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A \(q\)-analog of Ljunggren’s binomial congruence. (English. French summary) Zbl 1355.05053

Proceedings of the 23rd international conference on formal power series and algebraic combinatorics, FPSAC 2011, Reykjavik, Iceland, June 13–17, 2011. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 897-902 (2011).
Summary: We prove a \(q\)-analog of a classical binomial congruence due to Ljunggren which states that \[ {ap \choose bp} \equiv {a \choose b} \] modulo \(p^3\) for primes \(p \geqslant 5\). 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 W. E. Clark [Int. J. Math. Math. Sci. 18, No. 1, 197–200 (1995; Zbl 0816.05009)].
For the entire collection see [Zbl 1239.05002].

MSC:

05A30 \(q\)-calculus and related topics
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

Zbl 0816.05009
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