Geodesic
Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 184-198

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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. 
@article{ZNSL_2012_404_a10,
     author = {D. B. Karp},
     title = {Positivity of {Toeplitz} determinants formed by rising factorial series and properties of related polynomials},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {184--198},
     publisher = {mathdoc},
     volume = {404},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a10/}
}
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D. B. Karp. Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 184-198. http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a10/