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<p>We prove the conjecture of Pore and Fathima on congruences modulo 20 for Andrews’ partition function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_725_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline{\mathcal{E}\mathcal{O}}(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="script">E</mi> <mi mathvariant="script">O</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> using elementary <i>q</i>-series manipulations and 5-dissections. We also obtain an infinite family of congruences modulo 20 for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_725_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline{\mathcal{E}\mathcal{O}}(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="script">E</mi> <mi mathvariant="script">O</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A conjecture of Pore and Fathima on congruences modulo 20 for Andrews’ partition function

  • 期刊论文

  • 发表日期：

    2025年3月4日

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  • Russelle Guadalupe

摘要

We prove the conjecture of Pore and Fathima on congruences modulo 20 for Andrews’ partition function \(\overline{\mathcal{E}\mathcal{O}}(n)\) E O ¯ ( n ) using elementary q-series manipulations and 5-dissections. We also obtain an infinite family of congruences modulo 20 for \(\overline{\mathcal{E}\mathcal{O}}(n)\) E O ¯ ( n ) .