Many papers have studied inequalities for Andrews and Paule’s broken k-diamond partition function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta _{k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or 2. In this paper, we derive an exact formula for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta _{k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Building on this result, we also derive an asymptotic formula for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Delta _{k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with an explicit error bound. Using this formula, we prove that for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and sufficiently large n, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta _{k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfies the Turán and Laguerre inequalities of any order and exhibits asymptotic complete monotonicity. Define <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n_k:=\max \left\{ \Big \lceil 8k^{3}+\frac{k+1}{12}\Big \rceil ,526\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo>:</mo> <mo>=</mo> <mo movablelimits="true">max</mo> <mfenced close="}" open="{"> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">⌈</mo> </mrow> <mn>8</mn> <msup> <mi>k</mi> <mn>3</mn> </msup> <mo>+</mo> <mfrac> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>12</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">⌉</mo> </mrow> <mo>,</mo> <mn>526</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we show that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Delta _{k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is log-concave for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ n\ge n_k $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Consequently, it follows that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Delta _{k}(a)\Delta _{k}(b)\ge \Delta _{k}(a+b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$a,b \ge n_k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>≥</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>.

Asymptotics and inequalities for the broken k-diamond partition function

期刊论文
发表日期：

2026年5月20日

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Ying Zhong

摘要

Many papers have studied inequalities for Andrews and Paule’s broken k-diamond partition function $\Delta _{k}(n)$ $Δ_{k} (n)$ when $k=1$ $k = 1$ or 2. In this paper, we derive an exact formula for $\Delta _{k}(n)$ $Δ_{k} (n)$ when $k\ge 1$ $k \geq 1$ . Building on this result, we also derive an asymptotic formula for $\Delta _{k}(n)$ $Δ_{k} (n)$ with an explicit error bound. Using this formula, we prove that for $k\ge 1$ $k \geq 1$ and sufficiently large n, $\Delta _{k}(n)$ $Δ_{k} (n)$ satisfies the Turán and Laguerre inequalities of any order and exhibits asymptotic complete monotonicity. Define $n_k:=\max \left\{ \Big \lceil 8k^{3}+\frac{k+1}{12}\Big \rceil ,526\right\} $ $n_{k} : = max \{⌈ 8 k^{3} + \frac{k + 1}{12} ⌉, 526\}$ . Furthermore, we show that $\Delta _{k}(n)$ $Δ_{k} (n)$ is log-concave for $k\ge 3$ $k \geq 3$ and $ n\ge n_k $ $n \geq n_{k}$ . Consequently, it follows that $\Delta _{k}(a)\Delta _{k}(b)\ge \Delta _{k}(a+b)$ $Δ_{k} (a) Δ_{k} (b) \geq Δ_{k} (a + b)$ for $k\ge 3$ $k \geq 3$ and $a,b \ge n_k$ $a, b \geq n_{k}$ .