The concavity and convexity of the Boros-Moll sequences
Summary: In their study of a quartic integral, \textit{G. Boros} and \textit{V. H. Moll} [J. Comput. Appl. Math. 130, No. 1--2, 337--344 (2001; Zbl 1011.33005)] 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 \[ \lim_{m\to +\infty }\frac{\mathrm{card}\{i\mid d_i(m)+d_{i+2}(m)< 2d_{i+1}(m), 0\leq i \leq m-2\}}{\sqrt{m}}=1. \]
scientific article; zbMATH DE number 6389247
scientific article