The Rodriguez-Villegas type congruences for truncated q-hypergeometric functions

In this paper, we establish a Rodriguez-Villegas type congruence for truncated q-hypergeometric functions. Using this result, we can confirm several conjectures of Guo and Zeng, such as∑k=0p−1(q;q3)k(q2;q3)k(q3;q3)k2≡(−3p)q1−p23(mod(1+q+⋯+qp−1)2), where p⩾5 is a prime, (a;q)n=(1−a)(1−aq)⋯(1−aqn−1), and (⋅p) denotes the Legendre symbol modulo p.