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q-Analogues of some supercongruences related to generalized Van Hamme-type supercongruences

  • Liton Karmakar
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Abstract

Recently, Jana and Kalita (Res. Number Theory 8 (2022), Article $54$) obtained certain supercongruences motivated by some generalized Van Hamme type supercongruences, specifically for an integer ℓ ≥ 2 and an odd prime p with p ≡ −1 pmodℓ, ∑n=0(pv + 1)/ℓ (−1)n (2 ℓ n − 1) (ℓ2 n2 − ℓ n + 1) ((−1/ℓ)n3)/(1)n3 ≡ (−1)(pv + p + 2)/ℓ p3v pmodp3v+1, and ∑n=0(pv + 1)/ℓ (2 ℓ n − 1) (2 ℓ2 n2 − 2 ℓ n + 1) ((−1/ℓ)n4)/(1)n4 ≡ − p4v pmodp4v+1. Employing the q-telescoping technique, similar to the q-WZ method, we here establish some supercongruences involving certain q-shifted factorials. As particular cases, we provide q-analogues of the above supercongruences.

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Cite

BibTeXRIS
BibTeX
@misc{karmakar2026qanalogues,
  title = {{$q$-Analogues of some supercongruences related to generalized Van Hamme-type supercongruences}},
  author = {Liton Karmakar},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29071},
  eprint = {2608.29071},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29071},
}
RIS
TY  - UNPB
TI  - q-Analogues of some supercongruences related to generalized Van Hamme-type supercongruences
AU  - Liton Karmakar
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29071
UR  - https://arxiv.org/abs/2608.29071
AB  - Recently, Jana and Kalita (Res. Number Theory 8 (2022), Article $54$) obtained certain supercongruences motivated by some generalized Van Hamme type supercongruences, specifically for an integer ℓ ≥ 2 and an odd prime p with p ≡ −1 pmodℓ, ∑_n=0^(p^v + 1)/ℓ (−1)^n (2 ℓ n − 1) (ℓ^2 n^2 − ℓ n + 1) ((−1/ℓ)_n^3)/(1)_n^3 ≡ (−1)^(p^v + p + 2)/ℓ p^3v pmodp^3v+1, and ∑_n=0^(p^v + 1)/ℓ (2 ℓ n − 1) (2 ℓ^2 n^2 − 2 ℓ n + 1) ((−1/ℓ)_n^4)/(1)_n^4 ≡ − p^4v pmodp^4v+1. Employing the q-telescoping technique, similar to the q-WZ method, we here establish some supercongruences involving certain q-shifted factorials. As particular cases, we provide q-analogues of the above supercongruences.
ER  -