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May 9, 2026International Journal of Number Theory0 citations

Real-rootedness of quartic Jensen polynomials associated with pod ( n )

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GDGuosi DengFLFrank Z. K. LiELEdward Y. S. Liu

Key Points

  • The aim is to demonstrate that quartic Jensen polynomials associated with partitions have four distinct real roots for all integers.
  • Established inequalities for quartic Jensen polynomials using arbitrary-precision error estimates.
  • Confirmed positivity of a class of third-order determinants related to these polynomials.
  • Generalized results on third-order Toeplitz matrices associated with specific partitions.
  • The quartic Jensen polynomial consistently has four distinct real roots for all integers.
  • Inequalities derived showed positive bounds for the associated partitions.
  • Positivity was confirmed for third-order determinants, corroborating the results for Toeplitz matrices.

Abstract

Let Formula: see text denote the number of partitions of Formula: see text into distinct odd parts. In this paper, we prove that the quartic Jensen polynomial associated with Formula: see text: Formula: see text has four distinct real roots for all Formula: see text. To achieve this, we establish several inequalities for Formula: see text by utilizing its arbitrary-precision error estimates. Furthermore, we confirm the positivity of a class of third-order determinants with entries given by Formula: see text, which can be viewed as a generalization of previous results on third-order Toeplitz matrices related to Formula: see text.

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Cite This Study

Deng et al. (2026) studied this question.

synapsesocial.com/papers/69fecfe9b9154b0b82876d8ahttps://doi.org/10.1142/s1793042126500983
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