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<p>In this paper, we will prove the distinct partition function satisfies the Laguerre inequality of order 2 and the determinantal inequality of order 3 conjectured by Dong and Ji. We also prove the Laguerre inequality of order 3 holds for the distinct partition function. Moreover, for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(4\le m\le 11\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation>, we conjecture the thresholds for the Laguerre inequalities of order <i>m</i> and the positivity of <i>m</i>-order determinants for the distinct partition function.</p>

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Laguerre Inequalities and Determinantal Inequalities for the Distinct Partition Function

  • 期刊论文

  • 发表日期：

    2026年3月4日

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  • Larry X. W. Wang,
  • Eve Y. Y. Yang

摘要

In this paper, we will prove the distinct partition function satisfies the Laguerre inequality of order 2 and the determinantal inequality of order 3 conjectured by Dong and Ji. We also prove the Laguerre inequality of order 3 holds for the distinct partition function. Moreover, for \(4\le m\le 11\) 4 ≤ m ≤ 11 , we conjecture the thresholds for the Laguerre inequalities of order m and the positivity of m-order determinants for the distinct partition function.