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<p>Let <i>p</i>(<i>n</i>) denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order <i>N</i> (for any fixed positive integer <i>N</i>) along with estimates for error bounds for the shifted quotient of the partition function, namely <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p(n+k)/p(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p(n+k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the multiplicative inverse 1/<i>p</i>(<i>n</i>), which is of independent interest.</p>

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Asymptotics for the reciprocal and shifted quotient of the partition function

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月13日

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  • Koustav Banerjee,
  • Peter Paule,
  • Cristian-Silviu Radu,
  • Carsten Schneider

摘要

Let p(n) denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order N (for any fixed positive integer N) along with estimates for error bounds for the shifted quotient of the partition function, namely \(p(n+k)/p(n)\) p ( n + k ) / p ( n ) with \(k\in \mathbb {N}\) k ∈ N , which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version \(p(n+k)\) p ( n + k ) and the multiplicative inverse 1/p(n), which is of independent interest.