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 Sigma(n-1)(k=0)(-1)(k)q(-(2k+1))[(2k)(k)](q) equivalent to (n/5)q(-left perpendicularn4/5right perpendicular) (mod phi(n)(q)), where (n/p) is the Legendre symbol and phi(n)(q) is the nth cyclotomic polynomial. As consequences, we deduce that Sigma(3am-1)(k=0)q(k) [(2k)(k)](q) equivalent to 0 (mod 1 - q(3a))/(1 - q)), Sigma(5am-1)(k=0)(-1)q(k-(k+1/2)) [(2k)(k)](q) equivalent to 0 (mod 1 - q(5a))/(1 - q)), for a, 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. (C) 2010 Elsevier Inc. All rights reserved.