Some $q$-supercongruences for truncated basic hypergeometric series
Volume 171 / 2015
Abstract
For any odd prime $p$ we obtain $q$-analogues of van Hamme's and Rodriguez-Villegas' supercongruences involving products of three binomial coefficients such as \begin{align*} \sum_{k=0}^{{(p-1)}/{2}} \bigg[{2k\atop k}\bigg]_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0 \pmod{[p]^2} \ \text{for}\ p\equiv 3 \pmod 4, \\ \sum_{k=0}^{{(p-1)}/{2}}\bigg[{2k\atop k}\bigg]_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^2 } &\equiv 0 \pmod{[p]^2}\ \text{for}\ p\equiv 2 \pmod{3}, \end{align*} where $[p]=1+q+\cdots+q^{p-1}$ and $(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$. We also prove $q$-analogues of the Sun brothers' generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial $q$-binomial identities including a new $q$-Clausen type summation formula.
