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.


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


[1] Adamczewski, B., Bell, J.P., Delaygue, É., Jouhet, F.: Congruences modulo cyclotomic polynomials and algebraic independence for \(q\)-series. Sém. Lothar. Combin. 78B, #A54 (2017) · Zbl 1405.11019
[2] Andrews, Ge, \(q\)-Analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher. Discrete Math., 204, 1-3, 15-25 (1999) · Zbl 0937.05014
[3] Andrews, Ge; Askey, R.; Roy, R., Special Functions (1999), Cambridge: Cambridge University Press, Cambridge
[4] Gorodetsky, O., \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon, Intern. J. Number Theory, 15, 9, 1919-1968 (2019) · Zbl 1423.11043
[5] Guo, Vjw; Zudilin, W., A \(q\)-microscope for supercongruences, Adv. Math., 346, 329-358 (2019) · Zbl 1464.11028
[6] Meštrović, R.: Wolstenholme’s theorem: its generalizations and extensions in the last hundred and fifty years (1862-2012). arXiv:1111.3057 (2011)
[7] Pan, H., Factors of some lacunary \(q\)-binomial sums, Monatsh. Math., 172, 4, 387-398 (2013) · Zbl 1371.11044
[8] Straub, A.: A \(q\)-analog of Ljunggren’s binomial congruence. In: DMTCS Proceedings: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, pp. 897-902. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2011) · Zbl 1355.05053
[9] Straub, A., Supercongruences for polynomial analogs of the Apéry numbers, Proc. Amer. Math. Soc., 147, 3, 1023-1036 (2019) · Zbl 1442.11039
[10] Warnaar, So; Zudilin, W., A \(q\)-rious positivity, Aequationes Math., 81, 1-2, 177-183 (2011) · Zbl 1234.11023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.