Some parametric \(q\)-supercongruences from a summation of Gasper and Rahman (Q6621288)

The authors establish some new parametric \(q\)-supercongruences by employing a quadratic summation formula due to \textit{G. Gasper} and \textit{M. Rahman} [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)] and the creative microscoping method developed by \textit{V. J. W. Guo} and \textit{W. Zudilin} [Adv. Math. 346, 329--358 (2019; Zbl 1464.11028)].\N\NThe corresponding supercongruences of these \(q\)-supercongruences provide variants of Van Hamme's supercongruences (J.2),\N\[\N \sum_{k=0}^{(p-1)/2}\frac{6k+1}{4^k}\frac{\left(\frac12\right)_k^3}{(1)_k^3}\equiv p(-1)^{(p-1)/2}\pmod{p^4}\ \mbox{for}\ p>3 ,\N\]\Nand (L.2), \N\[\N \sum_{k=0}^{(p-1)/2}\frac{6k+1}{(-8)^k}\frac{\left(\frac12\right)_k^3}{(1)_k^3}\equiv p\left(\frac{-2}p\right)\pmod{p^3}. \N\]\NHere \(\left(\frac{\cdot}p\right)\) denotes the Legendre symbol, while the shifted-factorial is given by \N\(\N(x)_n =x(x+1)\cdots(x+n-1)\N\)\Nfor \(n\geq1\) and \((x)_0=1\). The authors also obtain three families of Ramanujan-type formulas for \(\pi\).