Some symmetric q-congruences modulo the square of a cyclotomic polynomial

Abstract Z.-H. Sun [J. Number Theory 143 (2014), 293–319] proved a symmetric congruence ∑ k = 0 p − 1 ( α k ) ( − 1 − α k ) f k ≡ ( − 1 ) 〈 α 〉 p ∑ k = 0 p − 1 ( α k ) ( − 1 − α k ) f ˆ k ( mod p 2 ) , where p is an odd prime, α ∈ Q is p-integral, 〈 α 〉 p is the least non-negative residue of α modulo p and f ˆ k = ∑ j = 0 k ( − 1 ) j ( k j ) f j . This congruence implies several supercongruences of Rodriguez-Villegas. In this paper, we give a q-analogue of this congruence and prove some symmetric q-congruences, which also confirm two conjectures of Guo and Zeng [J. Number Theory 145 (2014), 301–316].