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<p>In this paper, we study the partition functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\overline{R_\ell ^*}(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <msubsup> <mi>R</mi> <mi>ℓ</mi> <mo>∗</mo> </msubsup> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which count the number of overpartitions of <i>n</i> where the non-overlined parts are <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ell \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-regular for a given <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\ell \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>. Using elementary techniques, as well as the theory of modular forms, we establish several new arithmetic properties, including infinite families of congruences for these functions.</p>

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New Arithmetic Properties for Overpartitions where Nonoverlined Parts are \(\ell \)-Regular

  • 期刊论文

  • 发表日期：

    2026年7月1日

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  • Hemjyoti Nath,
  • Manjil P. Saikia,
  • James A. Sellers

摘要

In this paper, we study the partition functions \(\overline{R_\ell ^*}(n)\) R ℓ ∗ ¯ ( n ) , which count the number of overpartitions of n where the non-overlined parts are \(\ell \) ℓ -regular for a given \(\ell \) ℓ . Using elementary techniques, as well as the theory of modular forms, we establish several new arithmetic properties, including infinite families of congruences for these functions.