Some q-analogues of supercongruences for truncated $$_3F_2$$ hypergeometric series

Abstract

In 2003, Rodriguez–Villegas found four supercongruences modulo $$p^2$$\n (p is an odd prime) for truncated $$_3F_2$$\n hypergeometric series related to Calabi–Yau manifolds of dimension $$d=3$$\n . One of them was already proved by Van Hamme in 1997. A q-analogue of Van Hamme’s supercongruence was given by the author and Zeng, and the author and Zudilin. In this paper, we give q-analogues of the other three supercongruences of Rodriguez–Villegas. As a conclusion, we also generalize half of Rodriguez–Villegas’s supercongruences to the modulus $$p^3$$\n case.