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<p>In 1997, Van Hamme proposed 13 supercongruences corresponding to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1/\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> series of the Ramanujan-type. Inspired by the recent work of V.J.W. Guo, we establish a unified <i>q</i>-analogue of Van Hamme’s (B.2), (E.2) and (F.2) supercongruences, which is also a refinement of some known results obtained by different authors. As a consequence, we give a parametric supercongruence related to the Euler polynomials.</p>

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A refined q-analogue of some congruences of Van Hamme

  • 期刊论文

  • 发表日期：

    2026年4月27日

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  • Chen Wang,
  • Yu-Chan Tian,
  • Kai Huang

摘要

In 1997, Van Hamme proposed 13 supercongruences corresponding to \(1/\pi \) 1 / π series of the Ramanujan-type. Inspired by the recent work of V.J.W. Guo, we establish a unified q-analogue of Van Hamme’s (B.2), (E.2) and (F.2) supercongruences, which is also a refinement of some known results obtained by different authors. As a consequence, we give a parametric supercongruence related to the Euler polynomials.