{"ID":23620899,"CreatedAt":"2026-09-18T06:47:37.825099056Z","UpdatedAt":"2026-09-18T06:47:37.825099056Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20653","arxiv_id":"2609.20653","title":"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.","short_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 conj...","url_abs":"https://arxiv.org/abs/2609.20653","url_pdf":"https://arxiv.org/pdf/2609.20653v1","authors":"[\"Matthew H. Y. Xie\",\"Philip B. Zhang\"]","published":"2026-09-17T16:31:27Z","proceeding":"math.CO","tasks":"[\"math.CO\"]","methods":"[]","has_code":false}