Brändén's conjectures on the Boros-Moll polynomials (Q2878679)
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scientific article; zbMATH DE number 6340082
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Brändén's conjectures on the Boros-Moll polynomials |
scientific article; zbMATH DE number 6340082 |
Statements
4 September 2014
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Boros-Moll polynomials
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log-concavity
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Sturm sequences
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Brändén's conjectures on the Boros-Moll polynomials (English)
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Let \(\{d_i(n)\}_{i=0}^n\) be the Boros-Moll sequence. In [J. Reine Angew. Math. 658, 115--131 (2011; Zbl 1278.30007)], \textit{P. Brändén} conjectured that the polynomials \(Q_n(x)=\sum_{i=0}^n \frac{d_i(n)}{i!}x^i\) and \(R_n(x)=\sum_{i=0}^n \frac{d_i(n)}{(i+2)!}x^i\) have only real zeros. This implies the 2-log-concavity and the 3-log-concavity of \(\{d_i(n)\}\). The authors prove both conjectures by showing that \(Q_n(x)\) and \(R_n(x)\) form Sturm sequences.
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