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

In this note, we prove the positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally in studying log-concavity of rising factorial series. Several conjectures concerning zeros and coefficients of a generalized form of those polynomials are advanced. Also polynomials whose generating functions are higher-order Toeplitz determinants formed by rising factorial series are considered. Three conjectures about these polynomials are made. All conjectures proposed are supported by numerical evidence. Bibliography: 16 titles.

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Correspondence to D. B. Karp.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 184–198.

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Karp, D.B. Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials. J Math Sci 193, 106–114 (2013). https://doi.org/10.1007/s10958-013-1438-y

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  • DOI: https://doi.org/10.1007/s10958-013-1438-y

Keywords

  • Generate Function
  • Numerical Evidence
  • Factorial Series
  • Related Polynomial
  • Toeplitz Determinant