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<p>The study of overpartitions in recent years has been used to great effect in various fields, including hypergeometric series, <i>q</i>-series identities, and mathematical physics. We investigate the limiting distributions of the number of parts in a family of overpartitions of <i>n</i>,&#xa0; introduced by Andrews, where parts are counted with two different weights. Using Andrews’ identities and the saddle-point method, we establish two central limit theorems (CLTs) for the number of parts as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\rightarrow \infty ,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> corresponding to these weightings. We also derive explicit formulas for the mean and variance in each case.</p>

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Limit theorems for Andrews’ restricted overpartitions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月17日

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  • Tapas Bhowmik,
  • Alex Cao,
  • Jack Frew,
  • John Lehman,
  • Wei-Lun Tsai

摘要

The study of overpartitions in recent years has been used to great effect in various fields, including hypergeometric series, q-series identities, and mathematical physics. We investigate the limiting distributions of the number of parts in a family of overpartitions of n,  introduced by Andrews, where parts are counted with two different weights. Using Andrews’ identities and the saddle-point method, we establish two central limit theorems (CLTs) for the number of parts as \(n\rightarrow \infty ,\) n → ∞ , corresponding to these weightings. We also derive explicit formulas for the mean and variance in each case.