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

中文翻译：

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$$ {\ varvec {q}} $$ q的同余项-二项式系数

我们讨论q古典一致性的-analogues \（\左（{\ {开始}阵{C} AP \\ BP \ {结束阵列}} \右）\当量\左（{\ {开始}阵{C} a \\ b \ end {array}} \ right）\ pmod {p ^ 3} \），用于质数\（p> 3 \）及其推广。特别地，我们证明了整数阶乘比的（q-模拟）的相关同余。