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<p>We establish that specific truncated basic hypergeometric series contain the factor <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_619_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Phi _n(q)^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>n</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_619_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Phi _n(q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the <i>n</i>-th cyclotomic polynomial. This finding can be interpreted as an extension of a result of Guo [‘Some <i>q</i>-supercongruences from Gasper’s Karlsson–Minton type summation’, Ramanujan J 60:825–835, 2023].</p>

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A q-supercongruence from finite q-binomial theorem

  • 期刊论文

  • 发表日期：

    2025年2月10日

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  • Sipra Maity,
  • Rupam Barman

摘要

We establish that specific truncated basic hypergeometric series contain the factor \(\Phi _n(q)^2\) Φ n ( q ) 2 , where \(\Phi _n(q)\) Φ n ( q ) is the n-th cyclotomic polynomial. This finding can be interpreted as an extension of a result of Guo [‘Some q-supercongruences from Gasper’s Karlsson–Minton type summation’, Ramanujan J 60:825–835, 2023].