Some \(q\)-supercongruences related to Van Hamme's (C.2) supercongruence (Q6111120)

The author employs the ``creative microscoping'' method introduced by \textit {V. J. W. Guo} and \textit {W. Zudilin} [Adv. Math. 346, 329--358 (2019; Zbl 1464.11028)], a \(q\)-congruence from \textit {V. J. W. Guo} and \textit {M. J. Schlosser} [Constr. Approx. 53, No. 1, 155--200 (2021; Zbl 1462.33006)], and the Jackson's \({}_6\phi _5\) transformation illustrated in Vol. No. 96 of the \textit {Encyclopedia of Mathematics and its Applications} [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)], to prove that, letting \(n>1\) be an odd integer, \([n]=\left(1-q^{n}\right) /(1-q)\) the \(q\)-integer, \((a;q)_{n}\) the \(q\)-shifted factorial, \(\Phi_{n}(q)\) the \(n\)-th cyclotomic polynomial, and \(0 \leqslant s \leqslant(n-1) / 2\), \[ \sum_{k=s}^{(n-1) / 2} [4 k+1] \frac{\left(q ; q^{2}\right)_{k-s}\left(q ; q^{2}\right)_{k+s}\left(q ; q^{2}\right)_{k}^{2}}{\left(q^{2} ; q^{2}\right)_{k-s}\left(q^{2} ; q^{2}\right)_{k+s}\left(q^{2} ; q^{2}\right)_{k}^{2}} \equiv[n] q^{(1-n) / 2} \quad \pmod {\Phi_{n}(q)^{3}} \text {,} \] which seems valid also modulo \([n] \Phi_{n}(q)^{2}\) (open problem) and extends a supercongruence given by \textit {L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)]. With further manipulation, the author supplies the following version: \[ \sum_{k=s}^{(n-1) / 2+s} [4 k+1] \frac{\left(q ; q^{2}\right)_{k-s}\left(q ; q^{2}\right)_{k+s}\left(q ; q^{2}\right)_{k}^{2}}{\left(q^{2} ; q^{2}\right)_{k-s}\left(q^{2} ; q^{2}\right)_{k+s}\left(q^{2} ; q^{2}\right)_{k}^{2}} \equiv[n] q^{(1-n) / 2} \quad \pmod {[n] \Phi_{n} (q)^{2}} \text {.} \] By using the \(q\)-Lucas theorem and via induction on \(\mathbb{N}\), the paper eventually establishes that, letting \(0 \leqslant s \leqslant(n-3) / 4\), \[ \sum_{k=s}^{(n+1) / 2+s} [4 k-1] \frac{\left(q^{-1} ; q^{2}\right)_{k-s}\left(q^{-1} ; q^{2}\right)_{k+s}\left(q^{-1} ; q^{2}\right)_{k}^{2}}{\left(q^{2} ; q^{2}\right)_{k-s}\left(q^{2} ; q^{2}\right)_{k+s}\left(q^{2} ; q^{2}\right)_{k}^{2}} q^{4 k} \equiv 0 \quad \pmod {\Phi_{n}(q)^{4}} \text {,} \] which generalizes a \(q\)-supercongruence provided by \textit {V. J. W. Guo} and \textit {M. J. Schlosser} [J. Differ. Equ. Appl. 25, No. 7, 921--929 (2019; Zbl 1426.33048)].