Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green–Krammer identity involving central q-binomial coefficients, such as∑k=0n−1(−1)kq−(k+12)[2kk]q≡(n5)q−⌊n4/5⌋(modΦn(q)), where (np) is the Legendre symbol and Φn(q)Φn(q) is the nth cyclotomic polynomial. As consequences, we deduce that∑k=03am−1qk[2kk]q≡0(mod(1−q3a)/(1−q)),∑k=05am−1(−1)kq−(k+12)[2kk]q≡0(mod(1−q5a)/(1−q)), for a,m⩾1a,m⩾1, the first one being a partial q-analogue of the Strauss–Shallit–Zagier congruence modulo powers of 3. Several related conjectures are proposed.