Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas

Abstract

For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$\sum _{k=0}^{\frac{p-1}{2}}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3\frac{1}{64^k}\equiv 0(\mathrm{mod}{p}^2)\text{for}p\equiv 3(\mathrm{mod}4),$$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \begin{align*} \sum_{k=0}^{\frac{p-1}{2}} {2k\brack k}_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0\pmod{[p]^2} \quad\text{for}\quad p\equiv 3\pmod 4, \\ \sum_{k=0}^{\frac{p-1}{2}}{2k\brack k}_{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}\quad\text{for}\quad p\equiv 2\pmod{3}, \end{align*} where $[p]=1+q+\cdots +q^{p-1}$, $(a;q{)}_n=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$, and ${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q$ denotes the q-binomial coefficient. Actually, our results give q-analogues of Z.-H. Sun's and Z.-W. Sun's generalizations of the above Beukers-like supercongruences. Our proof uses the theory of basic hypergeometric series including a new q-Clausen-type summation formula.

22 pages

References in corpus (2)

• Some q-analogues of supercongruences of Rodriguez-Villegas
• A q-analog of Ljunggren's binomial congruence