Congruences for $${\varvec{q}}$$-Binomial Coefficients

Source abstract

Abstract We discuss q -analogues of the classical congruence $$\left( {\begin{array}{c}ap\\ bp\end{array}}\right) \equiv \left( {\begin{array}{c}a\\ b\end{array}}\right) \pmod {p^3}$$ ap b p ≡ a b ( mod p 3 ) , for primes $$p>3$$ p > 3 , as well as its generalisations. In particular, we prove related congruences for ( q -analogues of) integral factorial ratios.