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: $$\sum _{k=0}^{(p-1)/2}(-1{)}^k(4k+1)A_k\equiv p(-1{)}^{(p-1)/2}(\mathrm{mod}{p}^3)\text{and}\sum _{k=0}^{(p-1)/2}{A}_k\equiv a(p)(\mathrm{mod}{p}^2),$$ where $p>2$ is prime, $$A_k=\prod _{j=0}^{k-1}(\frac{1/2+j}{1+j}{)}^3=\frac{1}{2^{6k}}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3\text{for}k=0,1,2,\dots ,$$ 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.

9 pages

References in corpus (2)

• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• On a $q$-deformation of modular forms

Cited by in corpus (1)

• Dwork-type supercongruences through a creative $q$-microscope