty }\frac{{\rm card}\{i|d_i(m)+d_{i+2}(m)< 2d_{i+1}(m), 0\leq i \leq m-2\}}{\sqrt{m}}=1.
 \]