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<p>We prove two <i>q</i>-supercongruences modulo the cube of a cyclotomic polynomial. One is a generalization of a supercongruence of the first author and Schlosser, and the other is a generalization of a result of Hu. Meanwhile, these two results may be deemed <i>q</i>-analogues of two supercongruences of Wang and Sun. Our proof makes use of a terminating very-well-poised <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(_6\phi _5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>6</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> summation, the method of creative microscoping devised by the first author and Zudilin, and the Chinese remainder theorem for polynomials.</p>

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Two q-supercongruences from a terminating very-well-poised \(_6\phi _5\) summation

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Victor J. W. Guo,
  • Xing-Ye Zhu

摘要

We prove two q-supercongruences modulo the cube of a cyclotomic polynomial. One is a generalization of a supercongruence of the first author and Schlosser, and the other is a generalization of a result of Hu. Meanwhile, these two results may be deemed q-analogues of two supercongruences of Wang and Sun. Our proof makes use of a terminating very-well-poised \(_6\phi _5\) 6 ϕ 5 summation, the method of creative microscoping devised by the first author and Zudilin, and the Chinese remainder theorem for polynomials.