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<p>Applying a very-well-poised <sub>6</sub><InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1968_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi _{5}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϕ</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation> summation formula, we prove several <i>q</i>-supercongruences with a parameter <i>s</i> in this paper. As a corollary, we obtain a new <i>q</i>-generalization of the (E.2) supercongruence of Van Hamme. In addition, we also present two families of Ramanujan-type series on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1968_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>.</p>

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A Further q-generalization of the (E.2) Supercongruence of Van Hamme

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • He-Xia Ni,
  • Su-Dan Wang

摘要

Applying a very-well-poised 6 \(\phi _{5}\) ϕ 5 summation formula, we prove several q-supercongruences with a parameter s in this paper. As a corollary, we obtain a new q-generalization of the (E.2) supercongruence of Van Hamme. In addition, we also present two families of Ramanujan-type series on \(\pi \) π .