Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials

Details Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials

2012-03-07

arxiv.org/abs/1203.1482v1

Mathematics

Classical Analysis and ODEs

11 pages; no figures

Scientific paper

In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.

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