Some q-supercongruences from the Gasper and Rahman quadratic summation

Abstract

We give four families of q-supercongruences modulo the square and cube of a cyclotomic polynomial from Gasper and Rahman’s quadratic summation. As conclusions, we obtain four new supercongruences modulo $p^2$ or $p^3$, such as: for $d \ge 2, r \ge 1$ with $\gcd (d,r)=1$ and $d+r$ odd, and any prime $p\equiv d+r\pmod {2d}$ with $p\geqslant d+r$,

$$\begin{aligned} \sum _{k=0}^{p-1}(3dk+r)\frac{ (\frac{r}{d})_k (\frac{d-r}{d})_k (\frac{r}{2d})_k^2(\frac{1}{2})_k}{k!^4(\frac{d+2r}{2d})_k}\equiv 0\pmod {p^3}, \end{aligned}$$

where $(x)_n=x(x+1)\cdots (x+n-1)$ is the Pochhammer symbol. We also propose three related conjectures on q-supercongruences.