A New q-Variation of the (C.2) Supercongruence of Van Hamme

Abstract

Long proved that Van Hamme’s (C.2) supercongruence is also true modulo $p^4$ for any prime $p>3$. By making use of the q-WZ method, the author and Wang gave a q-analogue of Long’s supercongruence. In this paper, employing the method of ‘creative microscoping’, introduced by the author and Zudilin in 2019, we obtain a generalization of this q-supercongruence. A limiting case of our result implies that, for $0\leqslant t\leqslant s\leqslant 10$ and any odd prime $p\geqslant 4\,s+1$ and integer $r\geqslant 1$,

$$\begin{aligned}{} & {} \sum _{k=s}^{(p^r-1)/2+s} \frac{4k+1}{256^k}{2k-2s\atopwithdelims ()k-s}{2k+2s\atopwithdelims ()k+s}{2k-2t\atopwithdelims ()k-t}{2k+2t\atopwithdelims ()k+t}\\{} & {} \qquad \equiv p^r\pmod {p^{r+3}}. \end{aligned}$$