Some congruences involving central q-binomial coefficients

Abstract

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 = 0 n − 1 ( − 1 ) k q − ( k + 1 2 ) [ 2 k k ] q ≡ ( n 5 ) q − ⌊ n 4 / 5 ⌋ ( mod Φ n ( q ) ) , where ( n p ) is the Legendre symbol and Φ n ( q ) is the nth cyclotomic polynomial. As consequences, we deduce that ∑ k = 0 3 a m − 1 q k [ 2 k k ] q ≡ 0 ( mod ( 1 − q 3 a ) / ( 1 − q ) ) , ∑ k = 0 5 a m − 1 ( − 1 ) k q − ( k + 1 2 ) [ 2 k k ] q ≡ 0 ( mod ( 1 − q 5 a ) / ( 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.