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

Zudilin, Wadim

Ann. Comb. 23, No. 3-4, 1123-1135 (2019).

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

Keywords:

congruence; \(q\)-binomial coefficient; cyclotomic polynomial; radial asymptotics

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