<p>In this paper, based on the authors’ recent work about a novel supercongruence on the truncated Appell series <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1204_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> with its two generalizations, we further establish a stronger generalization of this supercongruence by employing the Chinese remainder theorem. Moreover, inspired by recent research on supercongruences of the four truncated Appell series <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1204_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1204_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1204_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1204_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>, we give two new supercongruences on the truncated Appell series <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1204_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> with their <i>q</i>-analogues with the help of some classical transformation formulas.</p>

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More supercongruences for the truncated Appell series \(F_{1}\) and \(F_{4}\)

  • Xiaoxia Wang,
  • Wenjie Yu

摘要

In this paper, based on the authors’ recent work about a novel supercongruence on the truncated Appell series \(F_1\) F 1 with its two generalizations, we further establish a stronger generalization of this supercongruence by employing the Chinese remainder theorem. Moreover, inspired by recent research on supercongruences of the four truncated Appell series \(F_1\) F 1 , \(F_2\) F 2 , \(F_3\) F 3 , \(F_4\) F 4 , we give two new supercongruences on the truncated Appell series \(F_4\) F 4 with their q-analogues with the help of some classical transformation formulas.