Scholarlink

<p>We give some new <i>q</i>-supercongruences modulo the third, fourth and fifth powers of a cyclotomic polynomial, respectively. Two of them are partial <i>q</i>-analogues of supercongruences of Deines et al., and another one is a <i>q</i>-analogue of the following supercongruence due to Pan, Tauraso and Wang: for primes <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\equiv 2\pmod {5}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>2</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <Equation ID="Equ38"> <EquationSource Format="TEX">\(\begin{aligned} \sum _{k=0}^{p-1}(-1)^k {-\frac{2}{5}\atopwithdelims ()k}^5 \equiv \frac{p}{10}\Gamma _p(\tfrac{1}{5})^5\Gamma _p\big (\tfrac{2}{5}\big )^5 \pmod {p^5}, \end{aligned}\)</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> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> <msup> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mrow> <mo>-</mo> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mi>k</mi> </mfrac> </mfenced> <mn>5</mn> </msup> <mo>≡</mo> <mfrac> <mi>p</mi> <mn>10</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>1</mn> <mn>5</mn> </mfrac> </mstyle> <mo stretchy="false">)</mo> </mrow> <mn>5</mn> </msup> <msub> <mi mathvariant="normal">Γ</mi> <mi>p</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mstyle> <msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mn>5</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="IEq4"> <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> denotes the <i>p</i>-adic Gamma function. Our proof is mainly based on Jackson’s <InlineEquation ID="IEq5"> <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 the “creative microscoping method” developed in [Adv. Math. 346 (2019), 329–358].</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Some new q-supercongruences from Jackson’s \(_8\phi _7\) summation

  • 期刊论文

  • 发表日期：

    2025年12月8日

查看全文
  • Victor J. W. Guo

摘要

We give some new q-supercongruences modulo the third, fourth and fifth powers of a cyclotomic polynomial, respectively. Two of them are partial q-analogues of supercongruences of Deines et al., and another one is a q-analogue of the following supercongruence due to Pan, Tauraso and Wang: for primes \(p\equiv 2\pmod {5}\) p ≡ 2 ( mod 5 ) , \(\begin{aligned} \sum _{k=0}^{p-1}(-1)^k {-\frac{2}{5}\atopwithdelims ()k}^5 \equiv \frac{p}{10}\Gamma _p(\tfrac{1}{5})^5\Gamma _p\big (\tfrac{2}{5}\big )^5 \pmod {p^5}, \end{aligned}\) ∑ k = 0 p - 1 ( - 1 ) k - 2 5 k 5 ≡ p 10 Γ p ( 1 5 ) 5 Γ p ( 2 5 ) 5 ( mod p 5 ) , where \(\Gamma _p(x)\) Γ p ( x ) denotes the p-adic Gamma function. Our proof is mainly based on Jackson’s \(_8\phi _7\) 8 ϕ 7 summation and the “creative microscoping method” developed in [Adv. Math. 346 (2019), 329–358].