Armin Straub
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A $q$-analog of Ljunggren's binomial congruence
dmtcs:2962 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
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https://doi.org/10.46298/dmtcs.2962
A $q$-analog of Ljunggren's binomial congruence
Authors: Armin Straub 1
0000-0001-6802-6053
Armin Straub
1 Tulane University
We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $\binom{ap}{bp} \equiv \binom{a}{b}$ modulo $p^3$ for primes $p \geq 5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.
Formichella, Sam; Straub, Armin, 2019, Gaussian Binomial Coefficients With Negative Arguments, Annals Of Combinatorics, 23, 3-4, pp. 725-748, 10.1007/s00026-019-00472-5.
Liu, Ji-Cai, 2022, A Variation Of The q-Wolstenholme Theorem, Annali Di Matematica Pura Ed Applicata, 201, 4, pp. 1993-2000, 10.1007/s10231-022-01187-w.
Zudilin, Wadim, 2019, Congruences For $${\varvec{q}}$$-Binomial Coefficients, Annals Of Combinatorics, 23, 3-4, pp. 1123-1135, 10.1007/s00026-019-00461-8.