## Congruences for $$q$$-binomial coefficients.(English)Zbl 1431.11032

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

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 05A10 Factorials, binomial coefficients, combinatorial functions 11A07 Congruences; primitive roots; residue systems
Full Text:

### Online Encyclopedia of Integer Sequences:

a(n) = (12*n)!*n! / ((6*n)!*(4*n)!*(3*n)!).
a(n) = (6*n)!/((3*n)!*(2*n)!) * (n/2)!/(3*n/2)!.

### References:

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