The Concavity and Convexity of the Boros-Moll Sequences

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 {di(m)}i=0. We show that for any integer m > 6, there exist two positive integers t0(m) and t1(m) such that di(m)+di+2(m) > 2di+1(m) for i ∈ [0, t0(m)] ⋃ [t1(m),m−2] and di(m)+di+2(m) < 2di+1(m) for i ∈ [t0(m)+1, t1(m)−1]. When m is a square, we find t0(m) = m− √ m−4 2 and t1(m) = m+ √ m−2 2 . As a corollary of our results, we show that lim m→+∞ card{i|di(m) + di+2(m) < 2di+1(m), 0 6 i 6 m− 2} √ m = 1.