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<p>In 2019, the minimal excludant of a partition was defined by Andrews and Newman as the smallest positive integer that is not a part of the concerned partition. In this paper, we address a question asked by Andrews and Newman which is to generalise the concept of the minimal excludant. Further, we extend the results of a paper by Chern and some results related to gap-free partitions. For particular cases, our results have nice connections with Ramanujan’s <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1224_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> function and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1224_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> function. As an unexpected byproduct of this work, we obtain a generalisation of Fine’s identity related to the third order mock theta function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1224_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Psi (q).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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On a generalisation of the minimal excludant function

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • R. Sachdeva

摘要

In 2019, the minimal excludant of a partition was defined by Andrews and Newman as the smallest positive integer that is not a part of the concerned partition. In this paper, we address a question asked by Andrews and Newman which is to generalise the concept of the minimal excludant. Further, we extend the results of a paper by Chern and some results related to gap-free partitions. For particular cases, our results have nice connections with Ramanujan’s \(\psi \) ψ function and \(\phi \) ϕ function. As an unexpected byproduct of this work, we obtain a generalisation of Fine’s identity related to the third order mock theta function \(\Psi (q).\) Ψ ( q ) .