A common $q$-analogue of two supercongruences

Abstract

We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{and}\quad \sum_{k=0}^{(p-1)/2}A_k\equiv a(p)\pmod{p^2}, $$ where $p>2$ is prime, $$ A_k=\prod_{j=0}^{k-1}\biggl(\frac{1/2+j}{1+j}\biggr)^3=\frac1{2^{6k}}{\binom{2k}k}^3 \quad\text{for}\ k=0,1,2,\dots, $$ and $a(p)$ is the $p$-th coefficient of (the weight 3 modular form) $q\prod_{j=1}^\infty(1-q^{4j})^6$. We complement our result with a general common $q$-congruence for related hypergeometric sums.