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<p>Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\overline{R_{\ell ,\mu }}(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <msub> <mi>R</mi> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which enumerates the overpartitions of <i>n</i>,&#xa0; where no part is divisible by either <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\ell \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, for various integer pairs <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((\ell , \mu )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical <i>q</i>-series manipulations and dissection formulas.</p>

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Extending recent congruence results on \((\ell ,\mu )\)-regular overpartitions

  • 期刊论文

  • 发表日期：

    2025年10月21日

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  • Bishnu Paudel,
  • James A. Sellers,
  • Haiyang Wang

摘要

Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function \(\overline{R_{\ell ,\mu }}(n)\) R ℓ , μ ¯ ( n ) , which enumerates the overpartitions of n,  where no part is divisible by either \(\ell \) ℓ or \(\mu \) μ , for various integer pairs \((\ell , \mu )\) ( ℓ , μ ) . In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical q-series manipulations and dissection formulas.