Infinite log-concavity of the Boros--Moll sequences

Let $(d_i(n){)}_{i=0}^n$ be the Boros–Moll coefficient sequence. We prove that, for every integer $n\ge 1$, the polynomial $$M_n(x)=\sum _{i=0}^n(d_i(n{)}^2-d_{i-1}(n)d_{i+1}(n))x^i$$ 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.