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<p>Alladi interpreted the summation side of the first Rogers–Ramanujan identity as the generating function for the partitions which have no nodes below the Durfee square. These were named as primary partitions. We interpret the sum side of the first Rogers–Ramanujan identity as the generating function for the partitions which have no nodes to the right of the Durfee square. Although the idea looks quite similar to the Alladi’s interpretation but it gives rise to an important class of partitions which are characterized by the fact that all the parts up to the size of Durfee square are equal. We name these as <i>R</i>-partitions. These partitions along with another similar class of partitions named as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_834_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(R^\prime \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-partitions provide very elegant interpretations for Ramanujan’s seventh-order Mock theta functions. These interpretations include the <i>R</i>-partitions and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_834_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(R^{\prime }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-partitions with rank 0. This fact acts as a stimulus to obtain the generating function for <i>R</i>-partitions with crank 0, which in turn provides us an interesting identity.</p>

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A new class of partitions for seventh-order Mock theta functions

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Shruti Sharma

摘要

Alladi interpreted the summation side of the first Rogers–Ramanujan identity as the generating function for the partitions which have no nodes below the Durfee square. These were named as primary partitions. We interpret the sum side of the first Rogers–Ramanujan identity as the generating function for the partitions which have no nodes to the right of the Durfee square. Although the idea looks quite similar to the Alladi’s interpretation but it gives rise to an important class of partitions which are characterized by the fact that all the parts up to the size of Durfee square are equal. We name these as R-partitions. These partitions along with another similar class of partitions named as \(R^\prime \) R ′ -partitions provide very elegant interpretations for Ramanujan’s seventh-order Mock theta functions. These interpretations include the R-partitions and \(R^{\prime }\) R ′ -partitions with rank 0. This fact acts as a stimulus to obtain the generating function for R-partitions with crank 0, which in turn provides us an interesting identity.