A new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\)


Using Andrews’ multiseries generalization of Watson’s \(_8\phi _7\) transformation, we give a new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\), as well as its q-analogue. Meanwhile, applying the method of ‘creative microscoping’, recently introduced by the author and Zudilin, we establish some further q-supercongruences modulo \(\Phi _n(q)^3\), where \(\Phi _n(q)\) denotes the nth cyclotomic polynomial in q.

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We thank the anonymous referee for valuable comments.

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Correspondence to Victor J. W. Guo.

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The author was partially supported by the National Natural Science Foundation of China (Grant 11771175)

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Guo, V.J.W. A new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\pmod {4}\). Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00369-5

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  • Cyclotomic polynomial
  • q-Congruence
  • Supercongruence
  • Andrews’ transformation
  • Watson’s transformation

Mathematics Subject Classification

  • 33D15
  • 11A07
  • 11B65