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Two parametric q-supercongruences from Andrews' multi-series transformation

  • Liton Karmakar
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Abstract

Recently, Guo and Guo (Results Math $76:37, ~2021$) obtained curious q-analogues of the following two supercongruences conjectured by Guo and Schlosser \cite[Conjectures $12.6$ and $12.7$]{GuoSclssr ConstApprox 21}: ∑n=0(p−1)/3 (6n+1) (1/3)n4(2n)!/(n!)4(2/3)2n ≡ p pmodp3 for primes p≡ 1 pmod3, ∑n=0(p+1)/3 (6n−1) ((−1/3)n4(2n)!)/((n!)4(−2/3)2n) ≡ p pmodp3 for primes p≡ 2 pmod3, where (z)n=z(z+1)⋯(z+n−1) denotes the shifted factorial. In this paper, we establish a one-parameter extension of them. The main ingredients of our proofs are the creative microscoping method developed by Guo and Zudilin \cite{guo2019q} and Andrews' multi-series transformation.

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Cite

BibTeXRIS
BibTeX
@misc{karmakar2026parametric,
  title = {{Two parametric $q$-supercongruences from Andrews' multi-series transformation}},
  author = {Liton Karmakar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33249},
  eprint = {2610.33249},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33249},
}
RIS
TY  - UNPB
TI  - Two parametric q-supercongruences from Andrews' multi-series transformation
AU  - Liton Karmakar
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.33249
UR  - https://arxiv.org/abs/2610.33249
AB  - Recently, Guo and Guo (Results Math $76:37, ~2021$) obtained curious q-analogues of the following two supercongruences conjectured by Guo and Schlosser \cite[Conjectures $12.6$ and $12.7$]{GuoSclssr ConstApprox 21}: ∑_n=0^(p−1)/3 (6n+1) (1/3)_n^4(2n)!/(n!)^4(2/3)_2n ≡ p pmodp^3 for primes p≡ 1 pmod3, ∑_n=0^(p+1)/3 (6n−1) ((−1/3)_n^4(2n)!)/((n!)^4(−2/3)_2n) ≡ p pmodp^3 for primes p≡ 2 pmod3, where (z)_n=z(z+1)⋯(z+n−1) denotes the shifted factorial. In this paper, we establish a one-parameter extension of them. The main ingredients of our proofs are the creative microscoping method developed by Guo and Zudilin \cite{guo2019q} and Andrews' multi-series transformation.
ER  -