Some q-analogues of supercongruences of Rodriguez-Villegas

We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence∑k=0p−1(2kk)216k≡(−1p)(modp2), where (⋅p) is the Legendre symbol, has the following two nice q-analogues:∑k=0p−1(q;q2)k2(q2;q2)k2q(1+ε)k≡(−1p)q(p2−1)ε4(mod(1+q+⋯+qp−1)2), where (a;q)n=(1−a)(1−aq)⋯(1−aqn−1)(a;q)n=(1−a)(1−aq)⋯(1−aqn−1) and ε=±1ε=±1. Several related conjectures are also proposed.