<p>We present three <i>q</i>-supercongruences modulo the fifth power of a cyclotomic polynomial by using Jackson’s <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2804_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(_8\phi _7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>8</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>7</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> summation and Watson’s <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2804_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(_8\phi _7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>8</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>7</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> transformation, together with the creative microscoping method introduced by Guo and Zudilin (Adv Math 346:329–358, 2019). As conclusions, we give a partial <i>q</i>-analogue of a supercongruence of Barman and Saikia, and a complete <i>q</i>-analogue of the supercongruence: <Equation ID="Equ40"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2804_Article_Equ40.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="395" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \sum _{k=0}^{(p-1)/4}(16k+1)\frac{(\frac{1}{8})_{k}(\frac{1}{4})_{k}^{5}}{k!(\frac{7}{8})_{k}^{5}}\equiv -\frac{5p^3}{64}\Gamma _p(\tfrac{7}{8})^{6} \Gamma _p(\tfrac{3}{8})^{10} \pmod {p^5}, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </munderover> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msub> <msubsup> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> </mrow> <mn>5</mn> </msubsup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <msubsup> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>7</mn> <mn>8</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> </mrow> <mn>5</mn> </msubsup> </mrow> </mfrac> <mo>≡</mo> <mo>-</mo> <mfrac> <mrow> <mn>5</mn> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mn>64</mn> </mfrac> <msub> <mi mathvariant="normal">Γ</mi> <mi>p</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>8</mn> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </msup> <msub> <mi mathvariant="normal">Γ</mi> <mi>p</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> <mn>10</mn> </msup> <mspace width="10.0pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <msup> <mi>p</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2804_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\equiv 5\pmod {8}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>5</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a prime, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2804_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\((x)_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> is the Pochhammer symbol, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2804_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma _p(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the <i>p</i>-adic Gamma function.</p>

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Three New q-Supercongruences from Jackson’s Summation and Watson’s Transformation

  • Victor J. W. Guo,
  • Xing-Ye Zhu

摘要

We present three q-supercongruences modulo the fifth power of a cyclotomic polynomial by using Jackson’s \(_8\phi _7\) 8 ϕ 7 summation and Watson’s \(_8\phi _7\) 8 ϕ 7 transformation, together with the creative microscoping method introduced by Guo and Zudilin (Adv Math 346:329–358, 2019). As conclusions, we give a partial q-analogue of a supercongruence of Barman and Saikia, and a complete q-analogue of the supercongruence: \(\begin{aligned} \sum _{k=0}^{(p-1)/4}(16k+1)\frac{(\frac{1}{8})_{k}(\frac{1}{4})_{k}^{5}}{k!(\frac{7}{8})_{k}^{5}}\equiv -\frac{5p^3}{64}\Gamma _p(\tfrac{7}{8})^{6} \Gamma _p(\tfrac{3}{8})^{10} \pmod {p^5}, \end{aligned}\) k = 0 ( p - 1 ) / 4 ( 16 k + 1 ) ( 1 8 ) k ( 1 4 ) k 5 k ! ( 7 8 ) k 5 - 5 p 3 64 Γ p ( 7 8 ) 6 Γ p ( 3 8 ) 10 ( mod p 5 ) , where \(p\equiv 5\pmod {8}\) p 5 ( mod 8 ) is a prime, \((x)_k\) ( x ) k is the Pochhammer symbol, and \(\Gamma _p(x)\) Γ p ( x ) is the p-adic Gamma function.