Infinite log-concavity of the Boros--Moll sequences

Abstract

Let $(d_i(n))_{i=0}^n$ be the Boros--Moll coefficient sequence. We prove that, for every integer $n\ge1$, the polynomial \[ M_n(x)=\sum_{i=0}^n \bigl(d_i(n)^2-d_{i-1}(n)d_{i+1}(n)\bigr)x^i \] has only simple negative zeros, which strictly interlace those of the Narayana polynomial of the same degree. This proves a conjecture of Chen, Yang, and Zhang and, by Brändén's preservation theorem, settles the infinite log-concavity conjecture of Boros and Moll. The proof uses an expansion of the reversed and normalized form of $M_n(x)$ in derivatives of the Narayana polynomial, together with estimates for the weights and partial sums of the normalized derivatives.