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A family of $ q $-congruences modulo the square of a cyclotomic polynomial

  • Received: 01 January 2020 Revised: 01 April 2020
  • Primary: 33D15; Secondary: 11A07, 11B65

  • Using Watson's terminating $ _8\phi_7 $ transformation formula, we prove a family of $ q $-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo $ p^4 $ ($ p $ is an odd prime) and their $ q $-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

    Citation: Victor J. W. Guo. A family of $ q $-congruences modulo the square of a cyclotomic polynomial[J]. Electronic Research Archive, 2020, 28(2): 1031-1036. doi: 10.3934/era.2020055

    Related Papers:

  • Using Watson's terminating $ _8\phi_7 $ transformation formula, we prove a family of $ q $-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo $ p^4 $ ($ p $ is an odd prime) and their $ q $-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.



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